Faust 0 s Epistle to the Cliffordians
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
QUANTUM MECHANICS
A Co-development of Quantum Mechanics
and
Lagrangian/Hamiltonian Classical Mechanics,
with
Perspectives from Quantum Electronics
and
Allied Fields
Faust′s Epistle to the Cliffordians
Walter FaustNaval Research Lab (Retired)
November 9, 2012
Contents
1 Introduction 3
2 The discovery of QM 42.1 Concepts that survive from CM (and some that notably do not) . . . 62.2 Critical features, new and counterintuitive . . . . . . . . . . . . . . . 82.3 Early Observations Altogether Unexpected, Classically . . . . . . . . 92.4 Paths in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1
3 Systems of QM Notation 16
4 A Comparative Exposition on CM and QM 19
5 Hamilton's Wave Equation; the Eikonal 36
6 Hamilton's Place in History 416.1 QM Development of the Eikonal; the WKB Approximation . . . . . . 41
7 Orbital Angular Momentum, Intrinsic Spin 447.1 Types of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 447.2 Quantization with respect to Rotation . . . . . . . . . . . . . . . . . 457.3 Symmetry upon Exchange of Particles; Fermi and Bose Statistics . . 467.4 Experimental Discovery and Theoretical Account for Intrinsic Spin . . 467.5 Orbital Angular Momentum; Summation over Contributions to An-
gular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.6 Commuting Operators for Angular Momentum . . . . . . . . . . . . . 48
7.6.1 Commutators Connecting the Coordinates . . . . . . . . . . . 497.6.2 Raising and Lowering Jz . . . . . . . . . . . . . . . . . . . . . 497.6.3 Summing Multiple Degrees of Freedom . . . . . . . . . . . . . 50
8 The Dirac relativistic theory of the individual free particle (FermiStatistics): 51
9 The Dirac Equation 53
A The Gaussian Beam; the Fundamental Ray 58
B Initial development of the Lagrangian, following [GCM] 58
C Sturm Louisville Problem 60
D Mathematical Foundations 60D.1 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60D.2 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 61D.3 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2
References
[GCM] Goldstein, Classical Mechanics
[LLQM] Landau and Lifshitz, Quantum M echanics
[DVQM] Davydov,Quantum Mechanics
[DWIQM] Dicke and Wittke Introductory Quantum Mechanics
[CTQM] Cohen Tannoudji Quantum Mechanics
[SQM] Schi, Quantum Mechanics
[GTQM] Heine, Group Theory in Quantum Mechanics
[HRSP] Hehre, Radom, Schleyer, and Pople, Ab Initio Molecular Orbital Theory
[CS] Condon and Shortley, Theory of Atomic Spectra
[PDF] Attached pdf les
1 Introduction
Classical mechanics (CM) is appreciated in a relatively intuitive fashion. Darwin'sagencies, over millions of years, have organized our brains to deal with this mattereciently as a lady robin knows how to build a nest without having previously seenher mother build one.
But microscopes and accelerators are very recent inventions, so QM (quantum me-chanics) and SR (special relativity) don't come so naturally; GR remains a mysteryto most of us (general relativity; black holes represent an even more recent discov-ery). [GCM], incidentally, remarks that QM requires a much more violent recastingthan does SR. These recastings are found necessary, respectively, as material objectsget tiny and/or move very fast (and nally, in intense gravitational elds).
We benet from a succession of surges in human understanding, apparent successes.Regarding the associated brilliance: The innate gifts, the labors, of those who havemade the great strides certainly are not to be minimized. But one's chancy arrival
3
at a particularly productive perspective deserves emphasis as well; and this suggestshumility for most of us.
2 The discovery of QM
Historically, the entry into QM came through insights re the corpuscular nature oflight: Planck's treatment of black-body distribution (1900); Einstein's treatment ofthe photoelectric eect (1905); and Compton's treatment of x-ray scattering fromatoms (1923).
From another perspective, we had not fully arrived at QM until a wavelength wasassociated with the propagation of material particles; they emulate light. De Broglie(1924) rst associated wavelike properties, hence a wavelength λ = h/p, with motionof material objects (p the momentum).
The bridge between QM and CM is a narrow one. One indication is that many bookson the one topic scarcely refer to the other. And it develops that the applicable skills,the methodologies, are quite dierent.
From a modern perspective, Erwin Schrödinger may seem to have faced a staggeringchallenge akin to reverse engineering: to construct a microscopic model on the basisof macroscopic and limited microscopic information. The macroscopic model of CMhad already been brought to maturity, by Lagrange (1788) and Hamilton (1833), butin retrospect it was quite far removed from the ultimate microscopic model of QM.It is now appreciated that the assertions that ow, for a given dynamical system,from a valid QM model become indistinguishable from those of CM in the limit(from the CM view) that the action S becomes large relative to Planck's constanth ' 6.6×10−27erg·sec; or (from the QM view) in the limit of large quantum numbers.(Terms will be further dened as we progress). But this point, taken alone, aordslittle insight into Schrödinger's problem.
On the more supportive side, indications had arisen before Schrödinger, that themotion of matter manifests the character of wave propagation, hence that that thereis an associated wavelength.
Reports in this line, from Wikipedia:
1. The Old Quantum Theory of Bohr and Sommereld (1913/25) had laid out
4
integral numbers of wavelengths for an electron tracing a Kepplerian orbitabout a nucleus, accounting for discrete electronic energy levels.
2. de Broglie (1923/4) had associated the wavelength of a free particle with itsmomentum , as λ = h/p :
Following Planck's E = hν and Einstein's E = mc2, de Broglie equated thetwo, but withmc2 replaced, for material particles, bymv2, to obtain hν = mv2.Replacing ν by v/λ , (again adopting v instead of c for the velocity of a materialparticle), he obtained hv/λ = mv2 or h/λ = mv = p , rearranged to the now-familiar λ = h/p.
3. Schrödinger (1926) published an equation describing how the matter waveshould evolve (in time), the matter wave equivalent of Maxwell's equations,and used it to derive the energy spectrum of hydrogen.
Concepts due to Schrödinger were central, essential, and far-reaching:
(a) The wave function ψ of a system: a complex-number function (conveyingamplitude and phase) of the specic state in which the system is foundat a given instant. Thus
√−1 = i and complex algebra enter. Phase
naturally enters into the wave function for wave propagation or for theevolution of an entangled system (localized or an extended). The sense issimilar to that in electronics.
However, in electronics we write a voltage function of time as V eiωt, thenretain only the real part |V |cos(ωt). QM diers in that any complexquantity gets multiplied by its complex conjugate, or with a Hermitian,ie a real, operator M sandwiched between, to evaluate an observablequantity. This again yields a real number, but thus in a dierent fashionfrom electronics.
When ψ is an eigenfunction of each of a set of commuting operators M,the state often is written to display explicitly the corresponding value ofeach operator.
Since the system must show up somewhere in the space or hyperspaceinvolved (for the present we assume perfect detectors), the probabilitiessum or integrate to unity:
∫ψ∗ (q)ψ (q) dq = 1, where the variables q span
the space. Then ρ (q) = ψ∗ (q)ψ (q) denes the probability density and∫ρ (q) dq = 1.
5
(b) A set of operators characteristic of the system, such that when each isexercised with ψ, it conveys the value of an observable characteristic ofthe system. More specically, when the operator Λ is applied in ψ†Λψ,the result represents an average nding for the corresponding variable,given that state ψ. This is written as 〈λ〉 = ψ†Λψ (for now ψ† may betaken as the conjugate).
Apart from the position operators x, y, and z these operators typicallyare partial derivatives (henceforth we use ~ for h/2π ):
The energy i~∂
∂t;
The x -component of linear momentum −i~ ∂∂x
.
The z-component of angular momentum −i~ ∂
∂ϕz.
There remain other very essential features of the most general state function, theoperators, and their interplay, yet to be developed.
1. The wavefunction is burdened with all the information that can be knownabout the system, even in principle.
Conversely: A function which accommodates every measurable property of asystem is automatically complete in this sense. This is a critical point towardDirac's 4-component model for the SR electron.
2. Superposition, interference, and entanglement. These are developed at length,below. Schrödinger evidently understood these quite well, cf his tale of Schrödinger'sCat; he even employed the German word for entanglement, vershrankung.
3. Canonically conjugate observables, and their constraint by the uncertaintyprinciple.
2.1 Concepts that survive from CM (and some that notablydo not)
A dynamical system (with the term inclusive of photons and other boson force-carriers) which invites our study perhaps closed, perhaps subject to external inu-
6
ences; and the state of this system.
Observable variables associated with a state, such as position r, momentum p, energyE - which, beyond the very constituents of the system, characterize a particularconguration of the system (for canonically conjugate pairs, even these encountercircumscription, due to the principle of uncertainty).
Velocities are not measured in QM, for this would require two measurements ofposition nearby in time. The rst would contribute spread to the momentum, thusupsetting the second [LLQM].
Angular momentum nds an important place but displays its own peculiar featurein intrinsic spin.
Time, a parameter, simply tick-tocks onward, indierent to particular systems, untilwe encounter SR and GR.
The classical action S of a system survives if there is a single path which weighs over-whelmingly (more accurately a tight collection of neighboring paths). S survivesas the phase ϕ of the QM wave function ψ = ρeiϕ. More specically, it describes theaccumulation of phase along the arc of propagation of the central ray, in a geomet-rical optical picture (this corresponds to the WKB approximation, which considersonly the rst order in h). This path and its accumulating phase ϕ comprise theeikonal, as developed at length below. To restate: A multiplicity of nearby paths,lying within a suciently small neighborhood, interfere constructively because thelocal δ-variation (which begs for precise denition) of the phase ϕ = S is vanishinglysmall.
Prominent among the missing in QM are features of continuity: In CM, for any twofeasible values of a system property, any intermediate value also is typically feasible,at least for some limited neighborhood, and barring singularities, chaos, etc. In manyinstances of QM, energy levels, angular momentum, and ultimately spacetime itself,this property is lacking.
Connement in one parameter induces discreteness in another, cf Bohr's Old Quan-tum Theory of the atom, or a particle in a box (discrete energy levels).
In CM, rotation about an axis invokes and innite group. QM generally deals innite representations of axial rotation, none of which is faithful to the innite group.And twice around 2π is not equivalent to once around!
7
In CM, rotation about an axis invokes an innite group. QM generally deals in niterepresentations of axial rotation, none of which is faithful to the innite group. Andtwice around 2π is not equivalent to once around!
A number of entities suggested by symmetry have not been found in nature, thoughnot otherwise forbidden: magnetic monopoles; particles of half-integral spin, whichrespect Bose statistics (see later).
2.2 Critical features, new and counterintuitive
Interference: The frequency of visible light (∼ 6 × 1016 Hz) is so large, and henceλ = c/ν is so small (∼ 5×10−7 cm) even though c is rather large (∼ 3×1010 cm/sec)that visualization of interference calls for equipment not available on the old farm(no ir nor radio equipment). For massive particles, λ is yet much smaller.
Superposition: This principle interplays closely with interference, and is fully asimportant. LLQM (p 7) rates this as the chief positive principle of quantum me-chanics (uncertainty evidently is regarded as negative). If states ψ1 and ψ2 yieldtwo denite and dierent values of a given variable (i.e. they are eigenstates), thenthe (normalized) sum of the two, ψ+ =
√2(ψ1 + ψ2) will, upon any single ideal
measurement, yield exactly the rst or exactly the second eigenvalue.
Let us suppose that each of ψ1 and ψ2 carry distinct eigenvalues of the same setof commuting operators, and that each may be taken as a product over that set,eigenfunctions of those operators. Individual measurements entail a winner-take-allsort of random competition among such product terms, one term only scoring. Theterm that scores in an individual measurement will carry its peculiar correlation ofthe eigenvalues, all belonging exclusively to one or the other of ψ1and ψ2. The scoringprobabilities will respect their complex squares. The term pure is applied to suchsum-wavefunctions, as well as to those of a single term (cf mixed, below).
Matter waves: Cited above were the early insights of Bohr and Sommereld, deBroglie, and Schrödinger.
Interference of matter waves: Davisson and Germer (1927) observed electron dirac-tion from a Ni surface; see below. Ever more compelling demonstrations have sincearisen: Today neutron diraction from crystals is standard technique. In Sven Hart-mann's billiard ball experiment, entire atoms, from two distinct spatial paths, in-
8
terfered, as manifested in the emission1. In more recent work, interference has beenobserved even between two colliding aggregates of very low-pressure gas crystals(spatially-periodic arrays) in collision2
See below, after Paths in Space have been treated, the extended discussion of entan-glement of spatially separated entities, as represented by a superposed wavefunctionfollowing two quite distinct spatial paths:
2.3 Early Observations Altogether Unexpected, Classically
In 1927 at Bell Labs, Davisson and Germer, ring slow moving electrons at a crys-talline nickel target, discovered electron diraction: Distinct waves, scattered fromdierent units of the periodic crystalline structure, interfere at points upon a detectorsurface; it seems that the electrons are indeed suciently tiny, even for modest volt-ages. To restate: In electron diraction, a multiplicity of information wave-streamsstatistically describe the potential presence of individual electrons. These streamsjointly contribute to the probability that an electron will be detected at a given pointin the space of the subsequent paths, where a detector is provided. That probabilityof detection is computed as the complex square of the sum of a number of amplitudestreams reaching that given point. Only the net probabilities, not the amplitudes,can be observed directly.
Any such information stream, e.g. the one approaching the Ni surface or the severaldeparting, is variously known as a wave function ψ, a probability amplitude, awave vector, a state vector, etc. (These are quite dierent from spatial vectors,being similar only in spanning several dimensions over some set of basis functions.They inhabit a function or Hilbert space, reecting concepts of quasi-length andprojection, orthonormality, etc.)
