1. Classical Mechanics - 20%

v2 = v20 + 2a(x− x0) v in terms of x in uniform acceleration

L = L(q, q, t) = T − U Lagrangianddt

(∂L∂q

)= ∂L

∂qEL eqs of motion

S =∫ t1t0L(q(t), q(t), t)dt Action, minimize this integral

p = ∂L∂q

generalized/conjugate momentum

F = ∂L∂q

generalized force

H = pq − L = T + U Hamiltonianq = ∂H

∂pp = −∂H

∂qH eqs of motion

p+ ρgy + 12ρv2 = constant Bernoulli’s eq

Fcoriolis = 2mr×Ω Coriolis forceFcentrifugal = −mω × (ω × r) Centrifugal forceτ = r× F torqueL = r× p angular momentumI =

∑imir

2i =

∫r2dm =

∫r2ρ dV moment of inertia

ω =√

mgrI

ω =√

g`

frequency of pendulum

τ 2 = 4π2

GMsa3 Kepler’s third law

where a is dist from center to aphelionI = Icm +md2 parallel axis theorem(

cos θ − sin θsin θ cos θ

)2D rotation matrix

µ = m1m2

m1+m2reduced mass

Ueff(r) = U(r) + `2

2µr22 body effective potential (1D)

ω =√

km

frequency of harmonic oscillator

v = ω × r v in terms of omega and r (in vectors)Fb = ρfV g bouyant forcefbeat = |f1 − f2| beats per second for 2 superposed wavesNo energy conserved in inelastic collision?

cs =√γ Pρ

γmono = 5/3 γdia = 7/5 speed of sound

γP = B = −∆p/(∆V/V ) definition of bulk modulusf = cs

λfrequency of sound

cs =√

γkTmper particle

= 343 m/s for air speed of sound of ideal gas (and for air)

cs =√γgh+ γ(atm)

ρspeed of sound in water

A1v1 = A2v2 pipe flow (non-viscous)

v =√

Tµ

speed of waves on a string

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R = Dvν

= ρDvη

Reynolds number

where D=char length, v=char speedη =dynamic viscosity, ν = kinematic viscosityR < 1 laminar, R > 1 turbulentdPdz

= −ρg hydrostatic equilibriump/p0 = ρ/ρ0 = e−z/H H = p0

gρ0= 8.55 km pressure vs height

M d~vdt

=∑Fext + ~vrel

dMdt

rocket equationv2 = 2GM

Rescape velocity

F = ηAdvdy

viscous force

λn = 2Ln

n = 1, 2, 3 . . . standing waves in open tube (fixed string)λn = 4L

nn = 1, 3, 5 . . . standing waves in closed tube (one end fixed

string)P = IV = Fv power definitionsLtotal = Lglobal + Lintrinsic total angular momentum of a systemstable non-circular orbits exist only in simple har-monic and inverse-square-force potentials

Bertrand’s theorem

ω =√w2

0 − t2

4m2 frequency of underdamped SHO

Moments of inertia:

Iz = mr2

2solid disc or cylinder

I = 2mr2

5solid sphere

I = 2mr2

3hollow sphere

Icenter = mL2

12rod about center

Icenter = mL2

3rod about end

kepler orbits?normal modes?

2. E&M - 18%

ε = “permittivity” µ = “permeability”ε = ε0(1 + χe) electric susceptibilityµ = µ0(1 + χm) magnetic susceptibilityε = ε0εr dielectric constant/relative permittivity

n =√

εµε0µ0

= cvp

=√εrµr index of refraction

c = 1√ε0µ0

speed of light

v = 1√εµ

v in terms of EM stuff

Typical values:

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ε > ε0 almost always, with χe > 0 permittivityµ < µ0, χm < 0 diamagneticsfield diesµ > µ0, χm > 0 paramagneticsµ >> µ0, χm >> 0 ferromagnetics∫

(∇ ·A)dτ =∮

A · da divergence theorem∫(∇×A) · da =

∮A · dl curl theorem

Maxwell Eqs:∇ · E = 1

ε0ρ Gauss’s Law

∇× E = −∂B∂t

Faraday’s Law∇ ·B = 0 No monopoles∇×B = µ0J + µ0ε0

∂E∂t

Ampere’s Law

∇ ·D = ρfree Gauss’s Law in matter∇×H = Jfree + ∂D

∂tAmpere’s Law in matter

D = ε0E + P definition of electric displacement DH = 1

µ0B−M definition of auxiliary field H

In linear media:P = ε0χeE polarization fieldD = εE displacement fieldM = χmH magnetization fieldH = 1

