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.
Key Formula
P(x,t) = |Ψ(x,t)|² = Ψ*(x,t) × Ψ(x,t)
LaTeX: P(x,t) = |\Psi(x,t)|^2 = \Psi^*(x,t)\,\Psi(x,t)
| Symbol | Meaning | Unit |
|---|---|---|
| P(x,t) | Probability density of finding particle at position x at time t | m⁻¹ |
| Ψ(x,t) | Wave function (complex-valued) | Dimensionless (normalised) |
| Ψ*(x,t) | Complex conjugate of the wave function | Dimensionless |
Worked Example
Problem
A particle in a 1D box of width L has ground-state wave function Ψ₁(x) = √(2/L) sin(πx/L). Verify normalisation and find the probability of locating the particle in the left half (0 to L/2).
Solution
Step 1: Verify normalisation. ∫₀ᴸ |Ψ₁(x)|² dx = (2/L) ∫₀ᴸ sin²(πx/L) dx Using ∫ sin²(u) du = u/2 − sin(2u)/4: = (2/L) × [L/2] = 1 ✓ (normalised) Step 2: Find probability in left half. P(0 to L/2) = ∫₀^(L/2) (2/L) sin²(πx/L) dx = (2/L) × [L/4 − (L/4π)sin(π)] = (2/L) × L/4 = 1/2
Answer
Probability = 0.5 (50%), as expected by symmetry for the ground state
Properties of the Wave Function
| Property | Requirement | Reason | Consequence of Violation |
|---|---|---|---|
| Single-valued | One value per point | Physical uniqueness | Ambiguous probability |
| Continuous | No sudden jumps | Finite kinetic energy | Infinite momentum |
| Normalised | ∫|Ψ|² dx = 1 | Total probability = 1 | Non-physical state |
| Square-integrable | Finite over all space | Particle must exist | Non-normalisable |
| Smooth derivative | dΨ/dx continuous | Finite potential regions | Energy diverges |
Interactive Tools
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Related Terms
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.
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.
Atomic Orbital
An atomic orbital is a mathematical function describing the wave-like behaviour and probable location of an electron in an atom, representing a region of space where there is a high probability (typically 90–95%) of finding the electron. Orbitals are characterised by three quantum numbers (n, l, mₗ) and have distinct shapes: s-orbitals are spherical, p-orbitals are dumbbell-shaped, and d- and f-orbitals have more complex geometries. Atomic orbitals form the basis for understanding electron configurations, chemical bonding, and molecular orbital theory.
The term "wave function" reflects Schrödinger's wave-mechanical formulation of quantum mechanics (1926). The mathematical symbol Ψ (psi) is from the Greek alphabet. Max Born's probability interpretation of |Ψ|² was introduced in 1926 and earned him the Nobel Prize in 1954.