📐 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

Fundamental
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iℏ ∂ψ/∂t = Ĥψ

The fundamental equation governing quantum evolution

Variables: ψ: wave function, Ĥ: Hamiltonian operator, ℏ: reduced Planck constant

Time-Independent Schrödinger Equation

Fundamental
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Ĥψ = Eψ

Energy eigenvalue equation for stationary states

Variables: E: energy eigenvalue

Heisenberg Uncertainty Principle

Fundamental
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Δx Δp ≥ ℏ/2

Fundamental limit on simultaneous precision of position and momentum

Variables: Δx: position uncertainty, Δp: momentum uncertainty

Commutator

Operators
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[Â, B̂] = ÂB̂ - B̂Â

Measures non-commutativity of operators

Position-Momentum Commutator

Operators
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[x̂, p̂] = iℏ

Canonical commutation relation

Expectation Value

Measurement
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⟨Â⟩ = ⟨ψ|Â|ψ⟩

Average value of observable A in state ψ

Normalization Condition

Wave Functions
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⟨ψ|ψ⟩ = ∫|ψ(x)|² dx = 1

Total probability must equal 1

Infinite Square Well Energy

Infinite Well
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E_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 Well
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ψ_n(x) = √(2/L) sin(nπx/L)

Normalized eigenstates for infinite square well

Harmonic Oscillator Energy

Harmonic Oscillator
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E_n = (n + ½)ℏω

Quantized energy levels with zero-point energy

Variables: n: quantum number (0,1,2,...), ω: angular frequency

Ladder Operators

Harmonic Oscillator
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â± = (mω x̂ ∓ ip̂)/√(2mωℏ)

Raising and lowering operators for harmonic oscillator

Hydrogen Energy Levels

Hydrogen Atom
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E_n = -13.6 eV/n²

Energy levels of hydrogen atom

Variables: n: principal quantum number

Angular Momentum Eigenvalues

Angular Momentum
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L² |l,m⟩ = l(l+1)ℏ² |l,m⟩

Total angular momentum squared

Variables: l: orbital quantum number, m: magnetic quantum number

Spin Angular Momentum

Spin
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S² |s,m⟩ = s(s+1)ℏ² |s,m⟩

Intrinsic angular momentum

Variables: s: spin quantum number (1/2 for electrons)

Pauli X Matrix

Spin
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σ_x = [[0,1],[1,0]]

Pauli spin matrix for x-component

Density Matrix

Mixed States
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ρ = 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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