Susskind Quantum Mechanics Equations and definitions

This is a working formula sheet for the Susskind quantum mechanics subtree. It favors meaning over formal completeness.

States and amplitudes

A state expanded in an orthonormal basis:

|Ψ⟩ = Σ_j α_j |j⟩
α_j = ⟨j|Ψ⟩

Born rule and normalization:

P(j) = |⟨j|Ψ⟩|² = |α_j|²
Σ_j |α_j|² = 1

Related notes: Quantum state, Bra-ket notation, Wavefunction.

Observables and measurement

Eigenvalue equation:

M|m⟩ = m|m⟩

If M is measured and outcome m is obtained, the post-measurement state is the corresponding eigenstate or eigenspace.

Expectation value:

⟨M⟩ = ⟨Ψ|M|Ψ⟩

Related notes: Observables and eigenvalues, Measurement and state preparation.

Compatibility and uncertainty

Commutator:

[L, M] = LM - ML

If [L, M] = 0, the observables can share a complete eigenbasis and can be jointly specified. If they do not commute, sharp knowledge of one generally limits sharp knowledge of the other.

Robertson-style uncertainty relation:

ΔL ΔM ≥ (1/2) |⟨[L, M]⟩|

For position and momentum:

[X, P] = iħ
Δx Δp ≥ ħ/2

Related notes: Commutators and compatible observables, Uncertainty principle.

Time evolution

Time-development operator:

|Ψ(t)⟩ = U(t)|Ψ(0)⟩
U†U = I

General Schrödinger equation:

iħ d|Ψ⟩/dt = H|Ψ⟩

Expectation-value time dependence, when the operator has no explicit time dependence:

d⟨L⟩/dt = (i/ħ) ⟨[H, L]⟩

Related notes: Quantum time evolution, Schrödinger equation.

Combining systems

Composite basis:

|ab⟩ = |a⟩ ⊗ |b⟩
dim(S_A ⊗ S_B) = dim(S_A) dim(S_B)

Product state:

|Ψ_AB⟩ = |A⟩ ⊗ |B⟩

Generic composite state:

|Ψ_AB⟩ = Σ_ab ψ_ab |ab⟩

If the coefficients ψ_ab cannot be factorized as α_a β_b, the state is entangled.

Related notes: Tensor product states, Entanglement, Density matrix.

Particles and waves

Position wavefunction:

ψ(x) = ⟨x|Ψ⟩
∫ |ψ(x)|² dx = 1

Operators in the position basis:

X ψ(x) = x ψ(x)
P ψ(x) = -iħ ∂ψ/∂x

Particle Schrödinger equation in one dimension:

iħ ∂ψ/∂t = [-(ħ²/2m) ∂²/∂x² + V(x)] ψ

Related notes: Wavefunction, Schrödinger equation.

Harmonic oscillator

Classical potential:

V(x) = (1/2) mω²x²

Quantum Hamiltonian:

H = P²/(2m) + (1/2)mω²X²

Energy levels:

E_n = ħω(n + 1/2),  n = 0, 1, 2, ...

Ladder operators move between adjacent energy levels; the ground state cannot be lowered.

Related note: Quantum harmonic oscillator.