µB auxiliary field

V = N dΦdt

Induced voltage from Faraday’s LawF = 1

4πε0

q1q2r2

Coulomb’s Law

dB = µ0

4πIdL×rr2

Biot-Savart Law

B = µ0I2

r2

(r2+z2)3/2Magnetic field on axis of a circle of current

B = µ0I2πR

Magnetic field from infinite straight wiredF = I d`×B Force on a wire from a magnetic field

Boundary conditions:E⊥above − E⊥below = 1

ε0σ BCs for E

E‖above = E

‖below

D⊥above −D⊥below = σf BCs for D

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D‖above −D

‖below = P

‖above −P

‖below

B⊥above = B⊥below BCs for B

B‖above −B

‖below = µ0(K× n)

H⊥above −H⊥below = −(M⊥above −M⊥

below) BCs for H

H‖above −H

‖below = Kf × n

Q = CV U = 12CV 2 Capacitance and energy

C = εAd

Parallel plate capacitorV = Vb − Va = −LdI

dtU = 1

2LI2 Inductance and energy

Φ = LI inductance

L = µN2Al

Inductance of a solenoidZ = R Z = 1

iωCZ = iωL impedance for resistor, capacitor, inductor

ω = qBm

Cyclotron frequencycharged particle passes through a medium fasterthan v = c/n and emits blue light in nuclear re-actors

Cherenkov radiation

P = q2a2

6πε0c3Larmor formula for radiated power

ρ = EJ

resistivity

R = ρLA

resistance, L = length, A = areaJ = σE conductivity10−4 m/s electron drift velocityJ = I

A= σE = nevdrift equation for drift velocity

E‖ = 0 outsiede, ~E reversed upon reflection reflection off a conductor

E ∝ −µ0p0ω2

4π( sin θ

r)θ E field of oscillating electric dipole

B ∝ −µ0p0ω2

4π( sin θ

r)φ B field of oscillating electric dipole

E ∝ µ0m0ω2

4πc( sin θ

r)φ E field of oscillating magnetic dipole

B ∝ −µ0m0ω2

4πc( sin θ

r)θ B field of oscillating magnetic dipole

p = qd electric dipole momentm = IA magnetic dipole momentσ = −ε0

∂V∂n

surface charge on conductor vs VZload = Z∗source impedance matchingσb = P · n bound surface chargeρb = −∇ ·P bound volume chargeKb = M× n bound surface currentJb = ∇×M bound volume currentused to measure exponent in Coulomb law placeda charge inside a hollow charged sphere measuredhow far it moved (not much)

Cavendish-Maxwell experiment

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τ = µ×B torque on a dipoleU = −µ ·B dipole potential energyB = µ0µ

2πz3magnetic dipole field

L = µ0N2h2π

ln ba

inductance of a toroidE = 1

4πε0

px3 electric dipole field (approx)

E = 14πε0

p(x2+(d/2)2)3/2

electric dipole field (exact)

current flows in metal in perp B field, charge accu-mulates on one side, magnetic force balances newE field, determines sign of charge carriers

Hall effect

S = 1µ0

(E×B) Poynting vector

gauges?EM field transformations ?cross product identities?

3. Optics and Wave Phenomena - 9%

vphase = ωk

vgroup = dωdk

Phase velocity versus group velocitygroup velocity information travels at...

ω = ω01±vobs/cn

1∓vsource/cnDoppler shift (for sound waves in a medium)

Top signs moving towards each other1o

+ 1i

= 1f

Thin lens formula

sin θ = ∆lf≈ 1.22λ

dRayleigh Criterion (aperture)

f = R2

1f

= (n− 1)(

1R1

+ 1R2

)Focal length of mirror and lens

I0 cos2(θ − φ) Intensity of polarized light through polarizerym = mλD

ddouble slit interference, position of maxima

d sin θ = mλ m = 0,±1,±2, . . . same equation in terms of θw sin θ = mλ m = ±1,±2, . . . single slit diffraction, position of minimaIθ = 4I0 cos2(πd sin θ