Simplify momentarily to a case in which there are just two such streams. It will befound that if either stream is intercepted such as to assess its information content,that nal interference at the subsequent detector must be subject to destruction.An amplitude can, in full safety in this regard, contribute to such an assessment ofprobability only once, along a path of propagation. These features, counterintuitiveas they may seem, have been veried and elaborated in further experiments ranging
1Beach, Hartmann, and Friedberg, Low Pressure Crystals of Atoms, Phys. Rev. A25,2658(1982).
2Kasevich, Science 298, 15 Nov. 2002, p.1363.
9
on to recent times. (Aside, re in full safety: Recent experiments with weak mea-surements3 have rened this issue to read that there must be at least a chance ofsuch destruction if one is to obtain statistical information of such a nature whichchance may be minimized, apparently in a sort of nibbling.
Simplify momentarily to a case in which there are just two such streams. It will befound that if either stream is intercepted such as to assess its information content,that nal interference at the subsequent detector must be subject to destruction.An amplitude can, in full safety in this regard, contribute to such an assessment ofprobability only once, along a path of propagation. These features, counterintuitiveas they may seem, have been veried and elaborated in further experiments rangingon to recent times. (Aside, re in full safety: Recent experiments have rened thisissue to read that there must be at least a chance of such destruction if one is toobtain statistical information of such a nature, which chance may be minimized, ina sort of nibbling. Jaws dropping?)
Thus a beam of matter of a given momentum, encountering a periodic structure,behaves much like a monochromatic beam of light encountering a diraction grating.Even for modest momentum or KE values, the wavelength will be very small, asasserted above; and for high voltages, it will be smaller still.
2.4 Paths in space
A wave amplitude may, from its construction, follow a prescribed path, ie this featureis embedded therein. Under this general heading we will consider issues centeredabout the following three subtopics with several examples:
A The operation of a confocal laser.
B Spatially merging or separating paths of propagation.
C The eikonal ray for a free material particle, where wavefronts self-propagate inthe vein of Huygens.
The specic examples are:
A1 The fundamental mode of a confocal laser models also the most severely radially-conned axially symmetric optical mode of free space: ie an ideal ray. See
3Science 333, 5 August 2011, p.690.
10
Appendix A for the explicit eld amplitude E (r, z). In microwave terms itis a TEM00n mode, radially a Gaussian packet propagating along the z-axis,converging to a Raleigh waist, then diverging.
This leaves adjustable the tightness of focus. To quantify this, let the wave-fronts conform to two spherical surfaces (potentially coinciding with laser mir-rors) , each having radius of curvature b, located symmetrically to each side ofthe waist, at z = ±b/2 ; the center of curvature of each thus lies at the center ofthe other. The minimal spot radius, dening the waist, is w0 =
√bλ/2π ; and
the waist diameter is 2w0 =√
2bλ/π . Note that the waist-size is proportionalto the geometric mean of the wavelength λ and the axial length b. The Fresnelnumber of the waist is w0/bλ.
For green light, λ = 5 × 10−5 cm , and a reasonable b = 102 cm , the spotdiameter is 8× 10−2 cm = 0.8 mm.
A2 For the wavefunction of a particle path to express the varying wavefront cur-vatures which are essential to the ideal rays of Appendix A, there must be anappropriate correlation among axial position and radial position; phase, andamplitude, such as to produce the varying wavefront-normals. The embeddingof this path in the wavefunction thus cannot be represented in a simple productwavefunction of the nature ψ1 (x)φ1 (y). An integral of correlation must appearas an essential character, so that we have immediately a state of superposition.
Indeed, note in Appendix A that, with E(r, z) in polar form e<(lnE)ei=(lnE) wedo have a product, and that the factors dier in their dependencies upon r andz, as promised.
A3 Consider again a pair of concave mirrors arranged confocally, as for a laser (butwithout the gain medium).
Toward green light the mirrors are, say, 98% reecting, 1.5% transmitting,with 0.5% loss. Let a monochromatic beam of such light approach externally,from the left. If such a setup is created without further specications, themost likely result is the obvious one: that a small fraction of the incidentintensity will be found circulating between the mirrors, and perhaps a mere(0.0152 = 0.0025%) of that intensity will exit the cavity at the opposite end(The exact proportion hinges upon a geometric series... wildly varying phaseswith a strong tendency toward cancellation). However, if the mirror separationis given a special value near an integral number of wavelengths, a substantial
11
fraction of the incident energy will exit to the right; we have struck a Fabry-Perot resonance.
For charge-free material particles (thus excluding space-charge repulsion), thebeam shape expression given in Appendix A should yet describe the spatialpropagation of the center of mass, with a multiplying function on the internalvariables describing the internal dance.
Further, a Fabre-Perot demonstration is possible in principal with a sucientlymonochromatic particle beam (a very tall order!...very small λ) - but comparefor diculty the Davission Germer experiment of 1924. See the attached papersGundy resonant tunneling..., sent me by my former student Prof. MartinGundersen, in reply to this point.
A4 In a recent article4, helical wavefronts are illustrated. In these, axial and radialposition-variables are joined by θ , so that phase and amplitude depend uponthree coordinate variables [PDF] (see image below).
A5 For two co-travelling waves of distinct but nearby frequencies, the QM expec-tation energy is just the weighted average of the two hν ′s , but this may notdescribe well the detected signal. If the eective duration of the measurementis brief (as set by the integration time the detector rise/decay time, or a gateduration), a measurement sequence may exhibit a beat freqency. This actuallywas the rst experimental result (Javan, Bennet) from the rst gas laser, HeNeat 1.15 µm (running simultaneously in a number of distinct modes within the
4Willner, Wang , and Wong,A Dierent Angle on Light Communications, Science 337, 655, 10August, 2012
12
Doppler width of the Ne line). Dirac (DQM, section 27, p. 109) seems toexclude such a case.
The extreme instance of this beat-eect lies with femtosecond-pulse lasers.There exist a variety of techniques to induce a laser to run in the manner of ashort pulse circulating between two mirrors: synchronized optical gating; lossin a cell of saturable-absorber dye; repetitively-pulsed pumping of the gainmedium together with gain depletion by the circulating pulse (the two eectstrim respectively the leading and trailing pulse-edges... my contribution to theart). The output is a repetitive train of pulses; the Fourier representation ofsuch a train is a uniform comb of frequencies. A Ti:sapphire laser can supportthis with gain essentially across the spectrum of green light; correspondinglyit may produce pulses of ∼ 5x10−15sec at a rate ∼ 100Mhz . Rather thanclocking such pulses with a fast detector, one is then soon occupied in usingthem for a clock!
A6 (fond remembrances to Art Schawlow:) A generic atom and a generic laser cavityare each a sort of resonator, having characteristic frequencies. Considering a)losses from the mirror system or b) loss of excitation from an atom (emission,collisions), the frequencies may be taken to be complex number quantities, real+ imaginary (think of LaPlace transforms).
For an atom, the reciprocal periods are real, determined by the level-spectrum;commonly the lossy imaginary parts are relatively small. For the laser, thereal parts comprise a decorated comb; but the backbone of the comb, of TEMqmodes, is the most important feature (see femtoseconds above)... determinedby longitudinal properties, not transverse... and these TEMq modes are cir-cularly symmetric, radially Gaussians. The losses, the imaginary parts, assertthemselves in the transverse dependence, with the TEMq modes lowest andothers ascending quickly from them.
Consider the energy escaping through a circular hole drilled in one mirror asa manner of output (again, my contribution), somewhat smaller than the 1stFresnel zone, on the central axis. Reect upon a progression of increasing holesizes. See the gure from McCumber5. The (Huygens scalar) eld amplitude...for a typical mode can be written in the form f jlp(ρ)exp(−ilϕ) ; thus the losscurves are labeled by the angular index node, l, and by the radial node index,ρ. The superscript j on f distinguishes the hole-bearing mirror j = 1 and
5 Bell System Tech. J.,48,1919.
13
the 2nd mirror j = 2. Provided that the incremental loss of the fundamentall = 0, p = 0 through the hole is less than the excess loss of the next circularlysymmetric mode l = 0, p = 2 , the character of the low-lying modes will remainroughly the same; this is parallel to perturbation theory for energy levels ofatoms (See following section on Raw Perturbation Theory).
A7 Eects of gain depletion... referring, in general, to exhaustion of populationinversion on a real or potential laser emission line, through some competingprocess of stimulated emission:
1. Continuing the above discussion of increasing coupling-hole size: Whenthe loss of the fundamental exceeds that of the next mode l = 1, p = 0(lower than l = 0, p = 2, not of the same circular symmetry), laser opera-tion will make the jump, in appreciation of minimal loss; the fundamentalwill fail from gain depletion (cf. femtoseconds and trimming, above)...quite in contrast to the suggestion from perturbation theory. NB the laserprefers the lesser loss, not the greater output energy. Quantitively, if themirrors' Fresnel number is 1.0, this permits∼ 0.3% coupling (McCumber).For smaller F#, the discrimination among modes increases [MCCumberFig. 5], but at the price of increasing loss for the fundamental, l = 1, p = 0.
2. In a similar vein, consider a pair of atomic congurations connected by anumber of emission lines, such as the Ne 3p → 2s system. The preferredlines are likely not strong ones as familiarly observed in spontaneous emis-sion, for this would compete with lasing. Unfortunately, calculation of linestrengths for lines which are weak (but important in lasing) is relativelylikely to produce large fractional errors.
B1 Soon after Bennett et al. had demonstrated the rst gas lasers (1961), he aspiredto use a partially reecting mirror to achieve co-travelling beams, merging,from two distinct lasers running on very nearby frequencies then to exhibitan interference pattern racing across the face of a detector (a time-gated imageconverter). I dissuaded him, asserting that one would thereby simply haveconstructed a wavefunction of two additive components, initiating in distinctpaths, but summed at the detector. I regarded the result as obvious; opticswas viewed as an extension of the context of radio(or from the very earlymultimode single laser). For matter waves, it the potential parallel behaviorwas less obvious, even as late as 1962.
A full appreciation of path-carrying wavefunctions might have led me (as it did
14
not) to an earlier recognition of entanglement wherein initially co-local parti-cles or waves separate, subsequent to some point in propagation. This pictureholds even if, as the experiment terminates, functions ψ are localized in Rome,and functions φ in Tokyo: the measured variable values emerge correlated. Thedemonstration which I had considered trivial (cf Bennett above) may be con-strued as a converse to demonstrations of entanglement, which are popularlyconsidered much more remarkable: entities merging vs. entities separating.Most remarkable is that these features apply to particulate matter as well asto photons! Arguably we should have been prepared for this since de Broglie.
B2 Since Shrödinger's original work, many have felt deeply disturbed by the con-cept of superposition, beginning with Einstein, Podolsky, and Rosen (1935;Wikipedia Schrödinger's Cat, ref. 5). In fact, Schrödinger introduced his catas a yes indeed reply to Einstein et al., re Einstein's Spooky action at adistance.
Bohm and others pursued a class of hidden variable theories, essentially sup-posing that the entangled partners somehow knew right along which termin the correlation would hold for a given single exercise (as opposed to anensemble).
John Bell's original theorem on entanglement dealt with a system of two double-valued variables, say two spins 1/2, yielding magnetic subcomponents±1/2 . Thedeclared result is that that no hidden variable theory could produce correctlythe distribution of measured values.
A relatively recent article in Science dealt with three double-valued variables.Here it was claimed that no hidden-variable theory could produce correctly eventhe results of single exercises of the experiment. (Sorry, this is from memoryonly; I haven't retrieved the two actual publications, either for two or for threevariables.) Wikipedia characterizes Bell's work as controversial.
C1 Discussion of the eikonal ray, promised above, is deferred, pending an enabling(and lengthy) review of Lagrangian-hamiltonian classical mechanics. This con-veys a classical account for material wave-propagation along the central ray ofany single path, the furthest reach of CM toward QM.
C2 Conversely, a suitable degradation of QM by exercise of the WKB approxima-tion resurrects the classical picture of the eikonal ray.
15
3 Systems of QM Notation
There are three systems of notation to be mentioned, those of Heisenberg (matrixequations), of Schrödinger (wave equations), and of Dirac (bras to the left and kets tothe right, hence brackets). In each we have states, on the one hand, and operatorson the other. The matter of the adjoint, and its role in yielding exclusively realobservable values, is represented somewhat dierently among them.
D&L eqn 8.3 (for instance) uses appropriately a hybrid format, the spin spectrumbeing discrete (for starters, twofold) and the spatial part being continuous; moremuch later.
The sense of multiplication is perhaps most in need of discussion for matrix notation(This is ok for nite matrices, but ill-suited for continuous, rather than discrete,eigenvalues). State and state-adjoint vectors are treated as column and row vectors,the latter the conjugate of the former. Operators are treated as square matrices.Multiplication is standard matrix multiplication.
There are two important classes of operators (here we conveniently use the languageof matrices):
Unitary operators can be employed to accomplish a linear transformation from onebasis set to another such set (set pertains to the functional components citedabove).
Hermitian operators correspond to real-number observable quantities. Their eigen-values are the allowed values which observable quantities may attain in individualexercises, eg, one electron at a time (as opposed to statistical distributions of out-comes).
In general, the inverse of a matrix M of m rows and m columns is given by anotherm×mmatrix, determined element by element as follows: The i, j element is evaluatedas the determinant of a smaller, (m− 1)× (m− 1) matrix, obtained by the deletionfrom M of the ith row and the jth column. The resulting m×m is then divided bythe determinant of the original m×m.
The adjoint of any of these square matrices is the matrix of transposed complexconjugates.
The adjoint of a unitary matrix is equal to the inverse of that matrix; this is essential
16
to preserve ortho-normality with the change of basis set.
The adjoint of a Hermitian matrix is equal to the matrix itself,(Hba
)∗= Hab , ie
< (H) (real part) is symmetric, and = (H) (imaginary part) is antisymmetric, across(perpendicular to) the diagonal descending to the right. The diagonal elements arereal.
A state is represented by a single-column matrix, listing the the amplitudes (complexnumbers) for the functional components of that state in the basis set adopted. Theadjoint state, a single-row matrix (or vector), is built of the complex conjugates ofthe components. (In passing row→column and taking the conjugate, this emulatesthe adjoint-taking of a matrix.)
