| Concept | Formula | Notes |
|---|---|---|
| Wavefunction | Ψ(r, t) |
Contains all measurable information. |
| Probability density | ρ = |Ψ|² |
Born rule. |
| Probability current | j = (ħ/m) Im(Ψ* ∇Ψ) |
Conservation of probability. |
| Momentum‑space wavefunction | Φ(p) = (1/(2πħ)³/²) ∫ e^{-ip·r/ħ} Ψ(r) d³r |
Fourier transform. |
| Concept | Formula | Notes |
|---|---|---|
| Time‑dependent Schrödinger equation | iħ ∂Ψ/∂t = ĤΨ |
Fundamental evolution law. |
| Time‑independent Schrödinger equation | ĤΨ = EΨ |
Stationary states. |
| Free particle (1D) | −ħ²/(2m) d²Ψ/dx² = EΨ |
Second‑order ODE. |
| Concept | Formula | Notes |
|---|---|---|
| Position operator | x̂ f(x) = x f(x) |
Acts by multiplication. |
| Momentum operator | p̂ f(x) = −iħ (df/dx) |
Generator of translations. |
| Canonical commutation | [x̂, p̂] = iħ |
Core quantization rule. |
| Concept | Formula | Notes |
|---|---|---|
| Commutation relations | [Jᵢ, Jⱼ] = iħ εᵢⱼₖ Jₖ |
SU(2) algebra. |
| Eigenvalues | J² |j m⟩ = ħ² j(j+1)|j m⟩ |
Total angular momentum. |
| z‑component | Jz |j m⟩ = ħ m |j m⟩ |
Magnetic quantum number. |
| Clebsch–Gordan expansion | |j m⟩ = Σ C^{j m}_{j₁ m₁ j₂ m₂} |j₁ m₁⟩|j₂ m₂⟩ |
Adding angular momenta. |
| Concept | Formula | Notes |
|---|---|---|
| Hamiltonian | H = p²/(2m) + (1/2)mω²x² |
Quadratic potential. |
| Energy levels | Eₙ = ħω (n + 1/2) |
Equally spaced. |
| Creation/annihilation operators | a = √(mω/2ħ)(x + i p/(mω)) |
Algebraic solution. |
| Concept | Formula | Notes |
|---|---|---|
| de Broglie wavelength | λ = h/p |
Wave–particle duality. |
| Compton shift | Δλ = (h/(mₑ c))(1 − cos θ) |
Photon scattering. |
| Dirac equation | (β m c² + c α·p)Ψ = iħ ∂Ψ/∂t |
Relativistic fermions. |
| Concept | Formula | Notes |
|---|---|---|
| First‑order energy shift | Eₙ^(1) = ⟨n⁽⁰⁾| H' |n⁽⁰⁾⟩ |
Non‑degenerate. |
| Second‑order shift | Eₙ^(2) = Σₘ |⟨m|H'|n⟩|² / (Eₙ⁽⁰⁾ − Eₘ⁽⁰⁾) |
Corrections from other states. |