λ) Iθ for 2 slit interference

Iθ = Imax( sinαα

)2 Iθ for 1 slit diffractionwhere α = πa

λsin θ

product of the above 2 slit interference and diffractionreflected wave same phase wave at boundary with n1 > n2

reflected wave 180 phase shift wave at boundary with n1 < n2

λ = vν

wavelengthobject side image sidetan θB = n2

n1Brewster’s angle

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θB measured from vertical to incident raycomponent perp to surface is eliminated2d sin θ = nλ condition for constructive interference for Bragg

diffractiond is distance between latticesθ is angle from horizontal

mθ = θeyθob

= −fobfey

magnification for telescope

λn = λn

wavelength in materialreference wave interferes with wave reflected offobject, stores phase and amplitude info

hologram

path diff = 2d+ 12λn for each phase shift thin films interference

sin θ = λd

diffraction, angle to edge of central maxminimize blur = d+ dscreen using above eqn pinhole camera sharpness

4. Thermodynamics and Statistical Mechanics - 10%

e = WQinput

definition of efficiency

1− Tc/Th efficiency of Carnot cycle∆U = ∆S = 0 Carnot cycleadiabat steeper than isotherm on PV plotrectangle on US plotZ =

∑s e−Es/kT P (s) = e−Es/kT/Z Boltzmann distribution

U , N , and V entropy is a function of...S = kB ln Ω Approximate definition of entropyS = −kB

∑s ps ln(ps) Exact definition of entropy

∆S = Nk lnVfVi

change in entropy when U and N are fixed

Q =∫TdS relation between entropy and heat

12kT per quad dof Equipartition theoremλmaxT ≈ 3× 10−3m ·K Wien’s Law (for a blackbody)j∗ = σT 4 power per unit area radiated by blackbodyσ = 5.67× 10−8 Stefan-Boltzmann constantni = gi

e(εi−µ)/kT+1Fermi-Dirac stat, number fermions in state i

ni = gie(εi−µ)/kT−1

Bose-Einstein stat, number bosons in state i

g degeneracy, ε energy,µ chem potential ∂U

∂N

PV = NkT ideal gas lawk = 1.381× 10−23J/K Boltzmann’s constant3 dofs for monatomic ideal gas

6

3 at > 10K dofs for cold diatomic ideal gas5 at > 100K dofs for diatomic ideal gas7 at > 1000K dofs for hot diatomic ideal gas

vmax =√

2kTm

vmax for an ideal gas

v =√

8kTπm

v for an ideal gas

vrms =√

3kTm

vrms for an ideal gas

Uthermal = N · f · 12kT total thermal energy of ideal gas

∆U = Q+Won sys 1st law of thermodWon sys = −PdV work in quasistatic compressionQ = 0 adiabaticT = const isothermalW = NkT ln Vi

Vfwork in isothermal compression of ideal gas

V Tf2 = const V

f+2f P = const relations for adiabatic compression

t1/2 = ln 2k

half lifewhere k = decay ratenumber of indep events in some time Poisson distribution√N uncertainty of N-count P distribution

σ =√< (x− x)2 > standard deviation

aka RMS deviation from meanσ2 varianceC = heat needed

degree temp increaseheat capacity

c = C/m specific heat

CV = (∂U∂T

)V = Nfk2

heat capacity at constant volumeCP = (∂U

∂T)P + P (∂V

∂T)P = CV +Nk heat capacity at constant pressure

A = Pρw

scattering cross section

where P is probability and w is width∆L = αL∆T linear expansionenergy of highest QM state at T = absolute zero Fermi energy∆Q∆t

= −kA∆T∆x

Fourier’s law (heat conduction)

5. Quantum Mechanics - 12%

Time-Independent Perturbation TheoryE1n = 〈Ψ0

n|H ′|Ψ0n〉 first order correction to energy

|Ψ1n〉 =

∑m 6=n

〈Ψ0m|H′|Ψ0

n〉E0n−E0

m|Ψ0

m〉 first order correction to wavefn

E2n =

∑m6=n

|〈Ψ0m|H′|Ψ0

n〉|2E0n−E0

msecond order correction to energy

degeneracies?