Here the components for a state or its conjugate are the function-projections ofthat state onto the members of an an orthonormal complete set in the appropriatecomplete function-space, according to the theory of such spaces. The projectionconnecting a pair of functions is given by the integral over the space (real and/orspin, etc.) of the product of the functions, in accordance with the theory of completesets of functions (See Appendix C re the Sturm-Liouville problem or also googleDirichlet.).
In the Schrödinger picture, the two-factor product of ψ† and ψ, can describe a prob-ability distribution over the component basis set, ie a distribution over the real/spinspace (continuous or discrete) of the functional variables. Since the sum/integral ofprobabilities over all congurations must be unity, we have
∫ψ(q)†ψ(q)dq = 1.
LLQM p. 6, eqn. 2.1 give as the most general expression of this sort the double in-tegral
∫ ∫ψ(q)†ψ(q′)φ(q, q′)dqdq′; but this seems to convey interaction at a distance;
and it is entirely unfamiliar.
Some circumstances may call for density matrices, say when the external parametersthemselves cover some classical distribution; accordingly the system is described bya set of pure wavefunctions ψi, each given a statistical weight. Each is applied tothe operator matrix H as ψ†Hψ, with the weighted summation being performed onlyafterward essentially a classical weighting. Whereas the term pure is applied tosingle or to multiple-term wavefunctions without such further post-summation, thecorresponding term here is mixed. This topic will not be pursued here at greaterlength.
The adjoint of a column state-vector is the row vector of complex conjugates. The
17
adjoint H† of the operator H representing an observable (Hermitian, self-adjoint)obeys, for ψ, φ any wave amplitudes,
ψ†Hφ = φ†H†ψ = φ†Hψ = ψ†(Hφ) = (ψ†H)φ (1)
The value-distributions of observable quantities (or their expectation values) appearas triple products: in the order left-to-right, an adjoint state row vector, a squareHermitian matrix for some observable, and the state column vector itself. The ex-pression
∫ψ(q)†ψ(q)dq = 1 above corresponds to case of the unit matrix operator
(solid diagonal 1's). For the present we assume perfect detectors.
In Schrödinger's treatment, a state function is a complex function of observablequantities, expandable in some basis set for the function-space. The adjoint statefunction is just the complex conjugate. unless spin-matrices enter. An observable istreated by a specic corresponding operator. An observable distribution is describedas the sum/integral over a sandwich: adjoint state times observable operator Ωtimes state,
∫ψ(q)†Ω(q)ψ(q)dq. Each entity is a function of the system variables
(space/spin) and the time.
Operators may involve partial derivatives as noted earlier. Particularly important
observable operators are, for instance, x and px = −i~ ∂∂x
, these representing the
x-position of a particle, say, and the x-component of its linear momentum; also
the time t and the energy operator H = i~∂
∂t. Here ~ ≡ h
2π. These choices
accommodate symmetries of spacetime. Immediately we can spot anomalies fromthe perspective of CM: There appears no mass in the momentum, nor a velocity. Fora state ψ ∼ ei(kx−ωt), it will develop that momentum is represented in a wavelengthλ where k = 2π/λ ; and the velocity appears as that of a wave, λν = λω/2π .
In Dirac notation, the parallel is again a sandwich called a bracket: bra timesoperator times ket. It appears as 〈ψ |O|ψ〉. The variable set typically are eigenvaluesfor some set of operators sucient for completeness within a function-set.
Appropriate unitary operators U (matrices) can be used to perform rotations(transform linearly) in function (Hilbert) space through similarity transformationsU−1MU on the functional basis set for the states. Unitary operators preserve prob-abilities.
Not only are the diagonal elements of a Hermitian matrix H real, the basis can fur-ther be rotated to produce exclusively zeroes o-diagonal, to diagonalize H. The
18
corresponding basis functions then are recognized as eigenfunctions of the observ-able operator H, with the eigenvalues along the diagonal of H. The necessary U iscomposed, row by row, from the eigenfunctions of H. This unfortunately does notconvey a means for accomplishing diagonalization; the strategy is logically circular.
If two operator matrices, say energy and momentum, commute, the same basis setwill diagonalize them; they are simultaneously diagonalizable. Following Schur'sLemma of group theory ([?], p.100; Appendix D, p.418), the state-energies of asystem fall into blocks according to sets of commuting operators with simultaneouseigenfunctions. Rotations respecting the commuting set only rearrange the internalcontents of those blocks.
4 A Comparative Exposition on CM and QM
One should have CM adequately in hand before confronting the passage to, or theconstruction of, QM. Acquaintance is presumed with Lagrangian CM; but do seeAppendix B, for it is not trivial.
Do we have at the outset a fully reliable concept (doubtful!) of coordinate or ofmomentum, of canonically conjugate pairs? This entails a number of complica-tions that bedevil us in CM, eg holonomic constraints, or not (GQM refers to someof these as vicious). For the context of QM the latter issue fades. Rather it scarcelyarises; we are not dealing with balls rolling upon surfaces, etc.
LLQM emphasize that typically there are external conditions rigidly prescribed bymaterials whose own uctuations are not addressed as QM issues; such are staticor resonant electromagnetic elds, gratings, etc. Exceptions to this point are posed,eg, by ab initio quantum chemical calculations on isolated molecules, a la Born-Oppenheimer (HRSP).
Also, the issue of time-dependence in the energy H is handled quite dierently inQM, as opposed to CM.
1. Quantum chemical calculations can yield the shape, the energy, and the vibra-tions of an isolated molecule. Here H is typically independent of time; and thisis widely the case otherwise. This allows transformations of variables to makeall the coordinates cyclic (quite a boon; see later).
19
2. Raw perturbation theory: for H (t) = 0, such theory enables treatment of aproblem not subject to exact solution, on the basis of an initial exact solutionof a nearby problem. For H (t) 6= 0 , it further treats a wide diversity ofexternal inuences.
Following SQM Ch. VII: It is assumed that we have in hand the normalizedeigenfunctions un and eigenvalues En for H0 . If the set un is degenerate, itshould rst be diagonalized in H ′. We write the Hamiltonian H as the sum oftwo parts H0 and H
′ , where the Schrödinger equation forH0 has already beensolved exactly and H ′ is small, so that it invites expansion in a power series.
We replaceH ′ by λH ′ and express the perturbed eigenfunctions and eigenvaluesin terms of the parameter λ . We shall then have in the powers of λ thesuccessive orders of perturbation theory. We assume that the series for theperturbed ψ and and energy W are analytic in λ between 0 and 1 , and setλ = 1.
ψ = ψ0 + λψ1 + λ2ψ2 + λ3ψ3 + ..., (2)
W = W0 + λW1 + λ2W2 + λ3W3 + .... (3)
Substituting into the wave equation, we obtain an equation for each order ofλ , of which we consider here only the rst two, the zero and the rst orders.The zero order teaches us nothing new:
H0ψ0 = W0ψ0. (4)
H0ψ1 +H ′ψ0 = W0ψ1 +W1ψ0. (5)
We expand ψ1 in the basis set un. That is, ψ1 =∫na
(1)n un, where the new sym-
bol ∫n denotes summation/integration over states n (for discrete/continuous
cases; or perhaps some of each), as the set is discrete or continuous. Hereψ0 = um is any one of the unperturbed states, and W0 = Em.
Substitution into the 1st order equation just above gives∫n
a(1)n H0un +H ′um = Em
∫a(1)n un +W1um. (6)
We replace H0un in the 1st term, multiply through by u†k, and integrate overspace, applying the orthonormality of the basis. For the special case k = m ,we obtain W1 = H ′mm .
20
For the more interesting case k 6= m , the result is a(1)k =
H ′kmEm − EK
. The kth
state is not included unless the perturbation connects the initial state to it,which hinges upon the symmetry. The signs are such that the states repel.Rather than crossing, closer examination reveals that they exchange character(symmetry, etc). See the subsequent discussion of an output mirror with acentral hole.
Special cases:
(a) Resonance: Absorption/emission of energy from an EM eld subject tohν ∼ ∆E, some splitting of levels, may build over many cycles. Typicallyonly two energy levels experience such eects, since dephasing expressesitself rapidly for pairs even slightly o-resonance. Typically we do notattempt to re-model the atom/molecule in its entirety with an H (t) in-clusive of such dephasing.
In some very recent work with femtosecond light pulses of extreme strength,a new sort of remodeling is indeed found appropriate.
(b) Adiabatic approximation: For the experimental deection of a beam byan inhomogeneous magnetic or electric eld, the eld (seizing upon amagnetic dipole or a Stark polarization) varies so slowly that the sys-tem's energy follows the continuous variation of each energy level. Thecharacteristic frequencies of the variation are small compared to the levelspacings, so that they are quite o-resonsnce. (The usage of adiabaticin thermodynamics is functionally derivative from the usage here.)
(c) Born-Oppenheimer approximation: For isolated molecules, widely dis-parate time-scales for electronic and for nuclear motions enable treatmentof the elec- tronic changes as adiabatic with respect to the nuclear mo-tions. With the nuclei taken xed, one computes electronic energies. Withthe latter taken as aording a background potential, the nuclear motionsare then treated. (See [HRSP]. But the H2 molecule is unfavorable; Hpresenting a light nucleus, the vibrations do not separate cleanly fromelectronic energy intervals.)
(d) Sudden approximation: This is essentially opposite to adiabatic: Theeigenstates of the initial Hamiltonian, abruptly nding themselves in anew Hamiltonian, individually project onto collections of eigenstates of
21
the new Hamiltonian.
(e) Wentzel Kramers Brillouin approximation: Only the rst order of an ex-pansion in Planck's constant is used. This is useful in tunneling problems,where in an important region the energy is negative. This is employed foralpha decay, the escape of a He4 unit from a nucleus.
There remains in common with CM the matter of changes of variables. These canfacilitate solutions by discovery of invariants. Also, they can rectify issues of asym-metry among the independent variables. They are accomplished in CM by Legendretransformations. Corresponding to this in QM are canonical transformations ac-complished by the unitary operators discussed above.
We must take due warning as given by GCM that, as such transformations areapplied, the characters of the variables recognizable specically as positions andmomenta may be abandoned (eg, angles don't have dimension length). They mayevolve to more general coordinates and their time-derivatives. Yet their status asconjugate pairs will remain; this behavior is delivered by the nature of CM Legendretransformations and again by that of QM unitary transformations.
The Lagrangian in Classical Mechanics - The physics here may be consideredelementary, but the algebra gets a bit intricate. Therefore the initial develop-ment is consigned to Appendix B, this seeming preferable to the pretense thatwe all are already familiar with the matter.
Path integrals: The Lagrangian L, and its friends (units of energy), particularlyH (see below), enter into path integrals over time. These involve variation ofthe independent variables and t, and hence of L. There are two distinct typesof path integrals. Variations δ entail xed end-points qi, t; the variations areviewed as virtual . For variations ∆, even the end points vary in qi, t, andt varies as necessary for feasible motions (H conserved).
It is central to principles of variation of path integrals that these pairs qi,pibe regarded as independently variable; ie, in Lagrangian computations withL (qi, qi, t), the nature of qi as the time derivative of qi is in this sense ignored.The δ-variation of L is
δL =∑i
∂L
∂qiδqi +
∂L
∂qiδqi +
∂L
∂tδt =
∑i
piδqi + piδqi +∂L
∂tδt (7)
Seeking a more fully symmetrical choice, we invoke the Legendre transformation
22
to −H (qi, pi, t) = L (qi, qi)− qipi (nb: In the similar relations of thermodynam-ics, the minus sign on the left is not present).
In the variation
− δH =∑i
piδqi + piδqi +∂L
∂tδt− piδqi − qiδpi, (8)
the rst and fourth terms cancel.
Thus
δH = −∑i
∂L
∂qiδqi + qiδpi −
∂L
∂t=∑i
−piδqi + qiδpi −∂L
∂t, (9)
since∂L
∂qi= pi by Lagrange's equations.
We have transformed from L to a new energy variable H (qi, pi, t). The new
CM equations of motion are∂H
∂qi= −pi and
∂H
∂pi= qi. We note that these are
quite symmetrical. We nd that∂H
∂t= −∂L
∂t.
The total time derivative of H is
dH
dt=∑ ∂H
∂qiqi +
∂H
∂pipi −
∂L
∂t= −∂L
∂t. (10)
Thus H is independent of time unless there is an explicit dependence; thisfeature arises spontaneously.
Consider two arbitrary functions of these same variables, f (qi, pi, t) and g (qi, pi, t).
The Poisson bracket is dened in general as
f, gq,p ≡∑i
∂f
∂qi
∂g
∂pi− ∂f
∂pi
∂g
∂qi. (11)
Thusdf
dt=∂f
∂qiqi +
∂f
∂pipi +
∂f
∂t=∂f
∂t+ f,H (12)
(a very important result!).
23
We may recognizedH
dtabove as H,Hq,p +
∂H
∂t, where this Poisson bracket
vanishes. Unless the Hamiltonian depends explicitly upon the time, it is aconstant.
We assume that we have now in hand, from CM, mature concepts of generalizedcoordinates and momenta, as we approach QM.
We must yet develop further our appreciation of these pairs of independent vari-ables in QM, where we will recognize them as canonical conjugates, whosepairwise commutators are proportional to ~ (hence very small). Poisson brack-ets will serve as a point of entry on the question How might QM have beendiscovered in a more present-day spirit? The reference is to such very mod-ern questions as Now, really, is such and such quantity zero (cf. the neutrinomass), or is it just very small?.
Quantum Mechanics, the Commutator of Two Operators - For the correspond-ing QM development in terms of the commutator , we draw liberally fromLLQM, Ch. 11.
As asserted earlier, The wave function ψ determines completely the state of aphysical system in quantum mechanics.
