quantum mechanics

the results i double-check rather than trust from memory --; exact coefficients, sign conventions, the multi-term identities. ħ kept explicit.

jump to: angular momentum · pauli & rotations · oscillator · perturbation · operator identities · density matrix

angular momentum

• algebra: [Jᵢ, Jⱼ] = iħ εijk Jₖ; J² commutes with every Jᵢ
• spectrum: J²|j,m⟩ = j(j+1)ħ²|j,m⟩, Jz|j,m⟩ = mħ|j,m⟩, m = −j … +j
• ladders: J± = Jx ± iJy; [Jz, J±] = ±ħ J±; [J+, J−] = 2ħ Jz
• exact action (the bit people misremember): J±|j,m⟩ = ħ √(j(j+1) − m(m±1)) |j, m±1⟩
• addition: j₁ ⊗ j₂ = (j₁+j₂) ⊕ … ⊕ |j₁−j₂|; mix with Clebsch–Gordan ⟨j₁m₁ j₂m₂ | JM⟩, nonzero only if M = m₁+m₂

pauli & rotations

• product identity: (a·σ)(b·σ) = (a·b) I + i (a×b)·σ
• exponential (unit n̂): e^(iθ n̂·σ) = cos θ · I + i sin θ (n̂·σ)
• rotation operator: R(θ,n̂) = e^(−iθ (n̂·J)/ħ); for spin-½, e^(−i(θ/2) n̂·σ) = cos(θ/2) I − i sin(θ/2)(n̂·σ)
• spinor sign: R(2π) = −I --; a spin-½ state needs 4π to come back

harmonic oscillator

• H = ħω(a†a + ½) = ħω(N + ½), with [a, a†] = 1, N = a†a
• actions: a|n⟩ = √n |n−1⟩, a†|n⟩ = √(n+1) |n+1⟩, N|n⟩ = n|n⟩
• operators: x = √(ħ/2mω) (a + a†), p = i √(ħmω/2) (a† − a)
• coherent state: a|α⟩ = α|α⟩, |α⟩ = e^(−|α|²/2) Σ αⁿ/√(n!) |n⟩ --; minimum-uncertainty, not orthogonal

perturbation theory

nondegenerate, H = H₀ + λV:

• energy: Eₙ = Eₙ⁰ + λ⟨n|V|n⟩ + λ² Σ over k≠n of |⟨k|V|n⟩|² / (Eₙ⁰ − Eₖ⁰) + …
• state: |n⟩ = |n⁰⟩ + λ Σ over k≠n of ⟨k|V|n⟩/(Eₙ⁰ − Eₖ⁰) |k⁰⟩ + …
• degenerate case: diagonalize V inside the degenerate subspace; its eigenvalues are the first-order splittings
• fermi's golden rule (time-dependent): Γ(i→f) = (2π/ħ) |⟨f|V|i⟩|² ρ(E_f)

operator identities

• expansions: [A, BC] = [A,B]C + B[A,C]; [AB, C] = A[B,C] + [A,C]B
• hadamard: e^A B e^(−A) = B + [A,B] + (1/2!)[A,[A,B]] + (1/3!)[A,[A,[A,B]]] + …
• BCH: e^A e^B = exp( A + B + ½[A,B] + (1/12)([A,[A,B]] − [B,[A,B]]) − … )
• glauber (when [A,B] commutes with both A and B): e^A e^B = e^(A+B) e^(½[A,B])

density matrix

• ρ = Σ pᵢ |ψᵢ⟩⟨ψᵢ|, with ρ = ρ† ≥ 0 and tr ρ = 1
• expectation: ⟨A⟩ = tr(ρA)
• purity: pure ⟺ ρ² = ρ ⟺ tr(ρ²) = 1; mixed ⟺ tr(ρ²) < 1
• von neumann entropy: S = −tr(ρ ln ρ) = −Σ λᵢ ln λᵢ (λᵢ the eigenvalues)
• subsystem: ρ_A = tr_B(ρ_AB) (partial trace)
• evolution: iħ dρ/dt = [H, ρ]