7

i~∂Ψ∂t

= HΨ Ψ(t) = e−iHt/~Ψ(0) time dependent SEHΨ = EΨ Ψ(t) = e−iEt/~Ψ(0) time independent SE

H = − ~2

2m∇2 + V Hamiltonian

p = −i~∇ momentum operator[x, p] = i~ commutator of x and pσAσB ≥ | 1

2i〈[A,B]〉| generalized uncertainty principle

σxσp ≥ ~/2 Heisenberg uncertainty principleλ = h

p= 2π~

pde Broglie Wavelength

E = hc/λ energy in terms of wavelength|1, 1〉 =↑↑ triplet states (j = 1)|1, 0〉 = 1√

2(↑↓ + ↓↑)

|1,−1〉 =↓↓|0, 0〉 = 1√

2(↑↓ − ↓↑) singlet state (j = 0)

Pauli spin matrices:(0 11 0

)σx(

0 −ii 0

)σy(

1 00 −1

)σz

[AB,C] = A[B,C] + [A,C]B identity for [AB,C]P =

∫|Ψ|2dV =

∫Prdr how P relates to Pr

l = 0 wavefunctions with spherical symmetry?in scattering: incoming plane wave not substan-tially altered by potential

Born approximation

ψ(r1, r2) = A(ψa(r1)ψb(r2) + ψb(r1)ψa(r2)) bosonssym, do not obey Pauli exclusionPboson ∝ T 5/2

ψ(r1, r2) = A(ψa(r1)ψb(r2)− ψb(r1)ψa(r2)) fermionsantisym, obey Pauli exclusion

E = n2π2~2

2ma2 energy levels in square well

ψn(x) =√

2a

sin(nπax) wavefunctions in square well

p = ~k p in terms of kψ0 = 1√

πa3e−r/a0 Hydrogen ground state wavefunction

H ′ = eEextz perturbed H for Stark effect

Might not need:d〈Q〉dt

= i~〈[H,Q]〉+ 〈∂Q

∂t〉

[Li, Lj] = i~Lkεijk angular momentum commutators

6. Atomic Physics - 10%

8

Electron quantum numbersn = 1, 2, 3, . . . principle quantum number, radial wavefn` = 0, 1, . . . , n− 1 orbital quantum number, θ wavefnL2ψ = ~2`(`+ 1)ψm` = −`,−`+ 1, . . . , `− 1, ` magnetic quantum number, φ wavefnLzψ = ~m`ψs = (1/2) electron spinS2|ψ〉 = ~2s(s+ 1)|ψ〉ms = ±(1/2) z-component of electron spinSz|ψ〉 = ~ms|ψ〉j = |`− s|, . . . , (`+ s) = |`± (1/2)| total ang momentum quantum numberJ2|ψ〉 = ~2j(j + 1)|ψ〉mj = −j,−j + 1, . . . , j − 1, j z-component of total angular momentummj = m` +ms Jz|ψ〉 = ~mj|ψ〉∆` = ±1 ( 6= 0) ∆m` = 0,±1 electric dipole transition selection rules∆j = 0,±1 ∆ms = 0Wigner=Eckart theorem governs theseenergy level splitting when ext B field applied Zeeman effectelectrical analog of Zeeman effect Stark effect

an = 4πε0~2

µe2Zn2 ≈ (0.529A)

(meµ· n2

Z

)Bohr model radius

En = Z2µe4

8ε20h2

(−1n2

)≈ (−13.6 eV)

(Z2

n2 · µme

)Bohr model energy

Eγ = −13.6eV( 1n2i− 1

n2f) energy of radiated photon from transition

1λ

= R( 1n2f− 1

n2i) wavelength of radiated photon

R = 1.097× 107m−1 Rydberg constant

x ray spectrum from electrons fired at atoms:photon knocks our inner e−, replaced by outer e− Auger transitioncontinuous spectrum, radiation produced by de-celeration of a charged particle

Bremsstrahlung

K, L, M, N → nf = 1, 2, 3, 4 x ray series namespick L (S,P,D,F,G,H,... for L = 0, 1, 2, . . .) term symbolwhere J is between L+ S and |L− S|write spin multiplicity = 2S+1LJµ = γS = (−eg/2me)S intrinsic magnetic moment

9

g = 2 γe = 1.76× 1011 gyromagnetic ration for an electronphotons hitting metal knocks off electrons, lightshines on emitter, sending electrons to a collector,making a current, which stops at some negativevoltage (at V=0 some electrons still collected)

Photoelectric effect

dominant process up to 500 KeVminimum energy needed to free electron from ma-terial

work function

photon scatters off electron, changes wavelength,electron flies off

Compton scattering

∆λ = hmc

(1− cosφ) Compton effect equationφ is angle between scattered photon and horizon-talconfirmed Bohr’s quantized model, atoms couldonly absorb specific amounts of energy

Franck-Hertz experiment

dσdΩ

= ( kqQ2mv20

)2 1sin4(θ/2)