With q now representing the variable set qi, pi, we refer to ψ (q) and to an ob-servable f (q). The concept of a derivative of f (q) with respect to time cannotbe dened in quantum mechanics in the same way as in classical mechanics.(QM delivers only a distribution over probabilities for f .) is quite
We take 〈f (q, t)〉 ≡∫ψ† (q, t) f (q, t)ψ (q, t) dq. Then we dene the time
derivative of f as the derivative, with respect to time, of the mean value of f ,ie as
d〈f〉dt
=∂
∂t
∫ψ† (q, t) f (q, t)ψ (q, t) dq
=
∫∂ψ†
∂tfψdq +
∫ψ†∂f
∂tψdq +
∫ψ†f
∂ψ
∂tdq. (13)
Using then∂ψ
∂t=i
~Hψ, this becomes
d〈f〉dt
= − i~
∫ψ†Hfψdq +
∫ψ†∂f
∂tψdq +
i
~
∫ψ†fHψdq, (14)
24
since H = H† and∂ψ†
∂t=
(i
~Hψ
)†= − i
~ψ†H† = − i
~ψ†H.
Altogether ([A,B] ≡ AB −BA)
− i~
∫ψ†Hfψdq +
∫ψ†∂f
∂tψdq +
i
~
∫ψ†fHψdq =∫
ψ†∂f
∂tψdq +
i
~
∫ψ† (fH −Hf)ψdq (15)
andd〈f〉dt
=
∫ψ†∂f
∂tψdq +
i
~[f,H] . (16)
Comparison to the CM result above shows a correspondence between the CMPoisson bracket and the QM expression with the commutator:
H, f vs.i
~[H, f ] =
i
~(Hf − fH).
Specically, as we consider systems increasingly classical in character, the QM
expressioni
~(Hf − fH) =
i
~[H, f ] must evolve to accommodate a role con-
sistent with the CM expressioni
~H, f. Relative to the macroscopic systems
described by CM, ~ is extremely small, only 6.6× 10−27erg · sec. Here evolvemeans that the variables/parameters attaining macroscopic magnitudes come
to dwarfi
~. This is why QM was not discovered until early in the 20th century.
[LLQM] argue yet further the striking argument that This result (the above
relationship between H, f and i
~[H, f ]) is also true for any two quantities
f and g (real/Hermitian operators). The QM operator i [f, g] evolves to the
classicali
~f, g, which is duly tiny. They assert that This follows at once
from the fact that we can always formally imagine a system whose Hamiltonianis g!
The presence of the i in the equivalence above, together with the other quan-tities real/Hermitian, indicates that the commutator must be an imaginaryquantity. Consider the conjugate pair px, x , or any canonically conjugate pair
25
pi, qi. Since pi = − i~∂
∂qi,
i (piqi − qipi)ψ = i
(− i~∂ψ
∂qi
)=
1
~∂ψ
∂qi, (17)
which is duly real.
The uncertainty relation for a canonically conjugate pair is demonstrated as fol-lows, here according to [?]:
Consider the commutator [K,F ] = iM , where K, F , and M are Hermitianoperators, M and F a conjugate pair. In the most familiar context, M = ~,but Davidov's proof is more general; nb M need not be a constant.
We indicate the averages or expectation values of K, F , and M by 〈K〉, 〈F 〉,and 〈M〉 (eg 〈K〉 ≡
∫ψ†Kψ). Note that, in this usage, < and > are not
inequalities. Dening ∆K ≡ K − 〈K〉 and ∆F ≡ F − 〈F 〉, we note that theseobey the same commutation relationship as K and F : [∆K,4F ] = iM .
We shall consider an auxiliary integral over τ (spanning the space of ψ),depending upon an arbitrary parameter α:
I (α) =
∫|(α∆K − i∆F )ψ|2 dτ ≥ 0. (18)
The integrand, a complex square, is non-negative, hence also the integral.
Using the self-adjoint property of K and F , the integral is expanded to yield
I (α) =
∫ψ† (α∆K + i∆F ) (α∆K − i∆F )ψdτ (19)
=
∫ψ†(α2 (∆K)2 + iα (∆F∆K −∆K∆F )− (∆F )2)ψdτ (20)
= α2⟨(∆K)2⟩+ α 〈M〉+
⟨(∆F )2⟩ ≥ 0 (21)
=⟨(∆K)2⟩(α +
〈M〉2⟨(∆K)2⟩
)2
+⟨(∆F )2⟩− 〈M〉2
4⟨(∆K)2⟩ ≥ 0 (22)
where we have used the fact that [K,F ] = iM applies to the two cross-terms,
26
so that they yield αM . Then use that
limα→∞
I (α) =∞ (23)
limα→−∞
I (α) =∞ (24)
dI
dα
∣∣∣∣α=−
〈M〉2⟨(∆K)2⟩ = 0 (25)
min (I (α)) =⟨(∆F )2⟩− 〈M〉2
4⟨(∆K)2⟩ ≥ 0 (26)
⟨(∆F )2⟩ ⟨(∆K)2⟩ ≥ 〈M〉2
4(27)
positive or negative; and it has a minimum at α = −〈M〉 /2 〈∆K2〉, where theterm in () in equation 22 vanishes. It follows that 〈∆F 〉2−〈M〉2 /4 < ∆K2 >≥0 , or 〈∆F 〉2 < ∆K2 >≥ 〈M〉2 /4. We have arrived.
The most familiar case M = ~, K = x, and F = px, where 〈∆px〉2 〈∆x〉2 ≥~2/4.
This juncture invites a discussion of a hyper-space, conventionally called phasespace, pixellated as Harold has noted, into cells of volume element
∆x∆y∆z∆px∆py∆pz = ~3/8. (28)
That constant is really small! Fermi statistics allow only one fermion per cellof phase space (Fermions have odd half-integral spins; they anticommute.).This is the basis of the Thomas-Fermi model of atoms. Ultimately it explainswhy material objects consume space and support themselves against pressure.Each electronic orbital requires its own pixel. Squashing an electron in ∆xraises ∆px and hence the energy, which requires work. But note: Fermi statis-tics themselves msut be regarded as an experimental result (Pauli's claim notwithstanding).
27
With Minkowsky in mind, one might add another factor of 1/2for time and energy. However, it is not clear in CM that t andH have the proper status of a conjugate pair: Although theyobey an ~ relation in QM, they do not appear classically via aregular Lagrange transformation. In discussion of such trans-formations with use of a generating function F (on which morebelow), GCM p.243 states a relationship which he notes, with-out demonstration, to be reminiscent of the relativistic resultthat iH/c is conjugate to x4 = ict. The issue arises repeatedly,in various contexts.
Note that the Poisson bracket of CM is couched in derivatives of system param-eters. The commutator of QM is couched in operators, themselves derivatives,which work upon the system wave function.
Multiple Corrolaries of the Uncertainty Principle - Here we follow [DWIQM](Ch. 8). As noted earlier, operators for velocity measurements do not occur inQM for they would require nearby pairwise measurements of time and position.A rst measurement of position renders the momentum indenite, and hencethe position thereafter. We must work with expectation values for a particle ofmass m moving in one dimension, within potential V (x). Momentum take theplace of velocity
Eects of the Uncertainty Principle are so pervasive as frequently to dictatethe approach to calculations.
The relation between packet velocity and momentum - Preliminary: Be-gin with [px, x] = −i~ (and the trivial but soon to be signicant [x, px] = +i~).
Multiply [px, x] = −i~ rst from the left, then from the right, by px , to obtainp2xx− pxxP = −i~px and pxxpx − xp2
x = −i~px .
Summing these eqns., the cross-terms cancel: [p2x, x] = p2
xx − xp2x = −2i~px
An incidental irrelevacy: This generalizes to [pnx, x] = −ni~pn−1x .
Now for the expectation values:
d〈x〉dt
=d
dt
∫ψ†xψdτ =
∫ ∂ψ†
∂txψ + ψ†x
∂ψ
∂t
dτ, (29)
(No explicit time dependence in x; dτ the function space.)
28
The time-derivatives are∂ψ
∂ti = − i
~Hψ and
∂ψ†
∂t= − i
~ψ†H. Recall that H is
self-adjoint and that ψ† left operates on H (a bit Cliordian?). Thus
d〈x〉dt
= − i~
∫ (ψ†H
)xψ − ψ†x (Hψ)
dτ. (30)
Take H = p2x/2m+V (x) and using equation 30 above, and [H, x] = 0, we have
[H,V (x)] = 0, so that:
d〈x〉dt
=1
2i~m
∫ψ†−p2
xx+ xp2x
ψdτ
= − 1
2i~m
∫ψ†[p2x, x]ψdτ
= − 1
2i~m
∫ψ† (−2i~px)ψdτ =
〈px〉m
, (31)
giving the time derivative of the position expectation, the packet velocity, interms of the momentum expectation.
Newton's 2nd Law of Motion - Referring to Preliminary above, begin insteadwith the commutator [x, px] = i~. It follows immediately that [x, p2
x] = 2i~px.
In similar fashion to the above,d〈px〉dt
is evaluated:
d〈px〉dt
=d
dt
∫ψ†pxψdτ =
∫ ∂ψ†
∂tpxψ + ψ†px
∂ψ
∂t
= − i
~
∫ −ψ†Hpxψ + ψ†pxHψ
= − i
~
∫ψ† [px, H]ψdτ. (32)
The term p2x/2m in H commutes with px, so the commutator includes only
29
V (x).
d〈px〉dt
= − i~
∫ψ† [px, V (x)]ψdτ = − i
~
∫ψ† [−i~∂/∂x, V (x)]ψdτ
= −∫ψ†[∂
∂x, V (x)
]ψdτ
= −∫ψ†(∂
∂xV (x)− V (x)
∂
∂x
)ψdτ
= −∫ (
ψ†∂
∂x(V (x)ψ)− ψ†V (x)
∂ψ
∂x
)dτ
= −∫ (
ψ†∂V (x)
∂x+ ψ†V (x)
∂ψ
∂x− ψ†V (x)
∂ψ
∂x
)dτ
= −⟨∂V (x)
∂x
⟩(33)
The rate of change of the momentum expectation is equal to the expectationof the force; this should be familiar.
Conserved Flow of Probability [CTQM]We have identied ρ(q, t) ≡ ψ∗(q, t)ψ(q, t)as the probability density asserted by ψ at time t and at the position q inphase space; and we have noted that
∫ρ(q, t)dq = 1 . Though the integral is
xed,ρ(q, t) may vary locally, in time. But, because the integral is xed, we
expect an equation of continuity/conservation, in the nature of∂
∂tρ (q, t) +∇ ·
J (q, t) = 0.
Consider the Schrödinger equation
i~∂
∂tψ (q, t) = − ~2
2m∇2ψ (q, t) + V (q, t) , (34)
where ∇2 ≡ ∇ · ∇ . We discern immediately the divergence of a vector, thegradient.
∂
∂tρ (q, t) =
∂
∂t(ψ∗ (q, t)ψ (q, t)) =
∂ψ∗ (q, t)
∂tψ (q, t) + ψ∗ (q, t)
∂ψ (q, t)
∂t. (35)
The 2nd term is taken from ψ∗ times Schrödinger's equation, and the 1st termfrom −ψ right-multiplied by the complex conjugate of Schrödinger's equation;
30
since V (q) is real, it cancels out:
i~∂
∂tρ (q, t) = − ~2
2m
(∇2ψ∗ (q, t)ψ (q, t)− ψ∗ (q, t)∇2ψ (q, t)
)= − ~2
2m((∇ · ∇ψ∗ (q, t))ψ (q, t)− ψ∗ (q, t) (∇ · ∇ψ (q, t))) . (36)
But we need ∇·, the divergence, operating upon an expression entire.
We have, for any scalar s and any vector V the identities ∇ · (V a) = ∇ · V a+V · ∇a and ∇ · aV = ∇a · V + a∇ · V .
Adding∇ψ∗ (q, t) · ∇ψ (q, t)−∇ψ∗ (q, t) · ∇ψ (q, t) = 0, (37)
we obtain
i~∂
∂tρ (q, t) = ∇ ·
(− ~2
2m(∇ψ∗ (q, t)ψ (q, t)− ψ∗ (q, t)∇ψ (q, t))
), (38)
or
∂
∂tρ (q, t) = ∇ ·
(− ~
2mi(∇ψ∗ (q, t)ψ (q, t)− ψ∗ (q, t)∇ψ (q, t))
), (39)
so that the conserved current is the quantity in brackets
J =~
2mi(∇ψ∗ (q, t)ψ (q, t)−∇ψ (q, t)ψ∗ (q, t)) , (40)
where in the 2nd term we have commuted the two factors.
For any function ψ oered as a wave equation, J must be positive denite. Inthe development of Dirac's model for the relativistic free particle by [?], thisbecomes a critical issue.
The relation between Poisson Bracket and Commutator - [DWIQM] point outthat the arguments above, applied to the expectation value of an arbitrary op-erator Q, yield
d〈Q〉dt
=i
~〈[H,Q]〉 +
⟨∂Q
∂t
⟩, recovering our earlier result for the case that
H is one of the pair of operators, and giving the general result for the time-dependence of the expectation value of an operator. Unless H depends explic-itly upon time, energy is conserved.
31
The Wave Packet yielding the Minimal Uncertainty-Product - The develop-ment follows the lines of DWICQM p. 129+, but as a preliminary we conveya more general picture of Hilbert function theory:
Let there be functions f (τ) and g (τ) upon a common parameter space τ . Aconcept of adjoint is dened; and a quasi-length product can be dened,isomorphic to a geometrical length product (a dot product). We begin witha projected length product (units quasi-length squared), richly analogous to|F | cosϑ |G| for two vectors in ordinary space. It is expressed as an integralover the space τ : f · g ≡
∫f (τ)† g (τ) dτ . The quasi-length2 of f is f · f =∫
f † (τ) f (τ) dτ ; this is taken to be nite for each function.
Orthonormality is quickly appreciated, and an orthonormal basis set ri (τ);hence also the expansion of an arbitrary f (τ) onto such a function-set, whereinthe integrals above play the role of the dot product.
