Core Notes for Comps (Mike Nielsen, Luke Pickering)

7/30/2019 Core Notes for Comps (Mike Nielsen, Luke Pickering)

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Comprehensive Notes

Contents1 Mechanics 3

1.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 SHO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Waves 5

2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Quantum Mechanics 73.1 Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . 73.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Potential Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Electromagnetism 104.1 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Electrostatic Potential . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Electromagnetic Waves in a Vacuum . . . . . . . . . . . . . . . . 124.5 Waves in Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Waves in Conductors . . . . . . . . . . . . . . . . . . . . . . . . . 144.7 EM Waves in a Waveguide . . . . . . . . . . . . . . . . . . . . . . 15

5 Thermodynamics And Statistical Physics 155.1 The Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.3 Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.6 Functions of State Variables . . . . . . . . . . . . . . . . . . . . . 185.7 Classical Statistical Physics . . . . . . . . . . . . . . . . . . . . . 195.8 Bose-Einstein and Fermi-Dirac Statistics . . . . . . . . . . . . . . 20

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5.9 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Atomic Physics 216.1 Spectral Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Hydrogen Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3 Pertubation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 236.4 Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.5 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Electrons in Solids/Solid State Physics 267.1 Free Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 Semiconductor Transitions . . . . . . . . . . . . . . . . . . . . . . 287.4 pn Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8 Particle And Nuclear Physics 308.1 Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2 Nuclear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.3 Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.4 Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9 Probability And Statistics 339.1 Properties of distributions . . . . . . . . . . . . . . . . . . . . . . 339.2 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . 339.3 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 349.4 Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . 359.5 Propagation of Errors . . . . . . . . . . . . . . . . . . . . . . . . 36

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1 Mechanics

1.1 RotationsL = I

Moments of Inertia:I xx = y2 + x2 dm

Thin Rod length l

I = x2dm dm = dv = Adx=

l2

l2x2Adx = 1

12ml 2

Sphere of uniform density, radius R

I = x2 + y2 dm dm = dv = r 2 sin() drdd=

25

mR 2

Disc of radius R

I = R

0r 2dm dm = dv = 2r

=12

mR 2

1.2 SHO

mx = kxsolution is oscillatory with frequency 2 = km

x = A sin(t) + B cos(t)

For a damped oscillator

mx x + kx = 0solution of form

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x (t) = A exp( pt) + B exp(

pt)

where p is found by

mp2 p + k = 0

p = 2 4mk2m

3 distinct cases,

Overdamped: 2 > 4mk , The system returns to equilibrium exponentiallywithout oscillating.

Critically Damped: 2 = 4 mk , The system returns to equilibrium exponen-tially in the shortest possible time, without oscillating.

Under Damped: 2 < 4mk , p C , The system oscillates about equilibriumwith exponentially decaying amplitude.

Q Factor: Dened as

Q = 2 Energy Stored

Power loss per cycle

or in the high Q limit Q = .

1.3 Special RelativityLorentz Transforms for a boost into a frame S with aligned x axes moving atc relative to S

x = (x ct )ct = (ct x )y = yz = z

where = 1 2 1 / 2

Length contraction - an observer in S will see a moving object whose restframe is S contract to l = l Time dilation - an observer in the moving frame S will record a timeperiod t in the stationary frame S as t = t

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Relativistic Kinematics:

E rest = mc2 E total = mc 2E 2 = p2c2 + m2c4 p = mv

=pcE

v transform

v =dxdt

=dxdt

1dtdt

= c1 vc

vc

Kinetic Energy: In the non-relativistic limit

E = mc 2

1 +12

2

E mc2 +12

mv2

KE 2 Waves

2.1 Properties

Travelling waves of the form

cos(kx t) sin (kx t) exp (i [kx wt])

Standing waves require L = 2 k = L , where L is the length of themedium in which the wave is set up.Superposition: Different beats as sin(1 t)+sin ( 2 t) = 2 sin 1 + 22 t sin

1 22 t

Velocities: Phase velocity v p = k , Group velocity vg =k - the group veloc-

ity is the ow of physical quantities

Doppler effect: Classical waves f source = v+ vov+ vs

f observer , where v is thespeed of waves in the medium, and vo , vs are the speeds of the observer andsource relative to the medium, respectively.

The relativistic doppler effect is slightly different and arises from time dila-tion, as t = t this leads to f s =

1+ 1f o .

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2.2 Diffraction

Innitesimal Youngs Slits:

The path difference between the two slits is n = d sin() . So for x thanthe distance to the screen we can also approximate sin() = , this gives us

d = m

thus the angular spacing between fringes is

= d

we can nd the verticle distance between fringes as

x = R = Rd

Grating Equation: To have constructive interference at a point we mustimpose d sin() = m , generalising to N slits in a grating Nd sin() = Nm ,called the grating equation.

Innitesimal Grating: The transmission function is given by T (y) = Rect (y/ d/ 2 ) (y

nd)

Finite Width Grating: The transmission function for the diffraction gratingcan be expressed as the product of the grating width and the convolution of eachslit with a dirac comb.

T (y) = Rect (y/ d/ 2 ) { (y nd) Rect (y/ a / 2 )}

A Dirac comb transforms to another comb where the width is 2k 1d = d ,the transform of the single slit gives A0 sin( ) and the transform of the width

of the grating gives another sinc function, as in reciprocal space small featuresfrom real space result in larger features we can see that the sinc generated bythe individual slits will be much wider in the diffraction pattern than the sincgenerated by the size of the grating. Thus taking the comb and multiplying bythe small grating width generated sinc and then we can use the individual slitwidth generated sinc as an envelope.

