In quantum mechanics, wave function collapse, also called reduction of the state vector, is the process by which a wave function—initially in a quantum superposition of several eigenstates—reduces to a single eigenstate when a measurement yields a definite outcome.^[1] Collapse is one of the two ways quantum systems are commonly described as evolving in time; the other is the continuous deterministic evolution governed by the Schrödinger equation.^[1]

Visualization of quantum measurement regimes: POVM allows non-orthogonal outcomes, weak measurement minimally perturbs the state, and projective measurement collapses the state onto an eigenbasis.

Mathematical description

A quantum state may be expanded in a basis of eigenstates ${| ϕ_{i} ⟩}$ of an observable:

$| ψ ⟩ = \sum_{i} c_{i} | ϕ_{i} ⟩ .$

Here the coefficients $c_{i}$ are probability amplitudes, given by

$c_{i} = ⟨ ϕ_{i} | ψ ⟩ .$

When the observable is measured, the state is postulated to collapse to one of the eigenstates:

$| ψ ⟩ = \sum_{i} c_{i} | ϕ_{i} ⟩ \mapsto | ϕ_{i} ⟩ .$

The probability that the outcome corresponding to $| ϕ_{i} ⟩$ is obtained is

$P (i) = | c_{i} |^{2},$

with normalization

$\sum_{i} | c_{i} |^{2} = 1 .$

This collapse postulate is introduced to account for the fact that an immediately repeated measurement gives the same result.^[2]

Physical meaning

Wave function collapse connects the probabilistic quantum description with definite measurement outcomes such as position, momentum, or spin.^[3] For an individual event, only one outcome is observed, even though the pre-measurement state may be a superposition of many possibilities.

Examples include the double-slit experiment, where each particle is detected at a definite location although many events build up an interference pattern, and the Stern–Gerlach experiment, where each atom is observed in one of the discrete spin channels.^[4]

The measurement problem

The Schrödinger equation predicts a continuous evolution containing all possible outcomes in superposition, but an actual measurement yields only one definite result. This tension is known as the measurement problem of quantum mechanics.^[5]

To make predictions, orthodox quantum mechanics combines unitary evolution with the Born rule and the collapse postulate. Although this framework is extremely successful experimentally, the physical status of collapse remains debated.^[6]

Interpretations

Different interpretations of quantum mechanics treat collapse in different ways.

In the Copenhagen interpretation, collapse is taken as a fundamental part of measurement theory.
In the many-worlds interpretation, collapse does not occur; instead, all outcomes persist in different branches.
In objective-collapse theory, collapse is treated as a real physical process.
In approaches based on quantum decoherence, interaction with the environment explains why classical alternatives appear, though decoherence by itself does not select a single outcome.^[7]

History

The idea of wave function reduction appeared early in the development of quantum mechanics. Werner Heisenberg used it in 1927 in discussing quantum measurement, and John von Neumann gave it a systematic mathematical formulation in 1932.^[8]

Physics:Quantum Measurement collapse

Mathematical description

Physical meaning

The measurement problem

Interpretations

History

See also

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