Schrödinger Equation
The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics, describing how the quantum state (wavefunction) of a physical system evolves over time in response to the system's total energy operator, the Hamiltonian.
Time-dependent Schrödinger equation
| Symbol | Name | SI Unit | Description |
|---|---|---|---|
| i | Imaginary unit | dimensionless | √−1 |
| ħ | Reduced Planck constant | J·s | h / 2π |
| Ψ | Wavefunction | depends on system (e.g. m⁻³⁄² in 3D) | Encodes the complete quantum state |
| ℋ | Hamiltonian operator | joule (J) | Total energy operator: kinetic + potential |
Conditions of validity: Valid for non-relativistic quantum systems — particles moving at speeds much less than the speed of light. For relativistic quantum particles, the Schrödinger equation is superseded by the Klein-Gordon equation (spin-0) or the Dirac equation (spin-1/2). The equation as written also assumes a single, non-relativistic particle in an external potential; extensions (many-body Schrödinger equation, second quantisation) are required for systems of multiple interacting particles or for quantum field theory.
Time-independent Schrödinger equation
When the Hamiltonian does not explicitly depend on time, the wavefunction can be separated into spatial and time parts, Ψ(x,t) = ψ(x)e^(−iEt/ħ), leading to the time-independent form used to find stationary states (definite-energy states):
| Symbol | Name | Description |
|---|---|---|
| ψ | Spatial wavefunction | Time-independent part of Ψ |
| E | Energy eigenvalue | The definite, measurable energy of that stationary state |
For a single particle of mass m in a potential V(x), the Hamiltonian operator is explicitly ℋ = −(ħ²/2m)∇² + V(x), giving the commonly seen expanded form:
Derivation context
Unlike Newton's second law, the Schrödinger equation cannot be derived from more fundamental classical axioms — it is itself a fundamental postulate of quantum mechanics, justified by its extraordinary empirical success rather than derived from prior principles. Schrödinger arrived at it in 1926 by analogy with the Hamilton-Jacobi formulation of classical mechanics and Louis de Broglie's 1924 hypothesis that particles have an associated wavelength λ = h/p. Schrödinger sought a wave equation consistent with de Broglie's relation and with the classical energy relationship E = p²/2m + V, arriving at the equation by substituting the differential operators E → iħ∂/∂t and p → −iħ∇ into that classical energy expression.
Worked example — particle in a box
For a particle of mass m confined to an infinite square well of width L (V = 0 inside, V = ∞ outside), solving the time-independent Schrödinger equation with boundary conditions ψ(0) = ψ(L) = 0 gives quantised energy levels:
For an electron (m = 9.109 × 10⁻³¹ kg) confined to a box of width L = 1 nm = 10⁻⁹ m, the ground state (n = 1) energy is:
E₁ = (1)² × π² × (1.0546 × 10⁻³⁴)² / (2 × 9.109 × 10⁻³¹ × (10⁻⁹)²) ≈ 6.0 × 10⁻²⁰ J ≈ 0.376 eV
This simple model already captures a defining feature of quantum mechanics: energy is quantised into discrete levels rather than forming a continuum, purely as a consequence of confining the wavefunction to a finite region.
Historical note
Erwin Schrödinger (1887–1961) published the equation in a series of papers in 1926, building on Louis de Broglie's 1924 matter-wave hypothesis. Schrödinger shared the 1933 Nobel Prize in Physics with Paul Dirac "for the discovery of new productive forms of atomic theory." Max Born's probabilistic interpretation of |Ψ|² (1926) was initially controversial — Schrödinger himself favoured a more literal, physical-wave interpretation — but became the standard understanding within a few years and earned Born the 1954 Nobel Prize in Physics.
Related equations
- Quantum Mechanics domain hub [PENDING FULL BUILD]
- Newton's Second Law — the classical analogue
- Reduced Planck constant (Constants Reference)
Primary Sources
- Schrödinger, E. (1926). "Quantisierung als Eigenwertproblem." Annalen der Physik, 384(4), 361–376.
- Born, M. (1926). "Zur Quantenmechanik der Stoßvorgänge." Zeitschrift für Physik, 37(12), 863–867.
- Griffiths, D.J. (2018). Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, Ch. 1–2.