EQ-SCHRODINGER · Quantum Mechanics
Schrödinger Equation (1D, time-dependent, free particle)
hbar**2*psi_xx/(2*m) + I*hbar*psi_t = 0
Derivative form
hbar**2*Derivative(psi(x, t), (x, 2))/(2*m) + I*hbar*Derivative(psi(x, t), t) = 0
Variables
variable
hbar
reduced Planck constant ℏ ≈ 1.0546e-34 J·s
- Object
- abstract_quantity
- Property
- ReducedPlanckConstant
- Context
- time_dependent
variable
m
particle mass
- Object
- point_particle
- Property
- Mass
- Context
- non_relativistic
variable
psi_t
partial time derivative of the wavefunction ∂ψ/∂t
- Object
- wavefunction
- Property
- WaveFunctionTimeDerivative
- Context
- time_dependent
variable
psi_xx
second spatial derivative ∂²ψ/∂x²
- Object
- wavefunction
- Property
- WaveFunctionSecondSpatialDerivative
- Context
- time_dependent
Axioms
constant_coefficients deterministic differential hermitian homogeneous isotropic linear linear_superposition non_relativistic probability_conserving quantum unitary
Assumptions
- Free particle (V = 0) or equivalent classical field
- Non-relativistic kinetic energy (|p| << mc)
- ψ is a complex scalar field, normalisable over the domain
- Time evolution is unitary (norm conserved)
Derivation
- Schrödinger, Ann. der Physik 79 (1926), 361 (original derivation via matter waves)
- Modern derivation: correspondence principle + E = p²/2m + symmetrisation
- Wick rotation t → −iτ converts the oscillatory kernel to the diffusive kernel of the heat equation (Feynman-Kac 1949)
References
- Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 1
- Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.1
- Feynman & Hibbs, Quantum Mechanics and Path Integrals (1965)