# Measurement in quantum mechanics

## Contents

In quantum mechanics, **measurement** concerns the interaction of a macroscopic measurement apparatus with an observed quantum mechanical system, and the so-called "collapse" of the wavefunction upon measurement from a superposition of possibilities to a defined state. A review can be found in Zurek,^{[1]} and in Riggs.^{[2]}

## Formulation

Measurement in quantum mechanics satisfies these requirements:^{[2]}:

- the wavefunction ψ (the solution to the Schrödinger equation) is a complete description of a system
- the wavefunction evolves in time according to the time-dependent Schrödinger equation
- every observable property of the system corresponds to some linear operator
*O*with a number of eigenvalues - any measurement of the property
*O*results in an eigenvalue of*O* - the probability that the measurement will result in the
*j*-th eigenvalue is |(ψ, ψ_{j})|^{2}, where ψ_{j}corresponds to an eigenvector of*O*with the*j*-th eigenvalue, and it is assumed that |(ψ, ψ)|^{2}= 1. - a repetition of the measurement results in the same eigenvalue provided the system is not further disturbed between measurements. It is said that the first measurement has
*collapsed*the wavefunction ψ to become the eigenfunction ψ_{j}.

Here (f, g) is shorthand for the scalar product of *f* and *g*. For example,

for a single-particle wavefunction in one dimension, with ‘*’ denoting a complex conjugate, and Ω the region in which the particle is confined.

This description is a bit elliptic in that there may be several states corresponding to the eigenvalue *j*, requiring some further elaboration.

## Paradox

The interpretation of measurement in quantum mechanics has led to a number of puzzles. The most famous illustration is Schrödinger's cat, in which a random quantum event like a radioactive decay is set up to kill a cat in a box. In the microscopic description, the cat is described by a superposition of "alive" and "dead" possibilities, and we have the peculiar result that all is in a state of suspense (the cat is neither alive nor dead, but a superposition of both) until we open the box to see what has happened.^{[3]} Is this uncertainty about us (the observers), or the cat? Can opening a box decide life or death?

## Notes

- ↑
W. Hubert Zurek (July, 2003). "Decoherence, einselection, and the quantum origins of the classical".
*Rev Mod Phys***vol. 75**: pp. 715*ff*. - ↑
^{2.0}^{2.1}Peter J. Riggs (2009). “§2.3.1 The measurement problem”,*Quantum Causality: Conceptual Issues in the Causal Theory of Quantum Mechanics*. Springer, pp. 31*ff*. ISBN 9048124026. - ↑
Erwin Schrödinger (John D. Trimmer, translator) (Original published in German in
*Naturwissenschaften*1935). "The present situation in quantum mechanics; a translation of Schrödinger's "cat paradox paper"".*Proc American Phil Soc***vol. 124**: pp. 323-388.