All Flashcards
0
Q
Ground state hydrogen energy
A
-E_1=ł^2 / (2 m a^2)=k^2 m e^4 / (2ł^2)
1
Q
Bohr radius
A
a=ł^2 / (k m e^2)
2
Q
Electric constant k=1/(4 pi e_0)
A
10^10 Н м^2/Кл
3
Q
How does the ground state energy of a hydrogen-like atom depend on Z?
A
E_1 ~ Z^2
4
Q
1 Henri in terms of seconds
A
H=V s^2 / C=В с^2 / Кл
5
Q
1 Ohm in terms of seconds
A
Ohm = V s / C = В с / Кл
6
Q
1 Farade in terms of other units
A
F = C / V = Кл / В
7
Q
h c = ? eV nm
A
1240
8
Q
ł c = ? eV nm
A
200
9
Q
k_b * 300 K = ? eV
A
0,026 = 1/40
10
Q
Mass of a proton
A
938 MeV/c^2
11
Q
Atomic mass unit (carbon/12)
A
931 MeV/c^2
12
Q
Rydberg constant
A
R = -E_1/hc = 1.1*10^7 m^(-1)
13
Q
Rydberg formula for hydrogen emission
A
1/lambda = R ( 1/m^2 - 1/n^2 )
14
Q
Fine structure constant
A
alpha = k e^2 / (łc) = 1/137
15
Q
Second order perturbation correction to energy levels
A
E_n^1 = E_n + + Sum ||^2 / (E_n - E_m)
16
Q
Wien displacement law
A
Lambda_max = 3 / T (mm)
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
18
Q
Energy-momentum vector
A
(E/c, p)
19
Q
Charge-current vector
A
(c rho, j)
20
Q
Wave four-vector
A
(omega, c k)
21
Q
Square of the energy-momentum vector = ?. Invariant? Conserved?
A
m^2 c^2 in any reference frames. Invariant and conserved.
22
Q
Relativistic energy in terms of the rest energy
A
E = gamma m c^2
23
Q
Relativistic momentum in a non-rest frame
A
p = gamma m v
24
Doppler frequency shift in classical waves
(v + v_rec) / (v - v_source) where v is the speed of waves
25
Relativistic Doppler frequency shift
Sqrt(1+beta/1-beta)
26
Compton wavelength shift
D_lamda=h/mc (1-cosO)
27
Table of quarks
u c t. 2/3
d s b. -1/3
e mu tau
ne nmu ntau
28
Proton and neutron quark formulas
p=uud
| n=udd
29
Fermi energy
```
k_f = (3 pi^2 n)^1/3
E_f = ł^2 k^2 / 2m
```
30
Density of states
g(E) = (V Sqrt(2) / (pi^2 ł^3)) m^(3/2) Sqrt(E)
31
Density of states at Fermi surface
g(E_f) = 3N / 2E_f
32
Solenoid inductance
L = mu_0 N^2 A / l
33
When moving with constant acceleration, what time does it take to go dx if the terminal velocity is v?
dt = 2dx/v
34
When moving with constant acceleration a, how does the square velocity change over distance dx?
d(v^2) = 2 a dx
35
How does the amplitude of driven oscillations depend on the frequency?
A ~ 1/|w^2 - w0^2|
36
The potential of an electric dipole
U = k (p,r) / r^3
37
Electric field of a dipole
E = k r (p,r) / r^5 - k p / r^3
38
Torque on a dipole
```
M = p x E
M = m x B
```
39
Radiation intensity of an oscillating dipole with amplitude p0 and frequency w.
~ p0^2 w^4 (sinO/r)^2
40
The total power emitted by an oscillating dipole
= mu0 p0^2 w^4 / (12 pi c), | c^2 times less for a magnetic dipole.
41
Energy of a capacitor
W = U^2 C / 2 = Q^2 / 2C
42
Energy of an inductor
W = I^2 L / 2
43
Time constants of RL and RC circuits
tau = L/R and RC
44
High pass filter
```
o-----||----------o
|
R
__|__
---
```
45
Low pass filter
```
o-----R----------o
|
C
__|__
---
```
46
Transition rules for l, m, j in hydrogen
```
Dm = 0, +1, -1
Dl = +1, -1
Dj = 0, +1, -1, except 0->0
```
47
Planck's formula for black body radiation
I(w) ~ łw^3/c^2 * 1/(exp(łw/kT) - 1)
48
How does the total power of the black body radiation depend on T?
dP/dA ~ T^4
49
Gauss theorem for magnetic field (Stokes theorem).