Schwartz's Inequality asserts that very generally (f · g)2 ≤ (f · f) (g · g) . Thisis analogous to the assertion that cosθ ≤ 1 in |F |cosϑ|G| above. Regardingthe expansion of f (τ) onto the full set ri(τ), we see that the projectiononto the single function g (τ) will typically miss a number of components,hence supporting the inequality. For such normalized functions, the equality isattained only if f (τ) is a unit complex number times g (τ) .
We consider now a single particle situated in 1-dimensional space, i.e. along thex-axis. The 'uncertainty' in the position of our particle must be given someexact meaning. For the square of the uncertainty we adopt the deviationfrom the mean, expressed as ∆x ≡ x − 〈x〉. The expectation for the squareddeviation is 〈∆x2〉 =
∫ψ†(∆x2)ψdτ =
∫(∆xψ)2dτ .
A similar expression for the squared deviation of the momentum px is⟨∆p2
x
⟩=
∫(∆pxψ)2dτ.
Then consider the Schwartz Inequality
⟨∆x2
⟩ ⟨∆p2
x
⟩≥∣∣∣∣∫ ψ†∆x∆pxψdτ
∣∣∣∣2 = |〈∆x∆px〉|2 . (41)
(Don't confuse here characters 〈 and 〉 with ≥ and the like; those bracketsframe expectations.)
32
The operator within the expectation on the right,
∆x∆px =1
2[∆x,∆px] +
1
2(∆x∆px + ∆px∆x) =
i~2
+1
2(∆x∆px + ∆px∆x) .
(42)
Note that ∆x∆px has one imaginary and one real (Hermitian) term. Conse-quently the complex square of the expectation, |〈∆x∆px〉|2 , is just the sum of
the square moduli of these two terms |〈∆x∆px〉|2 =~2
4+
1
4〈∆x∆px + ∆px∆x〉2.
The term ~2/4 is inescapable via choices of the parameters, so 〈∆x2〉 〈∆p2x〉 ≥
~2/4.
To attain the equality 〈∆x2〉 〈∆p2x〉 = ~2/4, the Schwartz Equality must be
achieved above, 〈∆x∆px + ∆px∆x〉2 /4 must vanish; DWIQM recognize twodistinct requirements here.
The 2nd condition is modied, relative to the above bracketed discussion ofnormalized functions, to f = αg , where α is some complex number. (DWIQMsays any complex number ??; well, surely not zero.)
This condition on f and g becomes ∆xψ = α∆pxψ, to which the adjoint isψ†∆x = α†ψ†∆px.
The condition for the Schwartz Equality |〈∆x∆px〉|2 = 0 may be written as∫ψ†(∆x∆px+∆px∆x)ψdτ = 0, whereupon the proportionality gives
∫ψ†(α†∆p2
x+
α∆p2x)ψdτ = 0, or (α† + α)
∫ψ† (∆px)
2 ψdτ = 0.
The integral must be positive denite (> 0); thus α must be pure imaginary.Any real part would give a result > 0.
Now it is possible to integrate ∆xψ = α∆pxψ.
∆xψ = α∆pxψ = α
[− i~∂
∂x− px
]ψ. The partials are unnecessary; and dx =
d(∆x).
∆xψ = α
[−i~ dψ
d(∆x)− pxψ
]. Multiply through by d(∆x)/ψ and transpose,
obtaining ∆xd(∆x) + αpxd(∆x) = −αi~dψ/ψ.
A logarithm is recognized for ψ , thence an exponential. Note pxd(∆x) = pxdx.
33
Thus d(∆x2) /2 + αpxdx = −αi~d (lnψ).
Recalling that α may be an arbitrary imaginary number, choose α =i
~,
consonant with quasi plane-wave character.
d(∆x2) /2+i(px/~)dx+ln (C) = d (lnψ), with C the constant of integration.
Recall that ∆x ≡ x− 〈x〉; and thus dx = d (∆x).
Note that neither the natural log nor the exp is allowed any net units withinthe argument.
Since the normalization condition is∫ +∞−∞ ψ†ψdx = 1, any imaginary part of C
would be irrelevant (self-cancellingonly a shift of phase); nor will the imag-inary part, the term in px, contribute to the integral. Thus we can produceaccord with the no net units principle by dividing the real term in d (lnψ)above by −2 〈∆x〉2 , without disturbing any normalization relative to the 2nd
term! The minus is essential for convergence.
d (ln <(ψ)) = lnC −d(∆x2)
4 〈∆x〉2. (43)
Re-attaching the imaginary part:
d (lnψ) = lnC −d(∆x2)
4 〈∆x〉2+i
~px∆x. (44)
Then < (ψ) ∆x = Ce−∆x2/4〈∆x〉2+i(px/~)∆x. For normalization purposes only,
omit again temporarily the imaginary term in px:∫ +∞
−∞<(ψ†)< (ψ) dx = C2
∫ +∞
−∞e−(∆x)2/2〈∆x〉2d (∆x) = 1. (45)
Note that 〈∆x〉 is a constant.
2C2 〈∆x〉∫ +∞
−∞e−(∆x/2〈∆x〉2)d (∆x/ (2 〈∆x〉)) = 1. (46)
Let u = ∆x/ (2 〈∆x〉). Then
2C2 〈∆x〉∫ +∞
−∞e−u
2
du = 1. (47)
34
The integral equals√π (See section 4). Thus C2 = (2π 〈∆x〉)−1, and C =
(2π 〈∆x〉)−1/2.
Finally, our minimal packet is:
ψ =1√
2π 〈∆x〉e
−(∆x)2
4 〈∆x〉2+i
~pxx
. (48)
Properties of the Gaussian Distribution - DWIQM p.132 diers in the powerof ∆x within the expectation in the constant C: There C is displayed as:
C = (2π 〈(∆x)2〉)−1/2.
The CRC Concise Encyclopedia of Mathematics gives the Gaussian Probabil-
ity Distribution as: Pr (x) =(σ√
2π)−1
e−(x−u)2/(2σ2), which is equivalent to
Pr (x) = (2πσ2)−1/2
e−(x−u)2/(2σ2) and similar to DWIQM's expression.
The resolution of this issue lies in a subtle, even deceptive, and rather inter-esting distinction between the Gaussian Probability Distribution Pr and anexpectation value given as an integral over dx.
Pr is an ordinary probability distribution, normalized to 1 under summationor integration, here over
∫d (instances) . The instances are spanned by dx ,
with x running from −∞ to +∞; but here this mathematician's x in Pr (x)is not the x of QM, nor does it bear units (length), or any particular geometricmeaning.
The integral for C2 above is dened with the dierential du of u = ∆x/ (2 〈∆x〉),running from −∞ to +∞.
In the normalization integral for C2, d∆x has geometric meaning, and henceunits = |length| . This unit and magnitude carry forth into the integral; butthis consideration is not reected in the
∫du as dened.
Hence there is need for a factor in C2 of some natural measure of length. 〈∆x〉is such a measure, and therefore the constant C2 is to be awarded an extrafactor ∆x. Thus C and ψ have been awarded an extra factor of
√〈∆x〉 ,
relative to Pr.
It is gratifying (and conrmatory) that the same factor of −2 〈∆x〉2, applied
35
to < (ψ), delivers this leading factor√〈∆x〉 in ψ and also brings accord with
the argument of the exponential such that the Gaussian form of the integralcan be recognized.
The error is assigned in that DWIQM, accepting the straightforward proba-bility identication with the Gaussian, have overlooked this last factor. Anexpectation value taken as an integral over a length x is not quite identical toa probability distribution over some unitless parameter.
The integral I ≡∫ +∞−∞ e−u
2du = 2
∫ +∞−0
exp−u2du ∝ C−2, because the integrand
is symmetric about u = 0. The problem is addressed with the double integralI2/4 =
∫ +∞−0
e−x2dx∫ +∞−∞ e−y
2dy (yes, ∝ C−4) , which is then taken to polar co-
ordinates as: I2/4 =∫ π/2
0dθ∫∞
0e−r
2rdr = π
∫∞0e−r
2rdr/2 = π
∫∞0e−r
22rdr/4.
With v = r2, I2/4 = π∫∞
0e−vdv/4 = π/4, so that I2 = π , and I =
√π. QED
5 Hamilton's Wave Equation; the Eikonal
Further toward our goal to convey the relationship between QM and CM, we drive foran intermediate objective, long-since promised: Via Hamilton's Principal Function S
and his Characteristic Function W with the recognized limitation∂H
∂t= 0 (ie a fully
isolated system), we show that all coordinates are cyclical. Thereupon we discover ascalar wave equation evolved from a context of QM.
The power of Legandre transformations and classical variational principles is furtherto be exploited. Here we hop along a series of points from GCM, but with longstretches of abridgement.
NB in general: lower case→ before transformation, upper case→ after. At somepoints hereafter the single symbol qi is used to represent the set qi, and pi for theset pi; but these points should be obvious.
The transformations discussed heretofore have the form qi → Qi and pi → Pi;these are termed point transformations (they don't scramble q's and p's). Butpresently others, more general canonical or contact transformations, interest us.It is required that there be some energy function K (Qi, Pi, t) (analogous to H), such
that Qi =∂K
∂Piand Pi = − ∂K
∂Qi
. The over dots represent time-derivatives.
36
Our variables must respect the variations δ∫ t2t1
(∑piqi −H (qi, pi, t)) dt = 0 and
δ∫ t2t1
(∑PiQi −K (Qi, Pi, t)
)dt = 0.
The two integrands need not be equal; rather they dier at most by a total timederivative of some arbitrary generating function F . We employ one of Hamilton'sfour functions F1,2,3,4. These contain in toto 4n + 1 quantities: the 2n originalcoordinates and momenta qi and pi; the 2n transformed quantities Qi and Pi ; plus t.Only 2n of the 4n are taken as the independent variables, the others being taken asconsequential. GCM discusses the four functions F1,2,3,4, diering in those choices,as F1 (qi, Qi, t), F2 (qi, Pi, t), F3 (pi, Qi, t), and F4 (pi, Pi.t). Of these only F2 (qi, Pi, t)concerns us now. It is obtained by a double Lagrange transformation; now treatingthe integrand for the variation:∑
piqi−H (qi, pi, t) =∑
PiQi−K (Qi, Pi, t)+d
dt
(F2 (qi, Pi, t)−
∑QiPi
). (49)
Expanding the nald
dt
∑, we note cancellation/replacement a' la Lagrange; and
withdF2
dtdeveloped, we obtain:
∑piqi −H (qi, pi, t) dt = −
∑QiPi −K (Qi, Pi, t) +
∂F2
∂qiqi +
∂F2
∂PiPi +
∂F2
∂t. (50)
Equating the coecients of our dotted independent variables qi, and Pi, we obtain
the transformation equations: pi =∂F2
∂qi, Qi =
∂F2
∂Pi, with the transformed energy
K = H +∂F2
∂t.
We now set up and justify the Hamiltonian-Jacobi equation for Hamilton's Principalfunction S, a case of F2 (qi, Pi, t) . We set the transformed Hamiltonian K ≡ 0, which
ensures that the new variables are constant in time: 0 =∂K
∂Pi= Qi and 0 = − ∂K
∂Qi
=
Pi. Also 0 = K = H +∂F2
∂t, so that 0 = H (qi, pi, t) +
∂F2
∂t. From lines just above,
this may be written as the Hamiltonian-Jacobi equation: H
(qi,∂F2
∂qi, t
)+∂F2
∂t= 0
the i's now representing two index sets (as anticipated above), for both the q's andfor the p's.
37
This is a partial dierential equation in n + 1 variables, the qi and t . There mustbe (justication!) a solution for F2 (qi, Pi, t), which we name as Hamilton's Prin-cipal Function S (qi, Pi, t), of which GCM writes Of course this only provides thedependence upon the old coordinates and time; it would not appear to tell how thenew momenta are contained in S. Indeed the new momenta have not been specied,except that they must be constants. However, the nature of the solution indicateshow the new Pi are to be selected.
The solution must have n+1 constants, α1, α2, . . . , αn+1. However, the H-J equationcontains only derivatives of S, not S itself; hence one of the constants, say αn+1, isjust an additive one, irrelevant to the solution, and may be omitted.
Hence a complete solution for F2 can be written as S (qi, α1, α2, . . . , αn, t). We areat liberty to take these constants to be the new (constant) momenta: Pi = αi.This does not interfere with their connection with the initial values of the qi and piat time zero.
Congruent with our earlier essential dismissal of time dependence in H (at least for
the initial treatment of typical QM problems), we presently take∂H
∂t= 0; in this the
results are less than general. The H-J equation becomes, with S (qi, α1, α2, . . . , αn, t)
the solution for F2: H
(qi,
∂S
∂qi
)+∂S
∂t= 0. Immediately upon taking
∂H
∂t= 0, we
have achieved separation of variables! That is, the time variable can be separatedby assuming a solution for S of the form S (qi, α1, α2, . . . , αn, t) = W (αi, qi) − α1t,
whereupon we nd H
(qi,∂W
∂qi
)= α1.
Limiting (GCM p. 307) to cases H = constant = total energy, we have S (qi, Pi, t) =W (qi, Pi)− Et.
For the central geometrical-optical ray, then, (and for its neighborhood in the sense ofthe δ-variation), and in this special generalized coordinate set, we have a wavefrontin conguration space, with the phase S proceeding according to −α1t. We mayanticipate a wave equation arising.
Since W is independent of time, surfaces of constant W in conguration space havexed locations in this space.
A surface of constant S must at a given time coincide with some surface of constantW . However, the correspondence changes with time according to the last equation.
38
A surface of constant S surface moves at velocity u =dS
dt(constant S →locally
constant over the surface, →constant in time).
In the same time dt that the S-surface travels distance udt , it travels to a newW -surface W + dW , with dW = Edt. The change dW = |∇W | dS, also, so that
u =dS
dt= dW/ |∇W | / dW/E = E/ |∇W | . (51)
The Pi are just a stack of constants α.