Rutherford scattering

σ incident ring, Ω scattered solid angleshot α particles at gold foil, found concentratednucleus

σ = ( q2

4πε0mc2)2 Thomson scattering

σ is the cross section (area/solid angle)photons scattering off electronsphoton destroyed, e+ e− created, pair productionbecomes significant at 10 MeVNsc = Nincntarσ classical scattering relationNsc(into dΩ) = Nincntar

dσdΩ

(θ, φ)dΩ diff scat cross (experiment)dσdΩ

= bsin θ| dbdθ| diff scat cross (theory)

σ is cross section area of one targetΩ = A/r2 definition of solid angledΩ = sin θdθdφgamma ray photons from nuclear transitionx ray photons from electron transitionnucleus interacts with lower shell electron, eject-ing it, higher shell electron fills hole, x ray whenheavy nucleus, visible when H atom

internal conversion

λ = 10−10 ν = 1018 x rayλ = 10−12 ν = 1020 gamma ray

orbitals: go diagonal top right to bottom left

10

quantum numbers: row denotes n, col denotes lnumber of electrons a shell can hold determinedby ms and ml

s p d f g1 1st2 2nd 3rd3 4th 5th 7th4 6th 8th 10th 13th5 9th 11th 14th 17th 21st

Might not need:anomalous Zeeman effect?hyperfine splitting?

Enj = −13.6eVn2 [1 + α2

n2 ( nj+1/2

− 34)] energy levels of H w/ fine structure

7. Special Relativity - 6%

γ = 1√1−v2/c2

≥ 1 Lorentz factor

γ ≈ 1 + 12v2

c21γ≈ 1− 1

2v2

c2

ct′ = γ (ct− βx) x′ = γ (x− β(ct)) Lorentz transformations

L′ = Lproper

γlength contraction

T ′ = (Tproper)γ time dilation

ω = ω0

√1∓v/c1±v/c doppler shift for light

p = γmv relativistic momentum

E =√

(mc2)2 + (pc)2 = γmc2 relativistic energyblack hole if R = 2GM

c2Schwarzchild radius

ω = ω0

γ(1−β cos θ)general doppler shift

vAC = vAB+vBC1+(vABvBC/c2)

velocity addition

C is lab frame

11

8. Laboratory Methods - 6%

∆f = (∑n

i=1[(∆xi)(∂if(x1, . . . , xn))]2)1/2

error propagation∆ff

=√∑

(∆xixi

)2 error propagation if no cross-corellated terms

1 J ≈ 6× 1018 eV Joules to eV conversion(kB)(300 K) ≈ .02 eV room temp in eVλ = 750− 380 nm ν = 400− 790× 1012 Hz visible light in m, Hz, eVE = 2− 3 eV10−6m/s drift velocity of electrons in metalfunctions like y = ax straight lines on semilog plotfunctions like y = xa straight lines on log log plotτc = RC ≈ 63% RC time constant

9. Specialized Topics - 9%

QM version of normal modes, vibration in lattice phononcollection of Fermions, QM ideal gas Fermi gas3 quarks (such as p, n), hadron baryoneach quark = 1/3 baryon numberνe, e, νµ, µ, ντ , τ leptonse, µ, τ each have own number lepton numberlepton and corresp ν have +1, anti have -12 quarks (such as π+ pion, K− kaon), hadron mesonEbind/c

2 = (total mass of nucleons) −(mass of nucleus)

binding energy

defined as positive, corresponds to negative po-tential energyiron element with highest binding energy per nucleon2 protons, 2 neutrons α particlen→ p+ e− + νe β− decayp→ n+ e+ + νe β+ decayN-doped, P-doped semicond with more electrons, holesρ = ρ0(1 + α∆T ) resistivity as a fn of ∆Tα > 0 α for conductorsα < 0 α for semiconductorsquarks, leptons weak force affectsquarks, gluons, hadrons strong force affectscrystal lattice looks the same from every point Bravais lattice

12

can cover entire lattice, contains 1 point primitive unit cellsum density of each point (1/number sharedblocks)

calculate lattice point density

2 body centered point density4 face centered point densityproton and neutron, boson deuteronNo weak interactions preserve strangeness?

10. RandomdAdt

= −kA⇔ A = A0e−kt

sin θ ≈ tan θ when θ is small

1.414√

2

1.732√

3sin 2θ = 2 sin θ cos θ trig identitiescos 2θ = 2 cos2 θ − 1

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