Physics-Intro to Quantum Mechanics

Constants:

Masses of particles: electron, proton, neutron, in units of MeV/c ² 
m _e = 0.511 MeV/c ² 
m _p = 938.26 MeV/c ² 
m _n = 939.55 MeV/c ²

Approximations for the value of hc
h c = 12,400 eV A
ħ c = 1,973 eV A

Approximation of k boltzmann
k _b = 1/11,600 eV/K

Approximation of Coulomb constant and two electrons worth of charge.
k e ² =14.4 eV A

E ₀ = (k e ² ) / (2 a ₀ ) = (m k ² e ⁴ ) / (2 ħ ² ) = 13.6 eV

R = 1.097 x 10 ⁷ m ^-1

Relativity:

E = (m ² c ⁴ + p ² c ² ) ^1/2

Planck's Law:

u (λ) = n (λ) E (λ)
n (λ) = 8 π / λ ⁴ 
E (λ) = h c / [ λ (e ^{h c / λ k _b T} -1)]

Energy in an oscillator:

E _f = n h f
P (E) ∝ e ^{-E / k _B T}

Stefan's law:

R = σ T ⁴ 
σ = 5.67 x 10 ^-8 W / m ² K ⁴ 
R = c U / 4
U = ∫ u (λ) dλ

Wein's Displacement law:

λ _m T = h c / 4.96 k _B

Photons:

Energy, momenta and frequency of photons
E = p c
E = h f
p = h / λ
f = c / λ

Photoelectric effect:

e V ₀ = m v ² / 2 = h f - φ

Compton Scattering:

λ ' - λ = h (1 - cos (θ) / (m _e c)

Rutherford Scattering:

b = (k q _α Q cot (θ / 2)) / m _α v ² 
ΔN ∝ 1 / sin ⁴ (θ / 2)

Electrostatics:

F = k q ₁ q ₂ / r ² 
U = q ₀ V
V = k q / r

Hydrogen spectra:

1 / λ = R (1 / m ² - 1 / n ² )

Bohr atom:

E _n = k e ² Z / 2 r _n = - Z ² E ₀ / n ² 
E ₀ = k e ² / 2 a ₀ = m k ² e ⁴ / 2 ħ ² = 13.6 eV
h f = E _i - E _f 
r _n = r ₀ n ² 
r ₀ = a ₀ / Z
a ₀ = ħ / m k e ² = 0.529 A
L = m v r = n ħ

X - ray spectra:

f ^1/2 = A _n (Z - b)
K : b =1, L : b = 7.4

de Broglie:

wavelength, frequency, momentum, and energy of particles
λ = h / p
f = E / h
ω = 2 π f
k = 2 π / λ
E = ħ ω
p = ħ k
E = p ² / 2 m

Group and phase velocity:

v _g = dω / dk
v _p = ω / k

Heisenberg Uncertainty principles:

Δx Δp ≈ ħ
Δt ΔE ≈ ħ

Wave Function:

Ψ(x , t) = |Ψ(x , t)| e ^{i θ (x , t)}
P(x , t) = |Ψ(x , t)| ² dx

Schrodinger Equation:

-ħ ² δ ² Ψ / 2 m δx ² + V(x) Ψ(x , t) = i ħ δ Ψ / δ t;
Ψ (x , t) = ψ(x) e ^{-i E t / ħ} 
-ħ ² δ ² Ψ / 2 m δx ² + V(x)&psi(x) = Eψ(x)
∫ dx Ψ * Ψ = 1
Ψ _n (x) = (2/L) ^1/2 sin(n π x / L)
E _n (x) = π ² 
x _op = x
p _op = ħ δ / i δ x
< A > = ∫ δ x Ψ * A _op Ψ
A _op Ψ = a Ψ

uncertainty:

Δ A = ( <A ² > - <A> ²  ) ^1/2

Harmonic Oscillator:

Ψ _n (x) = C _n H _n (x) e ^{-m ω x 2 / 2 ħ} 
E _n = (n + 1/2) ħ ω
E = p ² / 2 m + m ω ² x ² / 2 = m ω ² A ² / 2 , Δ n = +- 1

Step potential:

R = (k ₁ - k ₂ ) ² / (k ₁ + k ₂ ) ² 
T = 1 - R
k = (2 m (E - V) / ħ) ^1/2

Tunneling

Ψ (x) ≈ e ^-αx 
T ≈ e ^-2αx 
T ≈ e ^{-2 ∫ αx dx}
α(x) = [(2 m [V(x) - E]) / ħ ² ] ^1/2

3-D square well

Ψ(x, y, z) = Ψ ₁ (x) Ψ ₂ (y) Ψ ₃ (z)
E = π ²

Spherically symmetric potential

Ψ _n,l,m (r, θ, φ) = R _{n l} (r) Y _{l m} (θ, φ)
Y _{l m} (θ, φ) = f _{l m} (θ) e ^{i m φ}