Schrödinger Equation
The Schrödinger equation is the fundamental equation of motion in non-relativistic quantum mechanics, describing how the quantum state (wave function) of a physical system evolves over time. Its time-independent form is used to find the allowed energy levels and stationary states of quantum systems such as atoms and molecules. Solutions to the Schrödinger equation yield wave functions from which all measurable properties of a quantum system, including energy eigenvalues, transition probabilities, and electron densities, can be derived.
Key Formula
iℏ × ∂Ψ/∂t = Ĥ × Ψ
LaTeX: i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)
| Symbol | Meaning | Unit |
|---|---|---|
| i | Imaginary unit (√−1) | Dimensionless |
| ℏ | Reduced Planck's constant (1.055 × 10⁻³⁴) | J·s |
| Ψ | Wave function of the system | Dimensionless (normalised) |
| t | Time | Seconds (s) |
| Ĥ | Hamiltonian operator (total energy operator) | Joules (J) |
Worked Example
Problem
For a particle in a one-dimensional infinite potential well of width L = 1.0 × 10⁻¹⁰ m, find the ground-state energy of an electron (n = 1).
Solution
Step 1: The time-independent Schrödinger equation for a particle in a box gives energy eigenvalues: Eₙ = (n²π²ℏ²) / (2mL²) Step 2: Substitute values for ground state (n = 1). n = 1, m = 9.11 × 10⁻³¹ kg, L = 1.0 × 10⁻¹⁰ m, ℏ = 1.055 × 10⁻³⁴ J·s Step 3: Calculate numerator. n²π²ℏ² = 1 × (3.1416)² × (1.055 × 10⁻³⁴)² = 9.870 × 1.113 × 10⁻⁶⁸ = 1.099 × 10⁻⁶⁷ J²·s² Step 4: Calculate denominator. 2mL² = 2 × 9.11 × 10⁻³¹ × (1.0 × 10⁻¹⁰)² = 1.822 × 10⁻⁵⁰ kg·m² Step 5: Divide. E₁ = 1.099 × 10⁻⁶⁷ / 1.822 × 10⁻⁵⁰ = 6.03 × 10⁻¹⁸ J
Answer
E₁ ≈ 6.03 × 10⁻¹⁸ J ≈ 37.7 eV (ground-state energy of electron in 1D box)
Forms of the Schrödinger Equation
| Form | Equation | Use Case | Output |
|---|---|---|---|
| Time-dependent | iℏ ∂Ψ/∂t = ĤΨ | Dynamic quantum systems | Time evolution of Ψ |
| Time-independent | ĤΨ = EΨ | Stationary states | Energy eigenvalues E |
| Particle in a box (1D) | Eₙ = n²π²ℏ²/2mL² | Quantum confinement | Discrete energy levels |
| Hydrogen atom | Ĥψ = Eψ (spherical) | Atomic orbitals | Quantum numbers n, l, m |
| Harmonic oscillator | Eₙ = (n + ½)ℏω | Molecular vibrations | Vibrational energy |
Interactive Tools
PhET Quantum Bound States
Solve the Schrödinger equation visually for various potential wells
Open ToolWolframAlpha Schrödinger Equation
Compute energy eigenvalues and wave functions symbolically
Open ToolBrilliant – Schrödinger Equation
Guided derivations and applications of the Schrödinger equation
Open Tool
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Related Terms
Wave Function
The wave function (denoted Ψ) is a mathematical function in quantum mechanics that completely describes the quantum state of a particle or system. Its squared modulus |Ψ|² gives the probability density for finding the particle at a given position and time, as interpreted by Max Born in 1926. The wave function must be continuous, single-valued, and square-integrable (normalised so that the total probability integrates to one), and it evolves deterministically according to the Schrödinger equation.
Quantum Mechanics
Quantum mechanics is the fundamental theory of physics that describes the behaviour of matter and energy at the scale of atoms and subatomic particles, where classical Newtonian mechanics breaks down. It introduces concepts such as quantisation of energy, wave-particle duality, and the probabilistic nature of physical observables. Quantum mechanics underpins modern technologies including semiconductors, lasers, MRI machines, and quantum computing.
Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that it is fundamentally impossible to simultaneously determine both the exact position and exact momentum of a quantum particle with arbitrary precision; the more precisely one is known, the less precisely the other can be known. This is not a limitation of measurement instruments but an intrinsic property of quantum systems arising from the wave nature of matter. A complementary relation exists between energy and time, and the principle has profound implications for atomic stability, electron orbitals, and the zero-point energy of quantum systems.
Named after Austrian physicist Erwin Schrödinger (1887–1961), who formulated the equation in 1926. "Equation" from Latin "aequatio" (making equal). Schrödinger was awarded the Nobel Prize in Physics in 1933, shared with Paul Dirac, for the discovery of new productive forms of atomic theory.