For simplicity we consider just the motion of a single particle three dimensions,moving in potential V (qi).
The gradient magnitude |∇W | is obtained from
3∑i=1
p2i + V (qi) =
1
2m
3∑i=1
(∂W
∂qi
)2
+ V (qi)
= E (52)
as (∇W )2 = 2m E − V (qi) **, the Hamilton-Jacobi Equation for this single par-ticle.
And nally
u =E
|∇W |=
E√2m (E − V )
=E√2mT
=E
p=
E
mv. (53)
The velocity of a point on a surface of constant phase S is inversely proportionalto the particle velocity. This may be familiar from the QM behavior of packets inregard to phase and group velocities.
For cases more complex than that of a single particle, a parallel development canbe indicated. GCM (pp 228,310) opens this issue via the Least Action Principle,with a ∆-variation of
∫ t2t1
∑i piqidt (not developed here in full). This leads via some
dierential geometry to a more general expression for dt in terms of an arc length,as dt = dρ/
√2T and thence (GCM p. 310) to S-surface velocity u = E/
√2T .
The scalar wave equation of optics, ∇2φ−n2/c2d2φ/t2 = 0, (n the index of refraction,c the velocity of light), is satised by a plane wave solution ϕ = φ0e
i(k·r−ωt). Thewave number k and the frequency obey k = 2π/λ = nω/c. We adopt the axis z; k0
for the wave number in vacuum. Adopting the z-axis, we may write ϕ = ϕ0eik0(nz−ct).
The index n is taken to vary slowly in space, accordingly distorting and bending thewave.
39
Seeking a solution as near to a plane wave as possible, we adopt φ = eA(r)+ik0L(r)−ct,where L is an eective optical path length, called the eikonal. (GCM is about towin Planck's constant out of this!)
GCM then evaluates the Laplacian ∇2φ ; this has several terms but is manageable.The lhs of the wave equation has real and imaginary parts, both of which must vanish.Given that n varies only slowly, the prominent term is that term in the imaginarypart which does not contain A . The result is (∇L)2 = n2 **, the eikonal equation ofgeometrical optics. The similarity to the Hamilton-Jacobi Equation does not implythat L and W are equivalent, only that they are proportional to one another.
Accordingly, S = W −Et must be proportional to the total phase of the light wavedescribed by equation above: k0 (L− ct) = 2π (L/λ0 − νt).
Hence the particle energy E and the wave frequency ν must be proportional;. . . wedenote the constant ratio by the symbol h, obtaining E = hν . The wavelength andthe frequency are connected by λν = u. Thus, by u = E/
√2mT obtained above
%%, we have λ = u/ν = (E/p) / (E/h) = h/p.
Next GCM produces the Schrödinger Equation!
The scalar wave equation, in an other-than-optical context, may be written as
∇2φ − 1
u2
d2φ
dt2= 0. If the time-dependence is taken as e−iωt , we obtain the time-
independent wave equation ∇2φ+4π2
λ2φ = 0 .
Corresponding to the wave amplitude in optics there will be some quantity ψ in thewave theory of mechanics which must satisfy an equation of the same form... . Butnow λ = h/p = h/
√2m (E − V ), so that we have the result
∇2ψ +8π2m
h2(E − V )ψ = 0, (54)
the time independent Schrödinger Equation. For subsequent ready recognition, werewrite this as
~2
2m∇2ψ + E − V = 0 (55)
or as
Eψ = i~∂ψ
∂t= − ~2
2m∇2ψ + V (x) = 0. (56)
With the eikonal ray for a material particle now appreciated, recognize that it follows
40
the behavior of the ideal ray discussed above in connection with the fundamentalmode of a confocal laser.
6 Hamilton's Place in History
GCM states (p.314) that The equivalence of the Hamilton-Jacobi and eikonal equa-tions was rst realized by Hamilton in 1834.
GCM notes here that some have suggested that Hamilton almost discovered QM,had he reected whether the Poisson Bracket was truly zero (see section 4 afterequation 12). Then he argues, to the contrary, that Hamilton lacked the authorityof experimental evidence, one infers. Authority scarcely seems needed in theculture of contemporary physics, to advance such a conjectural theory in such acontext.
Likely more to the point: There was no concept abroad, in Hamilton's epoch, ofDe Broglie's matter-waves. De Broglie himself only mentioned his idea briey,in a footnote to his thesis; he didn't amplify upon the implications. Finally, perGoogle/Davisson-Germer, the initial interest of DG was solely in the surface ofnickel.
Cf.. the remark in the Introduction that a ...chancy arrival at a particularly pro-ductive perspective deserves emphasis..., as well as the innate gifts of workers. Itis said that Newton was inspired by a falling apple. One might, another day, havebeen inspired by one's sister playing jacks on the sidewalk.
6.1 QM Development of the Eikonal; the WKB Approxima-tion
Hamilton developed the Characteristic Function W and the Principal Function S,thence the eikonal, from a classical perspective, developed above at length. Capturingthe wave aspect of propagation of a material object, it anticipates QM, aording afurthest reach of CM. From an opposite perspective, the Characteristic Function Wcan be produced via a sort of degradation* of QM (an approximation).
Let the functional dependences of the wave function ψ be expressed within its loga-
41
rithm [SQM, p.184]: That is, in the Schrödinger equation
E = i~∂ψ
∂t= − ~2
2m∇2ψ + V (r)ψ,
write ψ as Ae
i~W (r, t)
.
Dierentiating once with respect to t ; and, separately, twice with respect to r, oneobtains:
∂W
∂t+
1
2m(∇W )2 + V (r)− i~
2m∇2W = 0. (57)
Here the common factor Ae
i~W (r, t)
has been cancelled throughout.
In the classical limit*, ~ = 0, and this equation is the same as Hamilton's equation(earlier) for the Principal Function W :
∂W
∂t+H (r, p) = 0, (58)
where p = ∇W .
Taking ψ to be an energy eigenfunction, ψ = u (r)−iEt
~ , W can be writtenW (r, t) =S (r)−Et; so we have recovered Hamilton's Action, as well as his Principal Function.
And u(r) = Ae
iS (r)
~ . We recognize the kinetic energy, the total energy, and thepotential energy in the successive terms of [SQM eqn 28.2, p.185]:
1
2µ(∇S)2 − (E − V (r))− i~
2µ∇2S = 0. (59)
The WKB method obtains the rst two terms (one term beyond the classical ex-pression) of an expansion of S in powers of ~ , in the one-dimensional case (to whichwe now collapse).
With u(x) = Ae
i
~S (x)
, Schrödinger's equation reduces to one of two forms:
d2u
dx2+ k2 (x)u = 0, or
d2u
dx2− κ2 (x)u = 0, (60)
42
where respectively k2 > 0 , or κ2 > 0 , so that k and κ are always real, putting
k (x) =1
~√
2µ (E − V (x)), or κ (x) =1
~√
2µ (V (x)− E) in the respective cases.
We focus upon the case k > 0 , returning to the logarithm of u(x) [ie, again cancelling
the factor Ae
iS (x)
~ which appears in each term] and using primes for dierentiation:i~S ′′ − S ′2 + ~2k2 = 0.
We apply an expansion of S to rst-order in ~ : S = S0 + ~S1.
i~ (S ′′0 + ~S ′′1 )− (S ′0 + ~S ′1)2
+ ~2k2. (61)
Then separating powers of ~:
−(S ′0)2 + 2µ(E − V ) = 0 ... the terms with no factor of ~; and iS ′0S ′1 = 0, the termswith a single factor of ~; and terms of still higher powers of ~.
The rst, restated as S ′0 = ~k , integrates to
S0 (x) = ±~∫ x
0
k (y) dy, (62)
where the lower limit, a constant, is absorbed within A.
The second is rewritten as:
S ′1 (x) =i
2
S ′′0 (x)
S ′0 (x)
=i
2
k′
k∫ x
0
S ′1 (y) dy =i
2
∫ k(x)
k(0)
dk
k
S1 (x)− S1 (0) =i
2ln
(k (x)
k (0)
)S1 (x)− S1 (0) =
i
2(ln (k (x))− ln (k (0)))
S1 (x) =i
2ln (k (x)) , (63)
43
where we have equated S1 (0) withi
2ln (k (0)).
Schi writes We thus obtain to this order of approximation
u(x) =
A√ke±i
∫ x kdx for V < E
A√κe±
∫ x kdx for V > E
.′′ (64)
It must not be supposed that ~2 is so small that this approximation is suitable forall practical purposes. Schi points out the following: Since S0 is a monotonicincreasing function so long as k does not vanish, the ratio ~S1/S0 is small if ~S ′1/S ′0is small. Thus we expect the method to be useful in that part of the domain of xwhere |~S ′1/S ′0| 1.
Now the DeBrolie wavelength λ is 2π/λ, so that the inequality can be writtenλ
4π
∣∣∣∣dkdx∣∣∣∣, which means that the fractional change in k (or in the wavelength) in
the distanceλ
4πis small compared to unity.
It is apparent that the inequality condition is violated near the turning point of theclassical motion, where V (x) = E, k and κ are zero, and the wavelength is innite.
7 Orbital Angular Momentum, Intrinsic Spin
7.1 Types of Angular Momentum
An electron circulating about a nucleus exhibits orbital angular momentum l.
Fundamental unit particles have another sort of angular momentum, intrinsic spin,usually termed simply the spin s. This is of a sort quite dierent from l and willrequire extensive discussion (below). The totals for several electrons are S and L.The total electronic angular momentum of an atom is J . The above and other typesto follow are taken as measures applied to the unit ~ of angular momentum (oftentaken implicitly, unstated).
For nuclei, aggregates of protons and neutrons (today themselves considered as as-semblies of three quarks), the net intrinsic spin is denoted by I; The total angularmomentum of an atom, including nuclear spin and hyperne structure, is F .
44
A rigid molecule may rotate about an axis of symmetry, with orbital angular mo-mentum J . A methyl group CH3 may twist about the axial bond connecting it tothe rest of its parent molecule, either in hindered rotation or in a rocking motion;these both are classied as vibrations (!), though in the former case they carryangular momentum. Molecular spectroscopy entails several intermediate variables:the electronic angular momentum about a linear axis Λ, the angular momentum Pabout an axis of symmetry, etc (texts of G. Herzberg).
7.2 Quantization with respect to Rotation
The operator for linear momentum along axis x isi
~∂
∂xand a representative eigen-
function appears as the spatial factor φ0eik·r in the familiar propagating plane wave
ϕ = φ0ei(k·r−ωt).
The operator for angular momentum isi
~∂
∂θ, and the corresponding eigenfunction
is ϕ = φ0eijθ. But the analogy between x and θ is not complete. Consider special
relativity: Not only is there no specially directed axis, nor a special position alongan axis, there are more particularly no ducial marks as are found upon a ruler nospecial value for the dierence between two positions.
With regard to rotation, however, there is in SP , in contrast, a special irrotationalframe (consider centrifugal force). And for rotation about an axis, passage through2π is equivalent, in regard to all observable features/quantities, equivalent to norotation at all; certainly this constitutes a special dierence
between two positions. The j-value1
2, given θ = 2π , yields a phase factor of
φ0eijθ = φ0e
i1
22π
= φ0eiπ = −φ0. Thus −1 is the rst real eigenvalue as j is
increased from zero. The norm of the j = 1/2 state isφ†φ ; this includes factors(−1)(−1) = +1, ie unity, just as for θ = 0. And for any hermitian operator Ω , theexpectation value φ†Ωφ is again unchanged. Thus -1 is an acceptable eigenvalue forthe rotation operator, and j = 1/2 conveys an acceptable eigenstate. Any half-integralor integral j is acceptable.
45
7.3 Symmetry upon Exchange of Particles; Fermi and BoseStatistics
Half-integral intrinsic spin is associated with Fermi statistics, and integral values toBose statistics: For fundamental particles the Fermi type, the exchange of any twoidentical particles exhibits antisymmetry, a change of sign in the net wavefunction.For Bose entities, the sign remains the same under such exchange. For two particles,eg, using subscripts for the functions and superscripts for the particle coordinates, wehave Ψ1
aΨ2b = ±Ψ2
aΨ1b , Bose/Fermi; multiple identical Fermions entail determinant
forms. Though one describes a Fermion as having spin s , quantitatively the valueis ~s . In the limit of classical mechanics this vanishes as ~→ 0.
Certainly the proposition that Nature submits to this limitation is supported byall pertinent experiments, these representing an enormous weight of results. Pauliclaimed to have proved the proposition via SP, and stoutly defended his analysisthrough the rest of his life, against all opposition. In the subsequent decades, thematter has been studied in great depth. I. Duck and E. C. G. Sudarshan have writtena review at book-length, expressly claiming for themselves no new results (Pauliand the Statistics Theorem, World Scientic, NJ, 1997; 503 pp.). In PreliminaryRemarks on p. 3, they write Everyone knows the Spin-Statistics Theorem, butno one understands it, eg Feynman. On p. 4 one nds The Pauli result does notexplain the spin-statistics relation and cannot.
7.4 Experimental Discovery and Theoretical Account for In-trinsic Spin
Intrinsic spin was discovered experimentally (vs. predicted) perhaps originally viaZeeman's eect (1896): Spectroscopic lines were split by a magnetic eld. Particu-larly notable also is the Stern-Gerlach experiment (1922): Silver atoms were deectedin an inhomogeneous magnetic eld, those with Sz = +~/2 in one direction and thosewith Sz = −~/2 oppositely.
The following is from Google, on intrinsic spin:
According to the prevailing belief, the spin of the electron or some otherparticle is a mysterious internal angular momentum for which no concretephysical picture is available, and for which there is no classical analog.