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2.3 Refraction

By Fermats Principle, Light travels takes the shortest optical path between 2points, where optical path length is given by d = ln where l is the physicaldistance and n is the refractive index of the media. From this Snells law canbe derived

n1 sin(i ) = n2 sin(r )

3 Quantum MechanicsDe Broglie Relations:

p = k E =

3.1 Heisenberg Uncertainty PrinciplePosition-Momentum:

p x

2For a particle travelling through a nite width slit the HUP imposes a posi-

tion restriction on the particle in the plane of the obstruction, this results in amomentum uncertainty in that plane which gives rise to diffraction.

A conned particle has a nite x and thus a nite uncertainty in its mo-mentum.

Energy-time:

E t

2This gives rise to line widths in atomic spectra due to short level lifetimes.

3.2 Observables

Represented by Hermitian operators

Hdx = H dx Operators have a set of eigenfunctions and eigenvalues Hermitian operators non-degenerate eigen-functions are orthogonal andtheir eigenvalues real Energy operator is the hamiltonian H = p

2

2m + V (x)

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Representations:x-rep

x = x p = i p-rep

x = i p = p

Schrodinger Equations:Time Dependant

i t

= H

Time Independant

H = E

3.3 Scattering

Particles will scatter off potential steps and barriers. Unlike classically therewill be a reection coefficient for a potential step down.

Separating the total into componets to the left and right of the step

L = A exp(ik l x) + B exp( ik l x)R = C exp( ik r x)

by p2

2m = E kl = 2mE / 2 , kr = 2m (E V 0 )/ 2 . For a particle withE < V 0 the wavefunction within the step is exponentially decaying, Classicallythis is a forbidden region. As and must be continuous everywhere exceptat an innite discontinuity L = R , L = R at the barrier. This leads toconditions on the amplitudes such that

A + B = C kl (B A) = kr C

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The expressions for the transmission and reection coefficients are given by,where the wavevectors are introduced to keep the ux of particles constant

T = |C |2 kr

|B |2 kl

R = |A|2 kl

|B |2 kl

For a nite length barrier the expontentially decay component within thebarrier will give rise to another free particle wavefunction at the other side of the barrier. The amplitude of R at the other side of the barrier dictates theprobability of the particle tunnelling through the classically forbidden barrier.

3.4 Potential Wells

Innite Well: A well, half width a, with innite potential at a, a and nopotential within the well. The boundary conditions dictate that any must bezero at both edges of the well, solving the TISE under these conditions yieldsthe general solution

= A sin(kx) + B cos(kx )

Where k = 2mE / 2 is the free particle wavevector. From the applicationof the boundary conditions k is quantised for 2 casesA = 0 k =

n2a

n = 2 , 4, 6

B = 0 k =n2a

n = 1 , 2, 3

So we arrive at the solution

=cos(nx / 2a ) n oddsin(nx / 2a ) n even

With E =2 2 n 2

8ma 2 .

Finite Well: Using a similar argument to the potential step, split up thetotal into 3 components wavefunctions L , C , R . The boundary conditionsare now that the wavefunction is continuous over the potential steps these leadto the 2 simultaneous conditions

k tan( ka ) =

k cot( ka ) = where k = 2mE / 2 , = 2m (E V 0 )/ 2 . These can be solved graphically toyield the energy levels allowed.

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3.5 Harmonic Oscillator

A particle conned to a potential well dened by V (x) =12 Kx

2. So the TISEbecomes

2

2m 2ux 2

+12

Kx 2u = Eu

The solution is the set of Hermite polynomials denoted H n (x). The energylevels are given by E n = n + 12 . note that this system has a minimumenergy, called the zero-point energy when n = 0 .

4 Electromagnetism

4.1 Maxwells Equations

Integral form Differential Form

Gauss Law for E s E ds =10 d E =

o

Gauss Law for B s B ds = 0 B = 0Faradays Law c E dl = =

dBdt E =

Bt

Amperian Maxwell Law c B dl = 0I + 00dE

dt B = 0 j f + 00

Displacement and Demagnetizing eld:

D = E + P H =B M

Where P , M are the polarisation and magnetisation respectivly and =r 0 , r 0 . In an HIL medium

D = E B = H

These lead to alternate forms of Maxwells equations containing H and D .

Force on a particle: The force on a charged particle due to Electric andmagnetic elds is given by the Lorentz Force

F = q (E + v B )In a uniform magnetic eld this leads to cyclotron motion, as F v , B the

particle exhibits circular motion. Equating the Lorentz force to the requiredcentripetal force the frequency of rotation can be determined as

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cyc =qB

m

4.2 Electrostatic Potential

Potential energy of a discrete charge distribution is given by U = q04 0 iqir i

or U = q04 0 dqr in the continuous case. The Electrostatic potential is thepotential energy per unit charge thus V = U q0 . The work done by/against aforce is dened as W ab =

ba F dl , this can be used to work out V for a givenE

V a V b = b

aE dl V = E dl

Notice that the value is only dependant on the starting and end points notthe path taken. Conversley the Electric eld can be calculated by a knownpotential

E = V Method of Images: There can be no electric elds within a conductor as freecharges will move to counter it. This can be used to nd the charge distributionon a conducting surface, there can be no component of E tangential to thesurface. If another charge is placed near this surface, the charge distributionof the surface will change to obey this condition. By placing fake charges inthe conductor such that there is only a component of E perpendicular to thesurface the charge distribution can be found.

