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