46
However, on the basis of an old calculation by Belinfante [Physica 6 887(1939)], it can be shown that the spin may be regarded as an angularmomentum generated by a circulating ow of energy in the wave eld ofthe electron. Likewise, the magnetic moment may be regarded as gen-erated by a circulating ow of charge in the wave eld. This providesan intuitively appealing picture and establishes that neither the spin northe magnetic moment are internal - they are not associated with theinternal structure of the electron, but rather with the structure of theeld. Furthermore, a comparison between calculations of angular mo-mentum in the Dirac and electromagnetic elds shows that the spin ofthe electrons is entirely analogous to the angular momentum carried bya classical circularly polarized wave.
Modern experiments with very high-energy electron beams suggest no limitation onthe localization of the electron (hence vanishing size), given sucient momentum.
Spin is limited to half-integral (1/2, 3/2, 5/2 ...) or integral multiples of H , the formerseries belonging to Fermi Statistics and the latter to Bose Statistics (see later).
There are fairly simple rules (see later) for the addition of distinct contributions toorbital angular momentum L, absent spin S; and there are entirely parallel rules foraddition of spins S, absent L. When both are present, construction of the total angu-lar momentum gets more complicated. Depending upon the strength of spin-orbitcoupling, the models range from l, s coupling (Russel-Saunders) to j, j coupling, witha continuum in between.
7.5 Orbital Angular Momentum; Summation over Contribu-tions to Angular Momentum
Vectors describing orbital angular momentum, a term used for the more classicalsort r × p take on only positive integral values. They follow the structure of theSpherical Legendre Harmonics.
There are fairly simple rules (see further below) for the addition of distinct contri-butions (dierent degrees of freedom) to orbital angular momentum L, absent spinS; and there are entirely parallel rules for addition of spins S, absent L. Whenboth are present, construction of the total angular momentum is more complicated.Depending upon the strength of spin-orbit coupling, the models range:
47
1. l, s (Russel-Saunders) where spins s accumulated and l's accumulated rst,respectively to S and L, then S + L = J .
2. j, j coupling where s and l of each electron summed to j , then the j's summedto J .
3. Continuum of intermediate cases (low-lying states of Ne, for instance, requiresuch treatment).
Intrinsic spin ~S , with only modest values of S (excluding magnetic solids), vanishesin the classical limit ~→ 0 . But orbital angular momentum, of the nature r×p , mayin principle take arbitrarily large integral multiples of ~. Planck's constant entersonly in the commutation relation [Lx, Ly] = i~Lz (see below), and in two similarrelations with cyclic permutation of the indices. For large values, this occasions onlya relatively small fractional uncertainty product.
7.6 Commuting Operators for Angular Momentum
An angular momentum vectorJ (either integral or half-integral) corresponding tothe the scalar index j is limited to just two simultaneously diagonizable operators,one for the square of the vector, J2 = J2
x + J2y + J2
z , and one for a single one ofthe three axial components, conventionally Jz; the three scalar components do notcommute pairwise with one another. To restate: The simultaneously diagonalizablevariables belonging to a vector J corresponding to index j are limited to J2 andJz, with the pair written as J2, Jz; Jx and Jy are not welcome in this company.While Jy and Jz are indeed observables, they are excluded from the simultaneous-eigenfunction pair. The scalar index j, while it enumerates the eigenvalues, is notitself even an observable. Jz is indeed one of the eigenvalue pair, and it further servesas an index on 2j+ 1 substates. Whether the net J belongs to a single particle or toan assemblage; whether of spin-only (S), orbital-only (L), or mixed type; whether Jis integral or half-integral Jz takes on values from −J to +J , with uniform spacingsof one unit.
Demonstration of the above (from [CS] p. 43, eqn 6): Among several vector identitiesinvolving commutators is found: [AB,C] = A [B,C]−[C,A]B. To verify this, retreat
48
to component coordinate indices, as
AiBiCj − CjAiBi = Ai[BiCj − CjBi]− [CjAi − AiCj]Bi
= AiBiCj − AiCjBi − CjAiBi + AiCjBi. (65)
The 2nd and 4th terms cancel, QED.
Now let both A and B =J ; and let C = Jz . Since JJ = J2 , we have [J2, Jz] =[(JJ), Jz] = (JJ)Jz − Jz(JJ) = J2(Jz − Jz) = 0 , QED; as claimed, J2 commuteswith any single one of the coordinates of J . The partner to J2 is conventionallytaken as Jz.
7.6.1 Commutators Connecting the Coordinates
As noted above, the scalar components of an angular momentum vector J do notcommute pairwise. In fact we have the commutation relations
[Jx, Jy] = i~Jz, (66)
[Jy, Jz] = i~Jx, (67)
[Jz, Jx] = i~Jy. (68)
This generalizes the result given in the preceding section for L.
7.6.2 Raising and Lowering Jz
Consider the operators Jx± iJy , applying Jz to either of them from the left on bothsides of the net equality:
Jz(Jx ± iJy) = JzJx ± iJzJy= JxJz − iHJy ± i(JyJz − iHJx)= (Jx ± iJy)(Jz + H). (69)
(Note that this may be expressed via the commutator, as: [Jz, (Jx ± iJy)] = (Jx ±iJy)H).
Using case to distinguish the operators J2 and Jz from their eigenvalues j2 and jz(and using γ for any other eigenvalues), we exercise the operators on each side of the
49
prior equation upon a general wavefunction Ω(γ, j2, jz)
Jz (Jx ± iJy) Ω(γ, j2, jz
)= (Jx ± iJy) (Jz + ~) Ω
(γ, j2, jz
). (70)
Jz (Jx ± iJy) Ω(γ, j2, jz
)= (Jx ± iJy) (Jz + ~) Ω(γ, j2, jz)
= (Jx ± iJy) Ω(γ, j2, jz
)(jz + ~) . (71)
Writing Ω± (γ, j2, jz) for (Jx ± iJy) Ω (γ, j2, jz) , and noting that the eigenvalue and~, and constants, commute with everything, we nd:
JzΩ± (γ, j2, jz
)= (jz + ~) Ω±
(γ, j2, jz
). (72)
The notation suggests that the operators Jx ± iJy may be recognized as operatorspromoting/demoting jz by one unit ~. This is true unless the result Ω± (γ, j2, jz) is0 Zeroes occur when the + operator would promote an initial state Ω (j2, jz = j), orwhen it would demote an initial state Ω (j2, jz = −j). This is achieved because eachof Jx and Jy yield zero for a pure state of jz = ±j ; the result is quite satisfactory;there exist no states jz > +j nor jz < −j.
The construction does not yield normalized eigenstates; normalization must be pur-sued subsequently, as Ω±† (γ, j2, jz) Ω± (γ, j2, jz) = 1.
7.6.3 Summing Multiple Degrees of Freedom
Suppose now that we wish to map out the states belonging to the physical summationof two j-values, j1 and j2, representing dierent degrees of freedom (arbitrarily wetake j1 ≥ j2 , for specicity). For the substates of extremal jz (max or min), thesolutions are trivial product states.
The state of maximal jz is
Ωsum((j1 + j2)2 , , jz = j1 + j2) = Ω1(j21 , j1z = j1)Ω2(j2
2 , j2z = j2). (73)
And the state of minimal jz is
Ωdiff ((j1 − j2)2 , jz = j1 − j2) = Ω1(j21,j1z = j1)Ω2(j2
2 , j2z = −j2), (74)
recalling that j1 ≥ j2. The expressions suce also for j2 = j1, yielding respectivelyjz values of 2jz and 0.
50
Beginning with one of these extremal states, the other jz substates may be generatedstepwise, according to section 7.6.2 just above.
Altogether, we are thus armed, in principle, to synthesize states and substates forany simple (but see remarks above re j, j coupling) nite summation over orbitaland spin functions.
8 The Dirac relativistic theory of the individual free particle(Fermi Statistics):
(following [?], rather than Dirac):
Intrinsic spin seems to arise in intimate association with special relativity, for weshall nd that relativistic theory implies spin, whether or not high velocities areevident.
For an isolated free particle, the nonrelativistic hamiltonian is H = p2/2m .
The transition to quantum mechanics is achieved with the transcription H → i~∂
∂t
, and pµ →~i
∂
∂xµ.
This transcription is Lorentz covariant, since it is a correspondence between two
contravariant four-vectors pµ → i~∂
∂xµ.
This leads to the nonrelativistic Schrödinger equation, NRSE,
i~∂ψ (x, t)
∂t= − ~2
2m∇2ψ (x, t) (75)
Like position x, y, z , momentum is an observable. Hence ψ may be a function ofthis variable, but a relativistic theory calls for a four-momentum. From BDRQM:According to the special theory of relativity, the total energy and momenta transformas components of a contravariant four-vector pµ = (p0, p1, p2, p3) = (E/c, px, py, pz).
The length is∑3
µ=0 pµpµ = E2/c2−p ·p = mc2 ; this is the Klein-Gordon equation.
Bold indicates three-vector momentum and · the dot product of 3-vectors. Thelength of the four-vector equals the invariant rest-energy mc2.
51
Following this, it is natural to take as the hamiltonian of a relativistic free particleH =
√p2c2 +m2c4 , and to write for a relativistic quantum analogue to the NRSE
above the candidate RSE i~∂ψ
∂t=√−~2c2∇2ψ(x, t) +m2c4ψ.
Unfortunately this is unsatisfactory; it treats time and space asymmetrically, and Ifwe expand it, we obtain an equation containing all powers of the derivative operatorand thereby a nonlocal theory.
We undertake to justify squaring the operators an each side of our candidate RSE(DQM does not include the following justication). BDRQM observes that ourcandidate has the formAψ = Bψ , and equivalently Bψ = Aψ . Left-multiply the 1st equation by A andthe 2nd by B, to obtain respectivelyAAψ = ABψ and BBψ = BAψ , then subtract. Provided that A and B commute,as indeed they do here, we have indeed justied A2ψ = B2ψ as a valid equation.
We have thus obtained
− ~2∂2ψ
∂t2=(−~2c2∇2 +m2c2
)ψ. (76)
Now from BDRQM: This is recognized as the classical wave equation
ψ +mc
~2
ψ = 0, (77)
where ≡ ∂2
∂t2− ∇2 = gµν
∂
∂xµ∂
∂xν=
∂
∂xµ
∂
∂xµ. ∇ represents the gradient in 3
dimensions, and ∇2 the Laplacian; in a break with consistency, per BDRQM, represents the 4-dimensional Laplacian, of 2nd order.
As it develops, the multiplication to accomplish squaring yields a negative-energysolution, spurious relative to our initial intent; but that evolves into the divine giftof the antiparticle.
Our rst task is to construct a conserved current, since our equation in is asecond-order wave equation and is altered from the Schrödinger form... upon whichthe probability interpretation in the nonrelativistic theory is based. This we do in
analogy with the Schrödinger equation, rst left-multiplying
( +
(mc~
)2)ψ by
ψ∗, then left-multiplying the complex conjugate,
( +
(mc~
)2)ψ∗ by ψ; and nally
52
subtracting, to obtain: ψ∗ψ − ψψ∗, or ∂
∂xµ
(ψ∗
∂ψ
∂xµψ − ψ∂ψ
∗
∂xµ
)= 0, or
∂
∂t
(i~
2mc2
(ψ∗∂ψ
∂t− ψ∂ψ
∗
∂t
))+
~2im∇ · (ψ∗∇ψ − ψ∇ψ∗) = 0. (78)
Equation EQIT, equation quadratic in time
We would like to interpreti~
2mc2
(ψ∗∂ψ
∂t− ψ∂ψ
∗
∂t
)as a probability density ρ . How-
ever, this is impossible, since it is not a positive denite expression. For this reasonwe follow the path of history [refs. Schrödinger, Gordon, Klein, Dirac] and temporar-ily discard [Equation EQIT] in the hope of nding an equation in rst order of thetime derivative which admits a straight-forward probability interpretation as in theSchrödinger case. We shall return to Equation EQIT above, however.
Although we shall nd a rst-order equation, it still proves impossible to retaina positive denite probability density for a single particle while at the same timeproviding a physical interpretation of the negative energy root of the equation above.
Therefore our Equation in the Square , EQIS, or equivalently EQIT , also re-ferred to as the Klein-Gordon equation, remains an equally strong candidate for arelativistic quantum mechanics as the one which we now discuss.
9 The Dirac Equation
Continuing quotes from BDRQM: We follow the historic path taken by Dirac inseeking a relativistically covariant equation of the form of [the NRSE above] withpositive denite probability density. Since such an equation is linear in the timederivative, it is natural [or so it was to Dirac] to attempt to form a hamiltonianlinear in the space derivatives as well. Such an equation might assume the form:
The genius in this approach lies in the dierence between the treatment of thetime/energy/mass term and that of the momenta: The former is just once (ie lin-early) introduced, in the hamiltonian operator, whereas the momenta, resident in theψ∗ and ψ, attack twice, both from the left and the right; each item is linear, but theresult is quadratic for the momenta. The three components of momentum, px, py, pzwhich might threaten spurious cross-terms, are limited (i.e. cross-terms killed) by
53
the introduction of matrix operators, four-spinors; and similarly for cross-terms withβ.
The linear relativistic Schrödinger equation (LRSE) [using the standard operatorsfor energy and momentum] is then:
i~∂ψ
∂t=
~ci
(α1∂ψ
∂x1+ α2
∂ψ
∂x2+ α3
∂ψ
∂x3
)+mc2β ≡ Hψ. (79)
Note comparison: While the Schrödinger equation is in general chartered for depen-dence upon the momentum, for a non-relativistic free particle this is encompassedwithin a plane wave.
The coecients αi and β here cannot be numbers, since the equation would notthen be invariant even under a spatial rotation. Also,... the wavefunction ψ cannotbe a simple scalar. ... the probability density ρ = ψ∗ψ should be the the timecomponent of a conserved four-vector if its integral over all space, at xed t, is to bean invariant. Is anybody disturbed about the issue of simultaneity? I would omitthe words if its integral over all space at xed t.
