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0
Q

Ground state hydrogen energy

A

-E_1=ł^2 / (2 m a^2)=k^2 m e^4 / (2ł^2)

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1
Q

Bohr radius

A

a=ł^2 / (k m e^2)

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2
Q

Electric constant k=1/(4 pi e_0)

A

10^10 Н м^2/Кл

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3
Q

How does the ground state energy of a hydrogen-like atom depend on Z?

A

E_1 ~ Z^2

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4
Q

1 Henri in terms of seconds

A

H=V s^2 / C=В с^2 / Кл

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5
Q

1 Ohm in terms of seconds

A

Ohm = V s / C = В с / Кл

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6
Q

1 Farade in terms of other units

A

F = C / V = Кл / В

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7
Q

h c = ? eV nm

A

1240

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8
Q

ł c = ? eV nm

A

200

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9
Q

k_b * 300 K = ? eV

A

0,026 = 1/40

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10
Q

Mass of a proton

A

938 MeV/c^2

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11
Q

Atomic mass unit (carbon/12)

A

931 MeV/c^2

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12
Q

Rydberg constant

A

R = -E_1/hc = 1.1*10^7 m^(-1)

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13
Q

Rydberg formula for hydrogen emission

A

1/lambda = R ( 1/m^2 - 1/n^2 )

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14
Q

Fine structure constant

A

alpha = k e^2 / (łc) = 1/137

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15
Q

Second order perturbation correction to energy levels

A

E_n^1 = E_n + + Sum ||^2 / (E_n - E_m)

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16
Q

Wien displacement law

A

Lambda_max = 3 / T (mm)

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17
Q

Lorentz transformation matrix from x to x’

A

Gamma -gamma beta 0 0
-gamma beta Gamma 0 0
0 0 1 0
0 0 0 1

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18
Q

Energy-momentum vector

A

(E/c, p)

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19
Q

Charge-current vector

A

(c rho, j)

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20
Q

Wave four-vector

A

(omega, c k)

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21
Q

Square of the energy-momentum vector = ?. Invariant? Conserved?

A

m^2 c^2 in any reference frames. Invariant and conserved.

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22
Q

Relativistic energy in terms of the rest energy

A

E = gamma m c^2

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23
Q

Relativistic momentum in a non-rest frame

A

p = gamma m v

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24
Q

Doppler frequency shift in classical waves

A

(v + v_rec) / (v - v_source) where v is the speed of waves

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25
Q

Relativistic Doppler frequency shift

A

Sqrt(1+beta/1-beta)

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26
Q

Compton wavelength shift

A

D_lamda=h/mc (1-cosO)

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27
Q

Table of quarks

A

u c t. 2/3
d s b. -1/3
e mu tau
ne nmu ntau

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28
Q

Proton and neutron quark formulas

A

p=uud

n=udd

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29
Q

Fermi energy

A 
k_f = (3 pi^2 n)^1/3
E_f = ł^2 k^2 / 2m
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30
Q

Density of states

A

g(E) = (V Sqrt(2) / (pi^2 ł^3)) m^(3/2) Sqrt(E)

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31
Q

Density of states at Fermi surface

A

g(E_f) = 3N / 2E_f

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32
Q

Solenoid inductance

A

L = mu_0 N^2 A / l

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33
Q

When moving with constant acceleration, what time does it take to go dx if the terminal velocity is v?

A

dt = 2dx/v

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34
Q

When moving with constant acceleration a, how does the square velocity change over distance dx?

A

d(v^2) = 2 a dx

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35
Q

How does the amplitude of driven oscillations depend on the frequency?

A

A ~ 1/|w^2 - w0^2|

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36
Q

The potential of an electric dipole

A

U = k (p,r) / r^3

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37
Q

Electric field of a dipole

A

E = k r (p,r) / r^5 - k p / r^3

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38
Q

Torque on a dipole

A 
M = p x E
M = m x B
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39
Q

Radiation intensity of an oscillating dipole with amplitude p0 and frequency w.