4.3 Poynting Vector

EM Energy:

Energy Density in an Electric Field uE = 12 E D uE = 14 Re E D

Energy Density in a Magnetic Field uB = 12 B H uB = 14 Re B

H

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Poyntings Theorem:Poynting Vector

N = E HUsing energy conservation we know the EM energy initially in a volume

must either

Converted by ohmic heating 1

Stay Stored in the elds 2

Leave the volume 3Which leads to the condition that any energy leaving the eld must escape thevolume or be converted by ohmic heating.

t 12 (E D + B H ) d

2+ s N ds 3

+ E j f d

1= 0

this can be expressed as a differential

t

12

(E D + B H ) + N + E j f = 0

Useful Properties of the Poynting Vector:

Energy Flux N = c u k

Momentum Density = Nc2

Radiation Pressure Pressure absorb =

N

c k =12 Pressure reflect

4.4 Electromagnetic Waves in a Vacuum

Complex Notation:Faradays law

H = k EAmperes law

E =H k

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Product Averages

AB = 12

Re [AB ]

Wave Equation: From Maxwells equations we can derive wave equations forboth E , H

2E = 00 2

t 2E

2H = 00 2

t 2H

Where the speed of propagation is c = 1 0 0 , the solution for E is givenby E = E 0 ei ( k r t ) , H can be found by Faradays law and it thus alwaysperpendicular to E . For elds of this form can use a simple, complex notationto express Maxwells equations.

Wave Properties:Wave Impedence: From Faradays law the ratio of E ,H is constant, for a

wave propagating in the x direction

|E ||H |

= = zThe energy density for the electric and magnetic components are equal

thus uE = uH = 12 u, giving the time average energy density of the EM eldas

u = |E |

2

2=

|H |2

2

Wave Polarisation: For a wave propagating in the x direction, linear po-larisation is where the direction of the E eld is at a constant angle to the yaxis. Circular polarisation is where the E and H elds rotate around the axis of propagation maping out a circle in the y z plane.4.5 Waves in Dielectric

The speed of light in a medium is given by cm = 1 < c and thus the refractiveindex of a material is dened as n = ccm > 1. Maxwells equations give usconditions for how waves act as they cross a boundary between two dielectricmedia these result in some familiar laws

Angle of Incidence = Angle of Reection I = R

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Law of Refraction kT = n 2n 1 kI

Snells Law Sin (I ) = n 2n 1 Sin (T )

Brewster Angle T an (B ) = n 2n 1 At this angle there is no reected wave, all the energy is transmitted

across the boundary.

Total Internal Reection: If n 1n 2

2Sin 2 (

I ) > 1then the x component of

kT is complex, the I that results in n 1n 22

Sin 2 (I ) = 1 is called the criticalangle, c . This results in an exponential decay envelope in the second mediumand the wave does not propagate but is totally reected. note the above con-dition can only be satised if n1 > n 2 . Transmitted wave is descibed as beingevanescent, oscillatory with a quickly decaying amplitude.

E = E T0 ei (k tz zt ) e|k tx |x y

4.6 Waves in Conductors

Free charge is given by Ohms law j f = E which alters the dispersion relation of the waves to

k2

= 2

1 +i

, as

kis complex the wave within the conductor

is evanescent.

Good Conductors: In a good conductor Electric Current DisplacementCurrent . k can be expressed as

k = (1 + i) 2 = kc (1 + i)This leads to the skin depth of a good conductor, the depth at which the

amplitude of the transmitted wave has fallen by e as

=1k

c

=

2

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4.7 EM Waves in a Waveguide

The Transverse Electric Wave, Electric eld in waveguide(Width a, Height b)of form. By Maxwells equations for a conducting waveguide, the electric eldtangential to the surface of the conductor and the magnetic eld normal to theconductor must be zero. This means we need to quantise the wave such thatE t = 0 at z = 0 , b.

E = E 0y Sinm

bz ei (kx t ) y

Wave Equation leads to the dispersion law,

2 = k2c2 +m

bc

2

From Faradays Law the H Field can be determined,

H =E 0

0im

bCos

mb

z x + kSinm

bz z ei (kx t )

For wave to propagate down waveguide k2 > 0, if

c

2bm

wave cannot propagate. Long wavelengths cannot propagate along waveguide.

Wave Velocities

v p =c

1 mb c 2vg =

c2

v p

5 Thermodynamics And Statistical Physics

5.1 The Laws

Zeroth Law: If A is in thermal equilibrium with B and B is in thermal equilib-rium with C then A is in thermal equilibrium with C. Can use this to constructisotherms from the ideal gas law.

First Law: Conservation of energy states that, The change in internal energyof a system is the sum of total heat energy in less the work performed by thesystem. For nite changes this is given as U = W + Q and for inntesimalchanges as

dU = dW + dQ

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Second Law: Entropy of a system and its surroundings must increase forany process

S AB B

A

dQT

Third Law: The entropy of a system approachs 0 as T 0.5.2 Equations of State

Ideal GasP V = NkB T U =

32

NkB T

Van de Waals Gas

P + aN 2

V 2(V Nb) = NkB T U =

32

NkB T aN 2

V

where a characterises the inter-particle attractive force and b is the volumeof a particle.