To free the LRSE from these limitations, Dirac proposed that it be considered amatrix equation. There are several constraints upon the operator matrices and thestate vector, for physical acceptability:
1. It must give the correct energy-momentum relation. That is, each componentψσ of the wave equation must satisfy the Klein-Gordon equation:
− ~2∂2ψσ∂t2
=(−~2c2∇2 +m2c4
)ψσ. (80)
2. It must allow a continuity equation and a probability interpretation for ψ.
3. It must be Lorentz-invariant.
Matrices and vectors/spinors of no fewer than 4 dimensions suce.
Pursuing (1): We iterate the LRSE and inspect it, to determine the non-vanishingelements of α and β, and to apply the stated constraints:
− ∂2ψ
∂t2= −~2c2
2
3∑i,j=1
∂2ψ
∂xi∂xj+
~mci
3∑i=1
(αiβ + βαi)∂ψ
∂xi+m2c2β2 (81)
54
We may resurrect [the LRSE] if the four matrices αi, β obey the algebraαiαk + αkαi = 2δik ; αβ + βα = 0 ; α2
i = β2 = 1 .
What other properties do we require of these matrices αi, β , and can we explicitlyconstruct them?The αi and β must be hermitian in order that the hamiltonian (in [?]) be a her-mitian operator. Since α2
i = β2 = 1, the eigenvalues must be ±1. Also, from theanticommutation properties, αβ + βα = 0, that the trace of each αi and β is zero.For example αi = −βαβ and by the cyclic property of the trace (in general)
Tr (AB) = Tr (BA) (82)
Tr (αi) = Tr(β2αi
)= Tr (βαiβ)
= −Tr (αi) = 0. (83)
And from
αiβ + βαi = 0 (84)
β = −αiβαi (85)
Tr (β) = −Tr(βα2
)= −Tr (β) = 0. (86)
Since the trace is just the sum of the eigenvalues, the number of positive and negativeeigenvalues must be equal, and the αi and β , must therefore be even-dimensionalmatrices. The smallest even-valued dimension, N = 2 , is therefore ruled out, since itcan accommodate only the three anticommuting Pauli matrices σi plus a unit matrix.The smallest dimension in which the αi and β can be realized is N = 4, and that isthe case we shall study. In a particular explicit representation the matrices are
αi =
(0 σiσi 0
)and β =
(1 00 1
), (87)
where the σi are the familiar Pauli matrices, and the unit matrices in β stand for2x2 unit matrices.
Pursuing (2): To construct the dierential law of current conservation,
Pursuing (2): To construct the dierential law of current conservation, we rstintroduce the hermitian-conjugate wave functions ψ† = (ψ∗1, ψ
∗2, ψ
∗3, ψ
∗4) and left-
multiply the LRSE by [this row-vector] ψ† .
55
Next we form the hermitian conjugate of [the LRSE] and right-multiply by ψ. Thenwe subtract the latter equation from the former. The result is:
i~∂
∂t
(ψ†ψ
)=
~ci
3∑k=1
∂
∂xk
(ψ†αkψ
)(88)
or∂ρ
∂t+∇ · j (89)
[Equation of Conserved Probability Flow, ECPF], where we make the identicationof probability density
ρ = ψ†ψ =4∑
σ=1
ψ∗σψσ, (90)
and of a probability current [PC3C] with three components
jk = cψ†αkψ. (91)
Integrating ECPF over all space and using Green's Theorem,∂
∂t
∫d3x ψ†ψ = 0 .
The notation anticipates that the probability current j forms a vector if [ECPF]is to be invariant under three-dimensional space rotations. We must actually showmuch more than this. The density and current in [ECP¡ F] must form a 4-vectorunder Lorentz transformations in order to insure the covariance of the continuityequation and the probability interpretation. Also, the Dirac Equation [LRSE] mustbe shown to be Lorentz covariant before we may regard it as satisfactory.
Before delving into the problem of establishing Lorentz covariance of the Diractheory, it is perhaps more urgent to to see rst that the equation makes sense phys-ically. Note that these authors share my frequent issue of desiring to proceed inseveral directions at once!
We start simply by considering a free electron and counting the the number of solu-tions corresponding to an electron at rest. Equation [LRSE] then reduces to iH∂ψ/∂tsince the deBroglie wavelength is innitely large and the wavefunction is uniform overall space.
In the specic representation for α and β (given above), we can write down by
56
inspection four solutions:
e−imc2/h
t
(
10
)(
00
) , e−
imc2/ht
(
01
)(
00
) , e
imc2/ht
(
00
)(
10
) , and e
imc2/ht
(
00
)(
01
) ,
the rst two of which correspond to positive energy, and the second two to negativeenergy. The extraneous negative-energy solutions which result from the quadraticform of... [the Klein-Gordon equation] are a major diculty, but one for which theresolution leads to an important triumph [Ch. 5 of BDRQM]... antiparticles.
Regarding the positive-energy solutions, we wish to show that they have a sensiblenon-relativistic reduction to the two-component Pauli spin theory. To this end weintroduct an interaction with an external electromagnetic eld described by a 4-potential: Aµ = (Φ,A).
The coupling is most simply introduced by means of the gauge-invariant substitu-tion: pµ → pµ− e
cAµ made in classical relativistic mechanics to describe the interac-
tion of a point charge e with an applied eld.
The Dirac Equation modied for the eect of an EM eld is:
i~∂ψ
∂t=(cα ·
((p− e
cA)
+ βmc2 + eΦ)ψ. (92)
Equation 92 expresses the `minimal' interaction of a Dirac particle, considered tobe a (massive) point charge, with an applied electromagnetic eld. To emphasizeits classical parallel, we rewrite equation 92 with H = H0 + H ′, with H ′ = −eα ·A + eΦ. The matrix cα appears here as the operator transcription of the velocityoperator in the classical interaction expression for the interaction of a point charge:
H ′classical = −ecv ·A+ eΦ. The operator correspondence is vop = cα . This operator
correspondence, is again evident in equation ?? for the probability current.
BDRQM continue to establish the upper/lower two spinor components as the large/smallparts of the 4-component wave function.
57
A The Gaussian Beam; the Fundamental Ray
From Wikipedia: For a Gaussian beam (or ray), the complex E-eld amplitude isgiven by
E (r, z) =w0
w1
E0e
− r2
w21
− ikz − ikr2
2R (z)+ iς (z)
. (93)
Here E0 = |E (0, 0)| |; r is the radial distance from the center axis of the beam; and zis the axial distance from the waist. k = 2π/λ is the wave-number (radians/meter).w1 is the radius at which the amplitude drops to 1/e of its axial value. R (z) isthe radius of curvature of the wavefronts. ς (z) is the Gouy phase shift, an extracontribution to the phase that is seen in Gaussian beams.
B Initial development of the Lagrangian, following [GCM]
The standard treatment considers a set mi of mass points belonging to the system,under forces other-than-of-constraint Fi. Following D'Alembert's principle, weconsider a set of virtual displacements δri consistent with any constraints involved(constraints here taken as holonomic: such a constraint can be expressed by a relationfri = 0 ). The δri are given as functions of a (likely lesser) set of generalizedcoordinates qj which are taken as independently variable. (There are ways ofdealing with some non-holonomic systems, e.g. with Lagrange multipliers; but wedo not need these here.) The forces of constraint do no work.
D'Alembert's principle may be written as∑i
(pi − Fi) δri = 0. (94)
Color will aord a guide to the successive algebraic alterations of terms.
The 2nd term, corresponding to qj and to the the generalized force Qj, is
−Qjδqj =∑i
Fi
(∂ri∂qj
)δqj. (95)
The 1st term entails more extended eort:∑i
piδri =∑i
mriδri =∑i,j
mri∂ri∂qj
δqj. (96)
58
Employing the identity d(uv) = udv + vdu upon ri ,we obtain:∑i,j
d
dt
(miri
∂ri∂qj
)δqj −m[i]ri
d
dt
(∂ri∂qj
)δqj
. (97)
Exchanging derivatives wrt t and wrt ri on the right, and noting that ri = ddtvi, thisbecomes ∑
i,j
d
dt
(mivi
∂ri∂qj
)δqi −mivi
∂vi∂qj
δqj
(98)
With the (striking!) substitution∂ri∂qj
=∂ri∂qj
=∂vi∂qj
, eliminating ri entirely, in favor
of vi :∑
i,j
d
dt
(mivi
∂vi∂qj
δqj −mivi∂vi∂qj
δqj
), which in turn equals
∑j
(d
dt
(∂T
∂qj
)δqj −
∂T
∂qjδqj
).
Note that the sum on i has disappeared into T .
Recalling now that this last expression equals the 1st term of D'Alembert's principle,apply the 2nd term equal to −Qjδqj, and obtain∑
j
(d
dt
(∂T
∂qj
)− ∂T
∂qj−Qj
)δqj = 0. (99)
GCM recognizes these (plural because of j) as Lagrange's equations for such relativelygeneral cases as non-conservative forces.
In QM, the forces are generally conservative, i.e. derivable from a potential, so that
Qj = −∂V∂qj
; and, specically, they do not depend upon the velocities qj. Thus the
force term can be collapsed together with T to yield∑j
(d
dt
(∂L
∂qj
)− ∂L
∂qj
)δqj = 0, (100)
where L ≡ T−V . This conveys the more typically recognized Lagrange's equations.
59
C Sturm Louisville Problem
The ordinary dierential equation arising from the separation of thegeneral partial dierential equation
∇2ψ + k2ψ = 0 (101)
can be written in the form:
d
dz
(p (z)
dψ
dz
)+ (q (z) + λr (z))ψ = 0. (102)
This is called the Liouville equation.
The parameter λ is the separation constant or eigenvalue (in some cases,more than one separation constant appears...). Each of the functions p,q, r are characteristic of the coordinates used in the separation...
The Sturm-Liouville problem is essentially the problem of determiningthe dependence of the general behavior of ψ on the boundary conditionsimposed on ψ.
The solutions exhibit the features cited in the main text regarding com-plete sets of functions permitting determination of quasi-lengths, hencenormalization, projection; supporting Schwartz' Inequality, etc.6
D Mathematical Foundations
D.1 Hilbert Space
A Hilbert space, H, is an abstract vector space possessing an inner product which isa sesquilinear function 〈, 〉 : H×H → C (note that C is the complex numbers and ∗
is the conjugation operator) with the properties for all x, y, z ∈ H and α ∈ C (linear6Morse and Feshbach, Methods of Theoretical Physics, 1953, vol.1, p.719)
60
in second argument)
〈x, y〉 = 〈y, x〉∗ (103)
〈x, αy〉 = α 〈x, y〉 (104)
〈x, y + z〉 = 〈x, y〉+ 〈x, z〉 (105)
〈x, x〉 ≥ 0 (〈x, x〉 = 0 ⇐⇒ x = 0) (106)
these allow length and angle to be measured. The derived properties of the innerproduct are
〈αx, y〉 = 〈y, αx〉∗ = α∗ 〈y, x〉∗ = α∗ 〈x, y〉 (107)
〈x+ y, z〉 = 〈z, x+ y〉∗ = 〈z, x〉∗ + 〈z, y〉∗ = 〈x, z〉+ 〈y, z〉 (108)
Furthermore, Hilbert spaces are required to be complete, a property that stipulatesthe existence of enough limits in the space to allow the techniques of calculus to beused.
For a nite vector space where vectors are represented by a = aiei (Einstein sum-mation convention) the inner product of two vectors a and b could be dened by
〈a, b〉 ≡(ai)∗bi. (109)
For the case of a function space on an interval (−∞,∞) for two functions f (x) andg (x) the denition could be
〈f, g〉 ≡∫ ∞−∞
f ∗ (x) g (x) dx. (110)
D.2 Hermitian Operators
In general if H is a Hilbert space, and A : H → H is a continuous linear operatorthen A† then the Hermitian conjugate of A dened by
〈x,Ay〉 =⟨A†x, y
⟩∀ x, y ∈ H (111)
61
from this denition the following properties can be derived
A†† = A (112)(A†)−1
=(A−1
)†if A−1 exists (113)
(A+B)† = A† +B† (114)
(αA)† = α∗A† ∀ α ∈ C (115)
(AB)† = B†A† (116)
An linear operator A on the Hilbert space H is Hermitian if
A† = A. (117)
IfH is a nite dimensional vector space over the complex numbers then the Hermitianoperators can be represented by matrices Aij where
Aij = A∗ji. (118)
The same method is used to dene the Hermitian conjugate of a vector, x†, as thelinear map x† : H → C dened by
x†y = x† (y) ≡ 〈x, y〉 ∀ x, y ∈ H. (119)
The derived properties of Hermitian conjugates of vectors x, y ∈ H are
(x+ y)† = x† + y† (120)
(αx)† = α∗x† (121)
x†y =(y†x)∗. (122)
Using this notation we have expressions x†Ay ∈ C so that if H is a nite dimensionalvector space then x† would correspond to a row vector, y to a colume vectors, andA to a square matrix.
An linear operator U on the Hilbert space H is Unitary if
UU † = U †U = I (123)
where I is the identity operatory forH. The main characteristic of a Unitary operatorU is that is preserves the inner product so that
〈x, y〉 = 〈Ux, Uy〉 ∀ x, y ∈ H. (124)
62
Thus U preserved lengths and angles.
An linear operator A on the Hilbert space H is Normal if
AA† = A†A. (125)
D.3 Dirac Notation
If H is a Hilbert space denote the vectors in H by |ψ〉 , |φ〉 ∈ H and the Hermitianconjugates of the vectors by 〈ψ| and 〈φ| and the inner product by
〈ψ |φ〉 ≡ |ψ〉† |φ〉 . (126)
If A is a linear operator on the Hilbert space, A : H → H, we write
〈ψ |A|φ〉 ≡ |ψ〉† (A |φ〉) , (127)
〈ψ |A|φ〉 =⟨φ∣∣A†∣∣ψ⟩∗ . (128)
If A is Hermitian, A = A†, then
〈ψ |A|φ〉 = 〈φ |A|ψ〉∗ (129)
and〈ψ |A|ψ〉 ∈ R > 0 ⇐⇒ ψ 6= 0. (130)
63
Related Documents
