Missed questions (pGRE)

cyclotron motion, velocity and applied field, velocity MULTIPLE components —> path of particle?

helix

rope swinging in circle/half circle, with mass at end —> tension?

conservation of energy —> velocity —> centripetal force

hydrogen atom transitions n1 —> n2, wavelength of photon?

Bohr formula (E = -E0/n², E0=13.6eV) —> deltaE of photon —> then photon relation E = hc/lambda

hermitian observables —> operators must be —

hermitian

net force on object is zero —> ?

acceleration is zero

ions (+) or (-) —> electronic configuration?

remember to add or subtract from neutral version

LC circuit —> time to peak magnitude of current

use resonant freq w = 1/rt(LC) —> assume initially charged —> Q = Qosin(wt) —> derivative —> find time to max I

PV diagram for engine —> amt of work done?

take area of curve, handedness matters

during adiabatic process, PV^y = — ? y = ?

const, Cp/Cv

adiabatic process → given P,V values → how to find out mono-, di- etc

PV^y, ideally where V = 1, solve for y

monoatomic gas, y = -; diatomic gass, y = -

5/3; 7/5

splitting of spectral lines due to electric field perturbation?

Stark effect

given a type of image and magnification and that it’s a mirror → concavity and radius?

m = -s’/s → use optics equation 1/f = 1/s + 1/s’ to find focal pt → R = 2f in general

particle in electric field → travels to different point → find potential energy

pot energy U = qV —> change E to V using line integral

given relativistic particle → asked about energy, momentum, rest mass, etc

E² = p²c² + m²c^4 → go from there

Velocity ratio at apogee vs perigee in an elliptical orbit (given distances from focus)

r_pv_p = r_av_a

given perturbation and unperturbed eigenfunctions —> new eigenenergies/functions

1st order perturbation theory

charges placed on infinite plane and asked for forces or fields

method of images → pay attention to lengths and interactions between charges on the same side as well

blackbody → asking for T or lambda

use Wien’s: T = 3×10^-3 * lambda

redshift

Doppler shift approx → swap signs if blueshift

if hamiltonian commutes with operator

share eigenfunctions

rotating platform with an initial frame and final frame → finding any observable from that?

use conservation of angular momentum L = Iw, Li = Lf

Compton scattering — photon hits electron → wavelength shift

A photon scatters off a (usually free) electron → wavelength increases (energy decreases): Δλ=λ′−λ=(h/mc)​(1−cosθ) -→ also can take limiting theta → 0, should be no change then, because the photon is continuing with no deflection

binding energy of any ion?

E_bind = 13.6(Z/n)²

Energy conservation in thermodynamics

deltaU = Q - W;

Q and W, and deltaU values for an isolated system

Q = 0, W = 0 , deltaU = 0

Q and W, and deltaU values for an cyclic system

Q = W, deltaU = 0

Q and W, and deltaU values for an adiabatic system

Q = 0 → deltaU = -W

what is deltaS for a perfectly reversible process

0

deltaS of the universe must always

be greater than or equal to 0

behavior of system as temperature reaches absolute zero?

S →0 as T→0 given a perfect crystal which represents a single microstate; real substances may have residual entropy

efficiency of heat engine

e = W_out/Qh = 1 - Qc/Qh

Maximum possible efficiency between two temperatures

e = 1 - Tc/Th​​​ (carnot eff) → only possible for reversible engines

coefficient of performance

refridgerator: Qc/W; heat pump: Qh/W

max cop for ideal fridge

Tc/(Th-Tc); but slow process at this cop

reversible adiabatic process, what happens to entropy

deltaS = 0 (isentropic

irreversible adiabatic, what happens to entropy

deltaS > 0; i.e. free expansion

expectation value of observable

if the state is an eigenstate and asked to find expectation value

asked for expectation value → if the integrand is odd over symmetric limits

<A> = 0

expectation value Lx, Ly, Lz given spherical harmonic eigenfunctions

0; 0; mh_bar

expectation value L² given spherical harmonic eigenfunctions

l(l+1)h_bar²

expectation value Sx, Sy, Sz given spin-1/2 system

0, 0, ±h_bar/2

if operator acting on given wavefunction yields imaginary num → expectation value?