5.3 Reversibility

A process is reversible if once done can be undone without affecting anythingoutside the system. Any real process is irreversible, this can be explained sta-tistically: for any system consisting of N particles there are a multitude moreways that they can be congured which differ to the original conguration thatit is a statistical impossiblity that they will recongure unaided.

Quasi-Static Process: A Process which evolves slowly so that at each stagethe system is in thermodynamic equilibrium.

Adiabatic Process: Any process for which Q = 0 is called an adiabaticprocess, in this case the rst law is dU = P dV Work: For a system with dissapative forces any process is irreversible thusthe work done by an ideal gas is given by

W rev = P dV W irr > P dV Resevoirs: A system which is in contact with the system under analysis buthas many more particles such that its temperature and pressure are effectivelyxed.

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Constant Volume: For a system at constant volume the rst law becomesdU = dQ dene the heat capacity at constant volume dQ = C V dT thus atconstant volume reversible heating can take place.

5.4 Engines

A Heat Engine: A cyclic process which produces work by using heat owfrom a hot resevoir to a cold resevoir. The efficiency of a heat engine is denedas the useful work produced per heat energy used

=W outQ in

A Heat Pump/Fridge: A cyclic process which heats a hot object/cools acold object by using work. As the goal of a heat pump and a fridge are differenttheir efficiencies are dened slightly differently

HP =QoutW in

F =Q inW in

Carnot Cycle: A cyclic process made of two adiabats and two isotherms.This is a practicle cycle that can be used to create efficient heat engines andpumps. Since all heating and cooling done at constant temperature only requireone set of heat resevoirs.

Carnot Theorem: No engine operating between two heat resevoirs can bemore efficient than a Carnot engine. All Canot engines have the same efficiencygiven by

= 1 T H

T C

5.5 Entropy

Clausius Inequality: For any cyclic process dQT 0 where the equality isfor a reversible process and the inequality for an irreversible one.

Entropy: We dene another state variable entropy, such that an innitesimalchange in entropy is given by dS = dQT . For any irreversible path betweentwo points in state space we can also dene a reversible path, since these arevariables of state this becomes a cyclic process and from Calusius inequality wearrive at the Entropy form of the second law

S AB B

A

dQT

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5.7 Classical Statistical Physics

Microstate: The state of all the individual particles corresponding to a sys-tem

Macrostate: The state as described by state variables, there are many mi-crostates that correspond to the same macrostate.

Statistical Denition of Entropy: The Boltzmann denition of entropyis given by

S = kB ln()

where is the multiplicity of the system, the multiplicity for choosing M dis-tinguishable objects from N is given by (M ) = N !M !(N M )! .

Microcanonical Ensemble: All macrostates, denoted by , are equiprobablethe microcanonical ensemble is dened by p = 1 . The system is contrained

from losing particles or energy to its surroundings thus M = N i r i , whereM = U/ 0 and the energy of each particle is = 0r for N distinguishableparticles.

Canonical Ensemble: The system is now allowed to exchange energy withits surroundings but the total particle number is still constrained. The prob-ability distribution for this ensemble is found to relate to the energy of themicrostate in questions and to have the form of the Boltzmann distribution

p =1Z

exp( / kB T )Where Z is the canonical partition function and is naively introduced so that

the distribution is normalised and is thus given by Z = p . The partitionfunction can be used as a link to the macroscopic thermodynamic state variables.The internal energy of a system is given by

U = [ln Z ]

= kB T 2

[ln Z ]T

The statistical physics equivalent to the free energy is given by F = U T S =kB T ln (Z ) where the usual thermodynamic relations can be used to nd otherstate variables.Generalised Entropy: The statisical physics generalised form of S is givenby S = kB ln ( p ).

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Density of states: Examining momentum states in 3D we see that each statein each dimension is spaced by p = h2L , where L is the connement in a givendimension. For the partion function, the summation over all momentum statesbecomes

Z tr =1

( p)3

n x

p

n y

p

n y

pexp[ ( p)]

which can be taken to integrals and integrated in polar coordinates to yield

Z tr =L3

h34 0 p2 exp[ ( p)] dp

this integral depends on if the particles in question are non-relativistic,( p) = p

2

2m , or ultra-relativistic, = pc. The full partition function also contains

a degeracy factor g which relates to the non-internal degrees of freedom.

