"Golden Rule"

The transition probability per unit of time

wf←iw_{f\leftarrow i}

for a transition from the state

ψi\bm\psi_i

to a group of states

F={ψf}F = \{\bm\psi_f\}

with energy

sin⁡Ei0\sin E_i^0

for a system characterized by the state density

ρ(E)\rho(E)

is given by:

wf←i=2πℏ∣⟨f∣H′∣i⟩∣Ei0≈Ef02⋅ρ(Ef0)w_{f\leftarrow i} = \frac{2\pi}{\hbar}|\braket{f|H'|i}|^2_{E_i^0 \approx E_f^0}\cdot\rho(E_f^0)

Dispersion (Born Approximation)

dσdΩ=∣f(ξ,η)∣2\frac{d\sigma}{d\Omega} = |f(\xi, \eta)|^2
f(ξ,η)=m2πℏ2∫ei(ki−kf)⋅r⋅v(r) d3rf(\xi, \eta) = \frac m{2\pi\hbar^2}\int e^{i(\bm k_i - \bm k_f)\cdot \bm r} \cdot v(\bm r)\: d^3r

For spherical symmetrical potential:

f(ξ,η)=2mℏ2K∫0∞sin⁡(Kr)⋅r⋅v(r)dr,∣K∣=2k⋅sin⁡(ξ2)f(\xi, \eta) = \frac{2m}{\hbar^2K}\int_0^{\infty}\sin(Kr)\cdot r\cdot v(r) dr, \quad\quad |K| = 2k\cdot\sin\left(\frac \xi 2\right)

Spherical box-potential:

V(r)={−V0r≤a0r>aV(r) = \begin{cases} -V_0 & r \leq a \\ 0 & r > a \end{cases}
f(ξ,η)=−2mV0ℏ2⋅sin⁡(Ka)−Kacos⁡(Ka)K3f(\xi, \eta) = -\frac{2mV_0}{\hbar^2}\cdot\frac{\sin(Ka) - Ka\cos(Ka)}{K^3}

Screened Coulomb Potential:

v(r)=−Ar⋅e−αrv(r) = -\frac A r\cdot e^{-\alpha r}
dσdΩ=(2mAℏ2(α2+4k2sin⁡2(ξ/2)))2\frac{d\sigma}{d\Omega} = \left(\frac{2mA}{\hbar^2\left(\alpha^2 + 4k^2\sin^2(\xi/2)\right)}\right)^2
σ=(Amℏ2)216πα2(α2+4k2)\sigma = \left(\frac{Am}{\hbar^2}\right)^2\frac{16\pi}{\alpha^2\left(\alpha^2 + 4k^2\right)}
When  α→0,dσdΩ→(Amℏ2)214(ksin⁡(ξ/2))4\textup{When}\; \alpha \rightarrow 0,\quad \frac{d\sigma}{d\Omega} \rightarrow \left(\frac{Am}{\hbar^2}\right)^2\frac 1{4\left(k\sin(\xi/2)\right)^4}

Periodic Potential

V(x)=0n(a+b)<x<n(a+b)+aV0n(a+b)+a<x<(n+1)(a+b)}V(x) = \begin{rcases} 0 & n(a + b) < x < n(a + b) + a \\ V_0 & n(a + b) + a < x < (n + 1)(a + b) \end{rcases}

Continuity Requirements:

cos⁡k1a⋅cos⁡k2b−k12+k222k1k2sin⁡k1a⋅sin⁡k2b=cos⁡(k(a+b)),V0<E\cos k_1a\cdot\cos k_2b - \frac{k_1^2 +k_2^2}{2k_1k_2}\sin k_1a\cdot\sin k_2b = \cos(k(a + b)), \quad V_0 < E
cos⁡k1a⋅cosh⁡κb−k12+κ22k1κsin⁡k1a⋅sinh⁡κb=cos⁡(k(a+b)),V0<E\cos k_1a\cdot\cosh \kappa b - \frac{k_1^2 +\kappa^2}{2k_1\kappa}\sin k_1a\cdot\sinh \kappa b = \cos(k(a + b)), \quad V_0 < E

Phase and group speed:

vf=ωk,vg=dωdk=dEdpv_f = \frac \omega k, \quad v_g = \frac{d\omega}{dk} = \frac{dE}{dp}

Effective mass:

m∗=(1ℏ2d2Edk2)−1m^* = \left(\frac 1{\hbar^2}\frac{d^2E}{dk^2}\right)^{-1}