A

<i> ~ p0^2 w^4 (sinO/r)^2</i>

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40
Q

The total power emitted by an oscillating dipole

A

<p> = mu0 p0^2 w^4 / (12 pi c),

| c^2 times less for a magnetic dipole.
</p>

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41
Q

Energy of a capacitor

A

W = U^2 C / 2 = Q^2 / 2C

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42
Q

Energy of an inductor

A

W = I^2 L / 2

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43
Q

Time constants of RL and RC circuits

A

tau = L/R and RC

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44
Q

High pass filter

A 
o-----||----------o
               |
               R
            \_\_|\_\_
              ---
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45
Q

Low pass filter

A 
o-----R----------o
               |
               C
            \_\_|\_\_
              ---
46
Q

Transition rules for l, m, j in hydrogen

A 
Dm = 0, +1, -1
Dl  =  +1, -1
Dj  =  0, +1, -1, except 0->0
47
Q

Planck’s formula for black body radiation

A

I(w) ~ łw^3/c^2 * 1/(exp(łw/kT) - 1)

48
Q

How does the total power of the black body radiation depend on T?

A

dP/dA ~ T^4

49
Q

Gauss theorem for magnetic field (Stokes theorem).

A

Int (B dl) = mu0*I

50
Q

Bio-Savart law

A

dB = mu0/4pi * [I dl,r] / r^2

51
Q

Solution of the poisson equation

A

V = k Int (rho/|r-r’|)dr

52
Q

Energy of a system of charges

A

W = Int (rho(r) V(r) dr) /2

53
Q

Boundary conditions for electric field

A

dEт = sigma/e_0

54
Q

Boundary conditions for magnetic field

A

dB|| = mu_0 [i x n]

55
Q

Bound charge density

A

Rho’ = -div P

56
Q

Bound current density

A

j’ = rot M

57
Q

Surface bound charge density

A

Sigma’ = (P,n)

58
Q

Surface bound current density

A

i’ = M x n

59
Q

Electric induction

A

D = e_0 E + P

60
Q

Magnetic field in terms of induction

A

H = B / mu_0 - M

61
Q

Maxwell equation for rotE

A

rot E = - dB/dt

62
Q

Maxwell equation for rotB

A

rot B = mu_0 j + e0 mu0 dE/dt

63
Q

Gauss theorem for electric induction D

A

div D = rho_0 – free charge

64
Q

Maxwell equation for rotH in medium

A

rot H = j_0 + dD/dt, where j_0 are the free currents

65
Q

Poynting vector

A

S = [E x B] / mu_0

66
Q

Average intensity of a plane electromagnetic wave

A

<i> = = c e0 E0^2 / 2</i>

67
Q

Entropy of the monatomic ideal gas

A

S = d(kT lnZ)dT = N k ln(VT^3/2 / N) + const

68
Q

Partition function and average energy relation

A

= -d(lnZ)/db, where b = 1/kT

69
Q

Free energy in terms of the partition function

A

F = -kT lnZ

70
Q

Partition function of the ideal gas

A

Z_N = V^N / (N! Л^3N),

where Л=h/Sqrt(2pi m kT)

71
Q

First law of thermodynamics

A 
DE = Q - W
dQ = T dS
dW = P dV
72
Q

Adiabatic isentropic (reversible) process equation

A

PV^g = const, where g=Cp/Cv=(n+2)/n

73
Q

Speed of sound in terms of bulk modulus

A

c = Sqrt (K/rho)

In ideal gas K = g P, g=Cp/Cv

74
Q

Commutator of x and p

A

[x,p] = ił

75
Q

Commutator of angular momenta

A

[Lx,Ly] = ił Lz

76
Q

Commutator of creation and annihilation operators

A

[a,a+] = 1

77
Q

Energy levels and eigenfunction in an infinite well

A

E_n = ł^2 pi^2 n^2 / (2 m a^2)

psi_n = Sqrt(2/n) sin(pi n x / a)

