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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Wavefunctions and modes}}
'''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.<ref>[https://www.britannica.com/science/wave-function Wave function – Britannica]</ref>

[[File:Quantum_wavefunction.svg|thumb|400px|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 <math>\psi(x,t)</math>, where:
* <math>x</math> represents position
* <math>t</math> represents time

The fundamental physical meaning is given by the Born rule:

<math>|\psi(x,t)|^2 = \rho(x,t)</math>

where <math>\rho(x,t)</math> is the probability density of finding the particle at position <math>x</math> at time <math>t</math>.<ref>[https://openstax.org/details/books/university-physics-volume-3 The Wave Function – OpenStax]</ref>

== Schrödinger equation ==

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

<math>i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi</math>

where:
* <math>\hbar</math> is the reduced Planck constant
* <math>\hat{H}</math> is the Hamiltonian operator

This equation determines how quantum states evolve over time.<ref>[https://www.britannica.com/science/Schrodinger-equation Schrödinger equation – Britannica]</ref>

== Normalization ==

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

<math>\int |\psi(x,t)|^2 dx = 1</math>

This ensures a consistent probabilistic interpretation.<ref>[https://openstax.org/details/books/university-physics-volume-3 Normalization of the Wave Function – OpenStax]</ref>

== Physical interpretation ==

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

* <math>|\psi|^2</math> gives measurable probabilities
* The phase of <math>\psi</math> influences interference effects
* Superposition of wavefunctions leads to quantum interference

This interpretation distinguishes quantum mechanics from classical physics.<ref>[https://plato.stanford.edu/entries/qm/ Quantum Mechanics – Stanford Encyclopedia of Philosophy]</ref>

== Wavefunctions and boundary conditions ==

Wavefunctions must satisfy physical constraints:

* Continuity of <math>\psi</math>
* 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.<ref>[https://openstax.org/details/books/university-physics-volume-3 Infinite Potential Well – OpenStax]</ref>

== 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.<ref>[https://www.britannica.com/science/quantum-mechanics-physics Quantum mechanics – Britannica]</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Wavefunction|1}}

Physics:Quantum Wavefunction - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template