Grand Canonical Ensemble: The system is allowed to exchange both en-ergy and particles with the surroundings, the probability of a given microstateis given by

p =1

Z exp[ ( N )]

where = T S N V . Similarly to the Canonical ensemble we can draw alink to thermodynamics, this time by the Grand Potential Function = kB T ln (Z ) = U T S N

5.8 Bose-Einstein and Fermi-Dirac Statistics

The Grand Partition Function is most simply expressed as a sum over all possiblevalues of all occupation numbers

Z =n 0 n 1

... exp r

n r ( r )

this factorises out as a product of the individual state occupation sums

Z = r n r exp[n r ( r )] where we express this as a product of in-dividual single particle state partition functions Z = r Z r . As possible oc-cupancy numbers depend on what type of particles the system is made up of there are two separate distributions for occupancy. For bosons the occupancysum runs from 0 , and for fermions, by the PEP, it runs from 0 1. Theaverage occupancy can then be determined from these partition functions byn r = n r n r pr,n r = 1

ln( Z r ) r resulting in,

nBE/FD =1

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From these distributions we can see that if ( r ) 1 the 1 in the de-nominator of the distributions becomes insignicant and thus these distributionsboth converge on the classical boltzmann distribution nB = exp [ ( r )].In this limit n r 1 therefore we expect the different properties of bosons andfermions to be less evident. nBE diverges for so we must impose

< min

5.9 Black Body Radiation

Modelling the system as a photon gas, photons are spin 1 and are thus bosons,massless and thus always ultrarelativistic. They have a degeneracy of g = 2 dueto two orthogonal polarisation states. The chemical potential of a photon gasis = 0 as partcles are not conserved. The occupancy distribution becomes

nBE n = 1exp[ r ]1We can nd the internal energy of this system by U = r n r r , it is more

insightful to change the integration variable to nd the spectral energy densitywant in the form u = U V = us () d by evaluating the integral we see theenergy density of the photon gas is given by

u =2k4B

15 3c3T 4

6 Atomic Physics

6.1 Spectral LinesWhen electrons make transitions between energy states the energy decit mustbe absorbed or emitted, usually as EM radiation. A transition between 2 levelswith energy E 1 and E 2 emits or abosrbes a photon of frequency = E 2 E 1h anygiven atoms has a myriad of energy levels and allowed transitions. These emis-sion spectra from atoms are used to classify atoms by their unique ngerprintof avaliable transitions. The emitted photons arent at a single frequency butrather the emission lines are broadened by various effects.

Radiative Transitions: A transition between 2 levels happens at a rate/probabilitygiven by Fermis Golden Rule which states that the transition rate is propor-tional to the matrix element of the interaction with the initial and nal states.

1 if | f |H | i |

2

For an incident, oscillating electric eld on an electric dipole the perturba-tion is given by H = P d E which to rst order gives the Electric Dipole

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approximation H eE 0 r . The rate is given by the product of the radial andangular integrals

e2 |E 0|2

0Rn f l f rR n i l i r

2dr 2

0

0Y n f l f r e rad Y n i l i sin() dd

The angular integral evaluates to 0 unless selection rules are obeyed. Sincethe parity of Y nl is given by (1)

l the angular integral must have a parity changebetween the states this gives the selection rule that L = 1.Intrinsic Broadening: Electrons in excited states have a nite lifetime gov-erned by the Einstein coefficients such that = 1A 21 , where A21 is the coefficientfor spontaneous decay of an energy level with no incident EM eld. By the HUPanything with a nite connement in time must also have a nite uncertaintyin energy given by t E thus = A 212 . Thus transitions with shorterlifetimes are broader than comparitvely long lived levels. The actual intensityprole is given by

I ( ) = I 0 2

2

( 0)2 + 2

2

In a real atom there are many levels and both the higher and lower levelsare broadend, so the total broadening of the spectral line is given by

21 =1

2 2+

12 1

Doppler Width: This has a greater broadening effect than the quantum

mechanical effect of the HUP. An effect due to the thermal velocity of theemitting particles, an emitter approaching an observer with velocity vx sees ashifted wavelength of 0 =

0 = x . Using the Boltzman distribution to nd

the number of atoms at a given velocity, P v ( ), the velocity distribution describesa gaussian about the thermal velocity, and thus the doppler broadening givenby the FWHM of the P v ( ) can be shown to be

D 0

=2c 2KT ln(2)m

which can easily be linked to a change in by d 0 = d 06.2 Hydrogen Atoms

The hydrogenic wavefunctions are valid for hydrogen and, with certain per-turbations, for other 1 electron system. They can be split up into radial andangular parts as tot = Rnl (r ) Y lm (, ). The 3D radial distribution is givenby 0 4 | nl |dr where nl = rR nl .For a hydrogen atom the principle quantum

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number n determines the energy of the unperturbed state. The angular mo-mentum l is under the constraint that n > l . The spectroscopic notation for astate of l = 0 , 1, 2, 3, 4... is given by s,p,d,f,g... continuing alphabetically as lgets larger. The projection of l onto the z axis is given by the quantum numberm l which can take values under the constraint |m l| < l . For the unperturbedhydrogenic wavefunction the energy is uniquely determined by n and thus thereis a i


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Stark Effect: In a external electric eld, , the electron orbitals about anatom will be distorted, this effect can be described by the pertubation H =

d , so for an electric eld in the zdirection the pertubation is given byH = ez z

6.4 Zeeman Effect

The Zeeman effect is the observed splitting of electronic levels, lifting the de-generacy on the magnetic quantum number, in the prescence of an externalmagnetic eld. The perturbation is given by the magnetic interaction betweena magnetic moment and the applied B eld. Considering spin the perturbationbecomes H mag = B = 2 B S B . For a magnetic eld only in the zdirectionwe have

E = 2 m s B Bz

as a result the degeneracy in mS z is lifted. The electrons orbit also producesa magnetic moment where the perturbation is given by H mag =

B L B . Againconsidering a eld in the z direction E = m l B Bz

as a result m lz is no longer a degenerate quantum number. The total mag-netic interactions for an electron is thus the addition of these two effects andthe Spin-Orbit interaction. There are two distinct regimes depending on thestrength of the external eld.

