📐 Equation Reference
A searchable database of all important equations in quantum mechanics. Find formulas by name, browse by category, and jump directly to the relevant course pages.
📐Equation Reference Database
Showing 16 equations
Time-Dependent Schrödinger Equation
Fundamentaliℏ ∂ψ/∂t = ĤψThe fundamental equation governing quantum evolution
Variables: ψ: wave function, Ĥ: Hamiltonian operator, ℏ: reduced Planck constant
Time-Independent Schrödinger Equation
FundamentalĤψ = EψEnergy eigenvalue equation for stationary states
Variables: E: energy eigenvalue
Heisenberg Uncertainty Principle
FundamentalΔx Δp ≥ ℏ/2Fundamental limit on simultaneous precision of position and momentum
Variables: Δx: position uncertainty, Δp: momentum uncertainty
Normalization Condition
Wave Functions⟨ψ|ψ⟩ = ∫|ψ(x)|² dx = 1Total probability must equal 1
Infinite Square Well Energy
Infinite WellE_n = n²π²ℏ²/(2mL²)Quantized energy levels for particle in a box
Variables: n: quantum number (1,2,3,...), m: particle mass, L: well width
Infinite Square Well Wave Function
Infinite Wellψ_n(x) = √(2/L) sin(nπx/L)Normalized eigenstates for infinite square well
Harmonic Oscillator Energy
Harmonic OscillatorE_n = (n + ½)ℏωQuantized energy levels with zero-point energy
Variables: n: quantum number (0,1,2,...), ω: angular frequency
Ladder Operators
Harmonic Oscillatorâ± = (mω x̂ ∓ ip̂)/√(2mωℏ)Raising and lowering operators for harmonic oscillator
Hydrogen Energy Levels
Hydrogen AtomE_n = -13.6 eV/n²Energy levels of hydrogen atom
Variables: n: principal quantum number
Angular Momentum Eigenvalues
Angular MomentumL² |l,m⟩ = l(l+1)ℏ² |l,m⟩Total angular momentum squared
Variables: l: orbital quantum number, m: magnetic quantum number
Spin Angular Momentum
SpinS² |s,m⟩ = s(s+1)ℏ² |s,m⟩Intrinsic angular momentum
Variables: s: spin quantum number (1/2 for electrons)
Density Matrix
Mixed Statesρ = Σ p_i |ψ_i⟩⟨ψ_i|Statistical description of quantum ensemble
Variables: p_i: classical probabilities
📊 Database Statistics
Total Equations
16
Categories
10
With Links
16
Fundamental
3
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