0

derivative form of momentum operator

-ih_bar d/dx

derivative form of Lz operator

double slit relation

dsintheta = mlambda

given a spinor, how to find the expectation values of Sx, Sy, or Sz

pauli matrix forms

for any normalized spinor that is spin-1/2, <S²> is

3/4(h_bar)²

total angular momentum J

J = L + S

For any angular momentum A=(Ax,Ay,Az), the commutation relations with each other are:

commutation relation for any angular momentum operator A², A_i

A² and A_i can be measured together

Bernoulli equation given two points

continuity equation (fluids) given two points

v1A1 = v2A2, v → fluid velocity

calculating probability of spherically symmetric wavefunction

(remove angular part if assumed to be normalized

Larmor formula describes power radiated by dipole in oscillating field

a is accel of charge

time average of cos²(wt) or sin²(wt) = ?

1/2

velocity of gas particles following the Maxwell distribution for ideal gases

v = rt(2kT/m)

Doppler shift for sound waves

the speed of a photon in any inertial reference frame is

c

mesons are the bound states of ?

a quark and an antiquarkt

total energy stored in a capacitor

E = 1/2CV² = Qo²/2C

given heat capacity and change in temp of a system, what is the change in heat?

deltaQ = mCpdeltaT

Find degeneracy of ground state from spin Hamiltonian

in case of heisenberg coupling, rewrite using total spin, for two spins, s = |s1 - s2| → pick s that minimizes E ~ s(s+1) → degen = 2s + 1

How do you determine degeneracy quickly?

degen = # of states related by sym that leaves H unchanged

degen of spherically sym Ham

if [H, J] = 0, degen = 2j + 1

hydrogen atom degen

degen = n²

effect of magnetic field on degen

degen in m is lifted

degen of two identical spins (i.e. two spin-1/2)

principal decay mode rule for hydrogen

delta(l) = ± 1, and note that decay means going down

radioactive decays follow which distribution

poisson, variance N

fractional error of poisson dist

sigma/N = rt(N)/N

rayleigh criterion

sin(theta) = 1.22lambda/D, D = diameter of observation

time constant of RC circuit

RC

self inductance formula

LI = fluxe

free electron laser

synchrotron radiation due to sinusoidal path of electrons; does not require electronic transitions in atoms

scaling for radio waves

mm to kmscal

scaling for visible light

400-700 nm

scaling of x-rays

0.01 to 10 nm

Euler lagrange equations are valid for systems with —>

time dependent potentials, with or without rotational symmetry, or acted on by conservative forces

order of magnitude of difference kind of fine structure corrections

hyperfine < Lamb shift < fine-structure

hyperfine structure

interaction of nuclear dipole moment with magnetic field from orbit of electrons

fine structure

from spin-orbit coupling, relativistic effects, and Darwin term

Lamb shift

interactions between e and vacuum that cause shift in s and p orbital energies for n = 2

when asked for the uncertainty of a given function

(deltaf(x,y))² = (df/dx)²(deltax)² + (df/dy)²(deltay)²

bandpass filter only allows signals to propagate that are

between two frequencies

inductors

suppress high freq signals

capacitors

suppress low frequency signals

resistors

restrain signals from “full strength”

time dilation (in terms of gamma)

t_new = gamma*t0, where t0 is in the moving frame

length contraction

L = L0/gamma, L0 measured in moving frame

when asked for total orbital angular momentum L

calculate L² if possible and take the sqrt

resonant frequency of something like an LC or RLC circuit is defined as

the frequency where the imaginary part of the total impedance vanishes

given an open/closed pipe question

take lambda to be something in terms of the length of the pipe (think distance between nodes). if asked for frequency, use f = c/lambda

when asked for torque produced by magnetic field

N = m x B

acceleration due to centrifugal force

Omega²R

potential in a conductor is

constant

potential inside a dielectric sphere that is solid and uniform in density

~ r

cyclotron radius

mv/qB