Strong External Field: We ignore H SO , thus j is not a good quantumnumber, as the L and S do not interact, and the Zeeman effect affects theintrinsic and orbital magnetic moments individually. For a 2 p ewhich hasl = 1 , s = 12 , there are 6 states for all possible combinations of m l = 1, 0 andm s = 12 .Weak External Field: j is still a good quantum number as E SO E zeeman ,thus can use m j to give the Zeeman shift as E = gJ B m j Bz .

6.5 Molecules

For single electron molecules can use LCAO(Linear Combination of Atomic Or-bitals) to create single electron molecular wavefunctions. By using the hydrogen

wavefunctions can construct two superpositional states for the molecular wave-functions

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g (r , R) =1

2 [1s (r a ) + (r b )] gerade evenu (r , R) =

1 2 [1s (r a ) (r b )] ungerade odd

where r a , r b are the distances of the electrons from nucleus A, B respectivly.Where g corresponds to a bonding state where the electron has a niniteprobablility of being between the two nuclei, u corresponds to an anti-bondingstate where the electron has a low probability of being between the nuclei anda zero probablility directly at the mid-point between the two. In molecules = |m l| is an important quantum number where states with = 0 , 1, 2, 3 arelabelled ,,, states respectivly.

Vibrational Modes: The form of the potential for intermolecular bonds canbe approximated by the morse potential V (x) = Deq (1 exp[ax ])2 . Expand-

ing the exponential to rst order gives a SHO potentials. Modelling vibrationsas SHO the energy is given by E v = 0 v + 12 , where v is the vibrationalquantum number v = E vhc = osc where osc =

02c . This is a valid approxi-

mation for heteronuclear molecules.For a more accdurate model the morse potential gives energies of v =

e v + 12 xe e v + 122 . This can be expressed in a similar form to SHO

but with a v depenedant frequency,

v = e 1 xe v +12

v +12

The selection rules state v =

1,

2,

3... where

1 is called the fundamental

transition, and higher are called the n th overtones. The fundamental transitionis by far the most important. At room temperature heternuclear moleculesundergo IR vibrational transitions, homonuclear molecules do not absorb.

Rotational States: Classically rotational energy is given by E = L2

2I , whereI = R 20 . The quantum mechanical operator is thus given by H rot = L

2

2I , callingthe rotational quantum number J we see H rot =

2 J (J +1)2I .

J = BJ (J + 1)

Where B =2

2hcI , can use B to nd I and thus R0 . The selection rule forthese transitions is given by J = 1. Pure rotational spectra are not seen inhomonuclear molecules.

Rotation Coupled Spectra: For a fundamental vibrational transition V =0 1, = v B u J u (J u + 1) B lJ l J l + 1 . J = J u J l splits thetotal transition into two branches, the P J = 1 and the R J = +1branch.

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Rotational transitions also couple to electronic transitions where the selec-tion rules become J = 0 ,

1 but not J = 0

0.

7 Electrons in Solids/Solid State Physics

7.1 Free Electron Model

Density of States: The FEA gives a k quantisation condition kn = nL sothe size of a k state in d dimesnional space is k = L

d . Find the numberof estates which can t between k k + dk this gives D (k) dk which canbe changed into the density of states at a given energy by a change of vari-ables D (k) dk = D (k)

dkdE

D (E )

dE . This method yields to following results for 1,2,3

dimensional Density of States

D1d (E ) = 8mL 2h2 1 E D2d (E ) =

4mL 2

h2

D3d (E ) =2

8mL 2

h2

3 / 2 E Can use these to determine the f , The Fermi Energy - the energy of the

highest occupied state/lowest unoccupied state at 0K , and the internal energy

at 0K , U 0 by

N = f

0D (E ) dE U 0 =

f

0ED (E ) dE

Fermi Wavevector: The wavevector of an free electron with Energy equal

to the Fermi Energy kf = 2m f

Bloch Theorem: The energy states of a 1e in a periodic potential can bechosen to have the form of a plane wave modulated by a function with theperiodicity of the lattice.

k (r ) = uk (r )exp( ik

r )

where uk (r ) = uk (r + R ) | k (r )|2 = | (r + R )|. The probability of theparticle particles position is equally distributed in any cell.

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Formation of Bands:

The Kronig-Penny model yields a band structure by enforcing boundaryconditions on Bloch electrons in a periodic potential dened by cells of nopotential with delta potentials bounding each cell.

by the LCAO(Linear Combination of Atomic Orbitals) we see that bring-ing two atoms close to each other results in energy states being split intotwo the so called Bonding and Anti-Bonding states, extending this toN atoms in a solid we can qualitatively see a bands of states forming.

What type of solid depends on the location of the Fermi level relative to thisband structure.

Metal - f is in a band - conduction states just above valence states.

S/C - f lies in a gap between bands but gap is small Insulator - same as an S/C but gap is large

Laue Condition: This is equivalent to the Bragg condition but in k space

G = k kwhere k , k are the incidient and scattered k s respectvely. P = Gmomentum transfered to the lattice.

Brillouin Zones: The locus of all points which obey the Laue condition andthus can be scattered by the crystal.The n th BZ is the locus of all points whichcan reach the origin by crossing (n

1) Bragg Planes/BZ surfaces. At BZ

surfaces electrons scatter - cannot travel through the crystal.

