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Quantum Mechanics — Foundations

Wave functions, operators, and the postulates of quantum mechanics

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quantum_quandary 25 terms Feb 11, 2026
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Terms 25

1
Wave Function (Ψ)
Mathematical description of quantum state; |Ψ|² gives probability density
2
Schrödinger Equation (time-dependent)
iℏ ∂Ψ/∂t = ĤΨ; governs how quantum states evolve over time
3
Schrödinger Equation (time-independent)
ĤΨ = EΨ; eigenvalue equation; E is allowed energy of stationary state
4
Hamiltonian Operator (Ĥ)
Total energy operator: Ĥ = −(ℏ²/2m)∇² + V; kinetic + potential
5
Eigenvalue / Eigenstate
Measurement of observable  on eigenstate gives definite eigenvalue a: Âψ = aψ
6
Superposition Principle
Quantum state can be sum of multiple eigenstates; collapses to one on measurement
7
Heisenberg Uncertainty Principle
ΔxΔp ≥ ℏ/2; position and momentum cannot both be precisely known simultaneously
8
Wave-Particle Duality
Quantum objects exhibit both wave and particle behavior depending on observation
9
de Broglie Wavelength
λ = h/p; every particle has associated wavelength; p is momentum
10
Planck's Constant (ℏ)
ℏ = h/2π ≈ 1.055×10⁻³⁴ J·s; fundamental constant setting scale of quantum effects
11
Photoelectric Effect
Light ejects electrons from metal only above threshold frequency; showed light is quantized (Einstein)
12
Photon
Quantum of electromagnetic radiation; E = hf; massless, travels at c
13
Quantization
Physical quantities (energy, angular momentum) take only discrete allowed values
14
Quantum Tunneling
Particle penetrates a classically forbidden energy barrier due to wave nature
15
Particle in a Box
Model with infinite potential walls; En = n²π²ℏ²/(2mL²); energy quantized
16
Harmonic Oscillator (quantum)
En = (n + ½)ℏω; zero-point energy ≠ 0; ladder operators
17
Spin
Intrinsic angular momentum of particles; spin-½ particles have ms = ±½
18
Pauli Exclusion Principle
No two fermions can occupy the same quantum state simultaneously
19
Fermi-Dirac Statistics
Statistical distribution for fermions (half-integer spin); obey Pauli exclusion
20
Bose-Einstein Statistics
Statistical distribution for bosons (integer spin); can occupy same state
21
Copenhagen Interpretation
Wave function represents probabilities; collapses on measurement; Bohr and Heisenberg
22
Many-Worlds Interpretation
All quantum possibilities occur in branching parallel universes; no collapse
23
Entanglement
Two particles share quantum state; measuring one instantly determines the other regardless of distance
24
Bell's Theorem
No local hidden variable theory can reproduce all quantum predictions; entanglement is real
25
Dirac Notation (Bra-Ket)
|ψ⟩ is ket (state); ⟨ψ| is bra; ⟨φ|ψ⟩ is inner product (probability amplitude)