Int (B dl) = mu0*I
50
Bio-Savart law
dB = mu0/4pi * [I dl,r] / r^2
51
Solution of the poisson equation
V = k Int (rho/|r-r'|)dr
52
Energy of a system of charges
W = Int (rho(r) V(r) dr) /2
53
Boundary conditions for electric field
dEт = sigma/e_0
54
Boundary conditions for magnetic field
dB|| = mu_0 [i x n]
55
Bound charge density
Rho' = -div P
56
Bound current density
j' = rot M
57
Surface bound charge density
Sigma' = (P,n)
58
Surface bound current density
i' = M x n
59
Electric induction
D = e_0 E + P
60
Magnetic field in terms of induction
H = B / mu_0 - M
61
Maxwell equation for rotE
rot E = - dB/dt
62
Maxwell equation for rotB
rot B = mu_0 j + e0 mu0 dE/dt
63
Gauss theorem for electric induction D
div D = rho_0 -- free charge
64
Maxwell equation for rotH in medium
rot H = j_0 + dD/dt, where j_0 are the free currents
65
Poynting vector
S = [E x B] / mu_0
66
Average intensity of a plane electromagnetic wave
= = c e0 E0^2 / 2
67
Entropy of the monatomic ideal gas
S = d(kT lnZ)dT = N k ln(VT^3/2 / N) + const
68
Partition function and average energy relation
= -d(lnZ)/db, where b = 1/kT
69
Free energy in terms of the partition function
F = -kT lnZ
70
Partition function of the ideal gas
Z_N = V^N / (N! Л^3N),
| where Л=h/Sqrt(2pi m kT)
71
First law of thermodynamics
```
DE = Q - W
dQ = T dS
dW = P dV
```
72
Adiabatic isentropic (reversible) process equation
PV^g = const, where g=Cp/Cv=(n+2)/n
73
Speed of sound in terms of bulk modulus
c = Sqrt (K/rho)
| In ideal gas K = g P, g=Cp/Cv
74
Commutator of x and p
[x,p] = ił
75
Commutator of angular momenta
[Lx,Ly] = ił Lz
76
Commutator of creation and annihilation operators
[a,a+] = 1
77
Energy levels and eigenfunction in an infinite well
E_n = ł^2 pi^2 n^2 / (2 m a^2)
psi_n = Sqrt(2/n) sin(pi n x / a)
78
Energy and Eigenfunction of a particle in the delta potential V=-A delta(x)
E = - mA^2 / 2ł^2
psi = Sqrt(mA)/ł exp(-mA |x| / ł^2)
79
Second Pauli matrix
0 -i
| i 0
80
Spin x eigenvectors
Up ~ 1, 1
| Down ~ 1, -1
81
Spin y eigenvectors
Up ~ 1, i
| Down ~ 1, -i
82
Raleygh scattering intensity
I ~ I0 w^4 a^6
83
Electric field of an infinite plane
E = sigma / 2e0
84
Electric field of an infinite wire
E = lambda / (2pi e0 r)
85
Electric field inside a capacitor
E = sigma / e0
86
Magnetic field of an infinite wire
B = mu0 I / (2pi r)
87
Magnetic field in the center of a circular loop current
B = mu0 I / 2R
88
Magnetic field inside a solenoid
B = mu0 n I
89
Magnetic field at distance r from the center of a toroidal solenoid ()inside it
B = mu0 N I / (2pi r)
90
Relation between phase and group velocities of quantum waves
Vgr = 2Vph since w~k^2
91
Corrections to hydrogen energy in increasing orders of alpha
```
Hyperfine (nuclear + spin-spin) a^4*me/mp
Lamb shift (vacuum fluctuations) a^5
Fine structure (relativistic+spin-orbit) a^4
```
92
Fine structure leaves energies in which spin degenerate?
In mj. It takes off the ml degeneracy
93
Which elements were created during nucleosynthesis in early Universe?
Up until 4Be and 7Li->4Be
94
Clausius-Klapeyron law
dP/dT = L / (T dv)
| where L is latent heat J/kg and v is specific volume v=1/rho
95
Diffraction law for double slits or gratings
D sinO = n lambda
96
Rayleigh diffraction criterion for circular aperture
D sinO = 1.22 lambda is first minimum
97
Bragg scattering formula
2d sinO = n lambda
98
Cyclotron frequency
w = qB/m
99
Cyclotron radius
r = mv/qB
100
Order or filling the electron orbitals
Growing n+l, then growing n
| 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s...
101
Harmonic oscillator ground state eigenfunction
Psi = (mw/pi ł)^1/4 exp(-m w x^2 /2ł)
102
Pythagorean triples
7 24 25
28 96 100
6 8 10
103
Focal length of a concave mirror
f = R/2
104
Focal length of a convex mirror
f=-R/2
105
Focal length of a lens with two radii
1/f = (n-1) (1/R1 - 1/R2)
| For converging lenses R2 is negative
106
Freezing temperature for vibrational modes
kT~łw, T~1000K
107
Freezing temperature of rotational modes
kT~ ł^2 /2I, T~1K
108
Photon-electron interactions in decreasing order
Photoelectric
Compton (dominant for keV-MeV photons)
Pair production near a nucleus (if E>2m_e)
109
Mass of electron in eV
0,5 MeV
110
Larmor formula (radiation power of an accelerating charge)
P = mu0 q^2 a^2 / (6pi c)
111
Third Kepler law
T^2 ~ a^3
112
BCC primitive cell volume
Octahedron a^3/2
113
FCC primitive cell volume
Parallelepiped a^3/4