Observations of K-P Dispersion:

1. E (k) is discontinuous at ka = n

2. Near k = 0 , E = 0 is not allowed

3. At zone boundaries E (k)has zero slope

7.2 Carriers

Electrons and Holes: When electrons are excited across a band gap they

leave behind an empty state, it is valid to model this empty state as a particlein itself called a hole. A is a not electron and thus has mh = m e , q h = + e.Particles which exhibit the Bloch wavefunction can be modelled as travellingthrough the solid as a free particle but with a modied and k dependant mass,called the effective mass. From the group velocity

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vg =d

dk=

1

dE

dk

F = mdvgdt

= 2d2E dk2

1

mdvgdt

At BZ zone boundaries as E (k) has zero slope m = , carriers cannottravel through the crystal with these values of k and are thus scattered into freestates.

Carrier Density: The effective Density of States near the band edges is givenby

N C/V = 2 mn/p kB T 2 2

3 / 2

And the carrier density at the band edges is given by

n/p = N C/V expC/V f kB T

(1)

the product np is found to be cosntant for a given f

np = N C N V exp gkB T

Intrinsic S/C: n i = pi , np = n2i = p

2i

Extrinsic S/C: In General the bulk S/C is neutral so n i + N A = pi + N d .Therefor for an n-type n = p + N d , n p and vice versa for p-type. In ann-type f is closer to c and vice versa for p-type.

Temperature Dependence:

Intrinsic region - high T - intrinsic carriers dominate Saturation region - dopants are fully ionised - device operating tempera-ture

Freeze-out region - extrinsic carriers frozen back to dopant atoms.

7.3 Semiconductor Transitions

Phonons - Horizontal 50 phonons to a vertical transition of g Photons - Vertical 1000 Photons to a horizontal transition of a BZ width

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The Coulomb term is the electric potential energy which would be requiredto bring the nucleus together from the point of view of 1 proton - BE =

ac z (z1)A 1 / 3 .The p s and n s must t into quantum energy levels, each has its own set of

energy levels, by the PEP they must ll up to higher energies. If there are moreneutrons than protons, for the same A the nucleus would have a higher bindingenergy if some of those neutrons were protons thus the asymmetry is given by BE = aa

(A2Z )2

A , note (A 2Z ) = ( N Z ).Again treating the p s and n sseparately, the total binding energy will bemore if theres an even number of particles. By the PEP they pair up in thesame spatial states but with opposing spin states, this results in a large overlapand thus a larger binding energy. The correction is given by BE = a pA 1 / 2 wherea p > 0 if both N, Z are even, a p = 0 is one is even and the other odd, and a p < 0if both are odd.

The total SEMF is given by

BE = av A a s A2 / 3 ac

z (z 1)A1 / 3 aa

(A 2Z )2

A+ a p

1A1 / 2

Shell Model: The SEMF is good for smooth trends in BE but some valuesof N, Z are found to have particularly high BE , these are called the magicnumbers. For N or Z or both = 2 , 8, 20, 28, 50, 82, 126... it is observed thatthese nuclei are particularly stable. These magic numbers correspond to a lledenergy level, analogous to the inert properties of the noble gases. The form of V (r ) required to solve the TISE to nd what number corresponds to a lledlevel is complicated.

Spin orbit coupling splits the 2 j + 1 degeneracy in j = l 12 states giving2l + 2 states for j = l +

12 and 2l for j = l

12 . These have different energies as BE l s , so for j = l + 12 , BE l, and for j = l 12 BE = (l + 1) .This predicts the experimentally determined magic numbers.

8.3 Decays

Gamma Decay: Due to EM forces and thus cannot change quark avor, onlyfor energy level decay from nuclear excited state to ground state.

X N Z X N Z + n Beta Decay: As this is due to the weak force quark avor can be changedand thus there are more possible decays. There are 3 process by which betadecay can make a nucleus more stable.

: n p + e + e + : p n + e+ + eEC : p + e n + e

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For odd A there is a single value with the minimum mass (maximum BE ),a given unstable nucleus will decay towards this with , + and EC . If thechange in mass is too small then there are some decays which lower Z whichcan only be achieved by EC and not by + . For even A then a loss of a p/nand thus a gain of a n/p will swap between N, Z being odd and even for eachdecay thus there are two curves one for even Z and one for odd Z . The lowestmass isotope on the odd Z curve can have multiple decay paths avaliable to itby having more than 1 isotope with less mass than it on the even Z curve. N, Z changes for the decays are

: X N Z X N 1Z +1 + e + e + : X N Z X N +1Z 1 + e

+ + eEC : X N Z + e

X N +1Z

1 + e

Alpha Decay: The emisson of a alpha particle from the nucleus to increasestability. To be possible mnuc > m this is at about A 150, for a decent m require A > 200. As A = 4 likely to be a decay chain until stability, asA decreases ratio of N Z 1 for stability so decay chains often also include emission. Consider the alpha particle as a separate entity within the nucleus andform a potential out of the coulomb repulsion and the Saxon-Woods residualstrong force attraction. The particle has to tunnel through this potential toescape the nucleus, we can derive a formula which gives good agreement withdata as to decay times. N, Z changes for the decay are

