QM Reminder
Dec 21, 2015
Outline
• Postulates of QM• Picking Information Out of Wavefunctions
– Expectation Values– Eigenfunctions & Eigenvalues
• Where do we get wavefunctions from?– Non-Relativistic– Relativistic
• What good-looking s look like• Techniques for solving the Schro Eqn
– Analytically– Numerically– Creation-Annihilation Ops
Postulates of Quantum Mechanics
• The state of a physical system is completely described by a wavefunction .
• All information is contained in the wavefunction
• Probabilities are determined by the overlap of wavefunctions
2| ba
Postulates of QM
• Every measurable physical quantity has a corresponding operator.
• The results of any individ measurement yields one of the eigenvalues n of the corresponding operator.
• Given a Hermetian Op with eigenvalues n and eigenvectors n ,the probability of measuring the eigenvalue n is
223* nn orrd
Postulates of QM
• If measurement of an observable gives a result n , then immediately afterward the system is in
state n .
• The time evolution of a system is given by
• .
Hdt
di
corresponds to classical Hamiltonian
Common Operators
• Position
• Momentum
• Total Energy
• Angular Momentum
r = ( x, y, z ) - Cartesian repn
),,( zyxii p
toptot iE
L = r x p - work it out
Using Operators: A
• Usual situation: Expectation Values
• Special situations: Eigenvalue Problems
rdAAspaceall
3*
A
the original wavefn
a constant(as far as A is concerned)
Expectation Values
• Probability Density at r
• Prob of finding the system in a region d3r about r
• Prob of finding the system anywhere
)()( rr
rd 3
13 rdspaceall
• Average value of position r
• Average value of momentum p
• Expectation value of total energy
rdrspaceall
3
rdspaceall
3 p
rdspaceall
3 H
Eigenvalue Problems
Sometimes a function fn has a special property
fnthewrt
constsomefn
OpOp
eigenvalue eigenfn
Since this is simpler than doing integrals, we usually label QM systems by their list of eigenvalues (aka quantum numbers).
Eigenfns: 1-D Plane Wave moving in +x direction
x,t = A sin(kx-t) or A cos(kx-t) or A ei(kx-t)
• is an eigenfunction of Px
• is an eigenfunction of Tot E
• is not an eigenfunction of position X
kekeix
tkxitkxix )()( P
)()( tkxitkxit eeiETot
xex tkxi )( X
Eigenfns: Hydrogenic atom nlm(r)
• is an eigenfunction of Tot E
• is an eigenfunction of L2 and Lz
• is an eigenfunction of parity
)(6.13)(1
2)4(
)(2
)()(
2
2
222
42
rn
Zr
n
emZ
rVm
rr
nlmnlmo
nlmnlmnlm
2PHETot
)()1()( 2 rr nlmnlm 2L
units eV
)()( 2 rmr nlmnlmz L
)()()( rr nlmnlm Parity
Eigenfns: Hydrogenic atom nlm(r)
• is not an eigenfn of position X, Y, Z
• is not an eigenfn of the momentum vector Px , Py , Pz
• is not an eigenfn of Lx and Ly
Where do we get the wavefunctions from?
• Physics tools– Newton’s equation of motion– Conservation of Energy– Cons of Momentum & Ang Momentum
The most powerful and easy to use technique is Cons NRG.
Schrödinger Wave Equation
Vm
VKEH 2
2p
Use non-relativistic formula for Total Energy Ops
toptot iE and
titH t ,, rr
titVm t ,,
2
2
rrp
titVm t ,,
22
2
rr
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians
Klein-Gordon Wave Equation
Start with the relativistic constraint for free particle:
Etot2 – p2c2 = m2c4 .
[ Etot2 – p2c2 ] (r,t) = m2c4 (r,t).
trcmtrcii t ,, 42222
p2 = px2 + py
2 + pz2
a Monster to solve
Dirac Wave EquationWanted a linear relativistic equation
[ Etot2 – p2c2 m2c4 ] (r,t) = 0
Etot2 – p2c2 = m2c4
Change notation slightly
toptot c
icEp
/0
p = ( px , py , pz )
~ [P42c2 m2c4 ] (r,t) = 0
P4 = ( po , ipx , ipy , ipz )
difference of squares can be factored ~ ( P4c + mc2) (P4c-mc2) and there are two options for how to do overall +/- signs
4 coupled equations to solve.
Time Dependent Schro Eqn
Hdt
di
Where H = KE + Potl E
tx,
titVm t ,,
2
2
rrp
titVm t ,,
22
2
rr
ER 5-5
Time Independent Schro Eqn
KE involves spatial derivatives only
If Pot’l E not time dependent, then Schro Eqn separable
tfxtx ,
/, iEtextx
ref: Griffiths 2.1
Sketching Pictures of Wavefunctions
xExxVm totx
2
2
2
xExxVm
ptot
2
2
xExxVKE tot
KE + V = EtotProb ~
xExxVm totx
2
2
2
To examine general behavior of wave fns, look for soln of the form
xikeA where k is not necessarily a constant (but let’s pretend it is for a sec)
totEVm
k
2
22
VEm
k tot 2
2
Sketching Pictures of Wavefunctions
KE
VEm
k tot 2
2
xikeA
If Etot > V, then k ReRe
~ kinda free particle
If Etot < V, then k ImIm
~ decaying exponential
2/k ~ ~ wavelength /k ~ 1/e distance
KE + KE
Techniques for solving the Schro Eqn.
• Analytically– Solve the DiffyQ to obtain solns
• Numerically– Do the DiffyQ integrations with code
• Creation-Annihilation Operators– Pattern matching techniques derived from 1D SHO.
Analytic Techniques
• Simple Cases– Free particle (ER 6.2)– Infinite square well (ER 6.8)
• Continuous Potentials– 1-D Simple Harmonic Oscillator (ER 6.9, Table 6.1, and App I)– 3-D Attractive Coulomb (ER 7.2-6, Table 7.2)– 3-D Simple Harmonic Oscillator
• Discontinuous Potentials– Step Functions (ER 6.3-7)– Barriers (ER6.3-7)– Finite Square Well (ER App H)
Numerical Techniques
• Using expectations of what the wavefn should look like…– Numerical integration of 2nd order DiffyQ
– Relaxation methods
– ..
– ..
– Joe Blow’s idea
– Willy Don’s idea
– Cletus’ lame idea
– ..
– ..
ER 5.7, App G
SHO Creation-Annihilation Op Techniques
xmpim
a ˆˆ2
1ˆ
xmpi
ma ˆˆ
2
1ˆ
22
2
1
2
1
2
ˆ)( xk
m
paa H
Define:
ipx ˆ, 1ˆ,ˆ aa
If you know the gnd state wavefn o, then the nth excited state is:
ona ˆ