78
Q

Energy and Eigenfunction of a particle in the delta potential V=-A delta(x)

A

E = - mA^2 / 2ł^2

psi = Sqrt(mA)/ł exp(-mA |x| / ł^2)

79
Q

Second Pauli matrix

A

0 -i

i 0

80
Q

Spin x eigenvectors

A

Up ~ 1, 1

Down ~ 1, -1

81
Q

Spin y eigenvectors

A

Up ~ 1, i

Down ~ 1, -i

82
Q

Raleygh scattering intensity

A

I ~ I0 w^4 a^6

83
Q

Electric field of an infinite plane

A

E = sigma / 2e0

84
Q

Electric field of an infinite wire

A

E = lambda / (2pi e0 r)

85
Q

Electric field inside a capacitor

A

E = sigma / e0

86
Q

Magnetic field of an infinite wire

A

B = mu0 I / (2pi r)

87
Q

Magnetic field in the center of a circular loop current

A

B = mu0 I / 2R

88
Q

Magnetic field inside a solenoid

A

B = mu0 n I

89
Q

Magnetic field at distance r from the center of a toroidal solenoid ()inside it

A

B = mu0 N I / (2pi r)

90
Q

Relation between phase and group velocities of quantum waves

A

Vgr = 2Vph since w~k^2

91
Q

Corrections to hydrogen energy in increasing orders of alpha

A 
Hyperfine (nuclear + spin-spin) a^4*me/mp
Lamb shift (vacuum fluctuations) a^5
Fine structure (relativistic+spin-orbit) a^4
92
Q

Fine structure leaves energies in which spin degenerate?

A

In mj. It takes off the ml degeneracy

93
Q

Which elements were created during nucleosynthesis in early Universe?

A

Up until 4Be and 7Li->4Be

94
Q

Clausius-Klapeyron law

A

dP/dT = L / (T dv)

where L is latent heat J/kg and v is specific volume v=1/rho

95
Q

Diffraction law for double slits or gratings

A

D sinO = n lambda

96
Q

Rayleigh diffraction criterion for circular aperture

A

D sinO = 1.22 lambda is first minimum

97
Q

Bragg scattering formula

A

2d sinO = n lambda

98
Q

Cyclotron frequency

A

w = qB/m

99
Q

Cyclotron radius

A

r = mv/qB

100
Q

Order or filling the electron orbitals

A

Growing n+l, then growing n

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s…

101
Q

Harmonic oscillator ground state eigenfunction

A

Psi = (mw/pi ł)^1/4 exp(-m w x^2 /2ł)

102
Q

Pythagorean triples

A

7 24 25
28 96 100
6 8 10

103
Q

Focal length of a concave mirror

A

f = R/2

104
Q

Focal length of a convex mirror

A

f=-R/2

105
Q

Focal length of a lens with two radii

A

1/f = (n-1) (1/R1 - 1/R2)

For converging lenses R2 is negative

106
Q

Freezing temperature for vibrational modes

A

kT~łw, T~1000K

107
Q

Freezing temperature of rotational modes

A

kT~ ł^2 /2I, T~1K

108
Q

Photon-electron interactions in decreasing order

A

Photoelectric
Compton (dominant for keV-MeV photons)
Pair production near a nucleus (if E>2m_e)

109
Q

Mass of electron in eV

A

0,5 MeV

110
Q

Larmor formula (radiation power of an accelerating charge)

A

P = mu0 q^2 a^2 / (6pi c)

111
Q

Third Kepler law

A

T^2 ~ a^3

112
Q

BCC primitive cell volume

A

Octahedron a^3/2

113
Q

FCC primitive cell volume

A

Parallelepiped a^3/4

GRE physics (1 decks)

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