X N Z X N 2Z 2 + 8.4 Nuclear Reactions

Fission: For a ssion reaction to be possible it is required that Q = BE products BE reactant > 0. Q can be calculated from the SEMF. From the SEMF we seethat for ssion to be possible Z

2

A > 0.7a sa c 18. During a ssion reaction the

nucleus will deform to eject the products, this deformation changes the BE sofor a nucleus to undergo spontaneous ssion the deformation must increase theBE . Changing from a sphere to an ellipsoid decreases the binding energy as-sociated with the surface tension and increases the Coulomb term. It can bederived that for a nucleus to undergo spontaneous ssion Z

2

A >2a sa c 51. Nuclei

for which this is true are very unstable and decay with the speed of a strongforce interaction, nuclei with 51 > Z

2

A > 18, ssion is possible but the nucleus

must be given energy. This can be given by s but the cross section for thesereactions is very small, it can also be done by injecting extra nucleons. Neutronsare the most suited as there is no Coulomb barrier for them to overcome.

Chain Reactions: Fission reactions often produce extra ns, these can beabsorbed by other nuclei to speed up their ssion reactions. the parameter m

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is used to characterise a possible chain reaction, m is the average number of neutrons produced per reactions taking into account unwanted reactions andneutrons escaping the reactive material without interacting. for m > 1 the totalreaction rate will increase - super-critical, m = 1 the reaction rate will stayconstant - critical, m < 0 the reaction rate will decrease - sub-critical.

Fusion: Similar to decay in reverse, for reactions bar addition of a neutron,reactants both have a net positive charge and thus there is a Coulomb barrier toovercome. This barrier can be overcome either by tunnelling or by the reactantshaving enough KE to overcome the potential. The product is a single nucleusin an excited state, this decays to the product we see from a fusion reaction.The energy release associated with the change in binding energy is carrier awayby the products or s.

9 Probability And Statistics9.1 Properties of distributions

Normalisation: A distribution is normalised if i P i = 1 for discrete P i or

p (x) dx = 1 for continuous p (x).Expectation Value: For a function on x, f (x) the expectation value of thatfunction under a distribution P i is given by f (x) = i f (x i ) P i for the discretecase and f (x) = f (x) p(x) dx. Thus the mean x = i x i P i | xp (x) dx.Variance: var x = (x )

2 = x2 x2 , the standard deviation =

var xCentral Limit Theorem: For a large N , the sum X = N j =1 x j is normallydistributed with X = N x and var X = N var x.

9.2 Binomial Distribution

P (r |N, P ) =N !

r ! (N r )!P r (1 P )

N r

Expectation Value

E (x) =

n

x =0 xP (x) =

n

x =1 xn!

x! (n x)!P x

(1 P )n

x

= nP n

x =1x

(n 1)!(x 1)![(n 1) (x 1)]!

P x1 (1 P )(n 1)(x1)

By the normalisation of the binomial distribution

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= nP n

x =0

n!x! (n x)!

P x (1 P )n x

E (x) = nP

Variance

2 = E x2 E (x)2 = E (x (x 1) + x) E (x)

2

By effectively carrying out the process above twice we arrive at

= nP n

x =0

n!x! (n

x)!

P x (1

P )n x

E (x) = nP

E (x (x 1)) =n

x =0x (x 1) P (n) = n (n 1)

2 = n (n 1) P 2 + nP (nP )2

2 = nP (1 P )

9.3 Poisson Distribution

P (n) =(t )n

n!et

Expectation Value

xd

dxex = xex =

k

kxk

k!

E (n) =

n =0n

n

n!e = e e

E (n) =

Variance

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x2 ddx

ex = x2ex =k

k (k 1) xk

k!

k

k2xk

k!= x2ex + xex

E n2 =

n =0n2

n

n!e = 2 +

2 =

9.4 Gaussian Distribution

G(x) =1

2 exp 12

x x

2

Expectation ValueStandard Integral

E (G (x)) =

xea (xb)2

dx = b a=

x

1 2 exp

12

x x

2

dx =1

2 x 2 2 = x

VarianceV ar (G (x)) = 2

Multiplication of Gaussian

G (x) =N

i =1

G (x) =N

i =1

1 2 exp

12

(x x i )2

2

= G0 exp 12

N

i =0

(x x i )2

2= G0 exp

122

Nx 2 2N

i=0xx i +

N

i =0x2i

= G0 exp N

22 x2 2xx +1N

N

i =0x2i

Taking x = N x as the mean of the separate gaussian parameters and absorb-ing everything not x dependent into the constant

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G (x) = G0exp 12

(x

x)2

2 /N

The standard deviation of this new Gaussian is = N .

9.5 Propagation of Errors

If we have a function, y (x), where we know x = x + x , what is error on y ? Bytaylor expanding,

E (y) = E y (x) +yx x = x

(x x) = y (x) +yx x = x

E (x x) = y (x)

V ar (y) = E [y (x) E (y (x))]2

= E yx x = x (x x)

2

2yyx x = x

2

E [x x]2 =

yx x = x

2

2x

In general

x = {x1 , x2 ...x n }, y (x)2y =

i

2x iyx i x = x

2

Lineary = ax 1 + bx2

y = 2x 1 + 2x 2Product, Ratio

y = x1 x2y =

x1x2

yy

=

2x 1x2

1

+2x 2x2

2

36

Core Notes for Comps (Mike Nielsen, Luke Pickering)

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