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Latest revision as of 19:52, 8 May 2026

Quantum wavefunction is a mathematical function that fully describes the quantum state of a physical system, encoding the probability amplitude for all measurable properties of the system.[1]

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

Mathematical definition

The wavefunction is typically denoted by ψ(x,t), where:

  • x represents position
  • t represents time

The fundamental physical meaning is given by the Born rule:

|ψ(x,t)|2=ρ(x,t)

where ρ(x,t) is the probability density of finding the particle at position x at time t.[2]

Schrödinger equation

The time evolution of the wavefunction is governed by the Schrödinger equation:

iℏ∂ψ∂t=H^ψ

where:

  • ℏ is the reduced Planck constant
  • H^ is the Hamiltonian operator

This equation determines how quantum states evolve over time.[3]

Normalization

The wavefunction must be normalized so that the total probability equals one:

∫|ψ(x,t)|2dx=1

This ensures a consistent probabilistic interpretation.[4]

Physical interpretation

The wavefunction itself is generally complex-valued and not directly observable. Instead:

  • |ψ|2 gives measurable probabilities
  • The phase of ψ influences interference effects
  • Superposition of wavefunctions leads to quantum interference

This interpretation distinguishes quantum mechanics from classical physics.[5]

Wavefunctions and boundary conditions

Wavefunctions must satisfy physical constraints:

  • Continuity of ψ
  • Continuity of its derivative (except at singular potentials)
  • Boundary conditions determined by the system (e.g., particle in a box)

These conditions lead to quantization of allowed energy levels.[6]

Applications

Wavefunctions are central to all areas of quantum physics:

  • Atomic and molecular structure
  • Quantum tunneling
  • Semiconductor physics
  • Quantum computing

They provide the fundamental link between mathematical formalism and experimental observations.[7]

See also

Table of contents (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ Wave function – Britannica
  2. ↑ The Wave Function – OpenStax
  3. ↑ Schrödinger equation – Britannica
  4. ↑ Normalization of the Wave Function – OpenStax
  5. ↑ Quantum Mechanics – Stanford Encyclopedia of Philosophy
  6. ↑ Infinite Potential Well – OpenStax
  7. ↑ Quantum mechanics – Britannica
Author: Harold Foppele

Source attribution: Quantum Wavefunction

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