Measurement in quantum mechanics
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, and in Riggs.
Measurement in quantum mechanics satisfies these requirements::
- 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, ψj)|2, where ψj 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 ψj.
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.
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. Is this uncertainty about us (the observers), or the cat? Can opening a box decide life or death?
- W. Hubert Zurek (July, 2003). "Decoherence, einselection, and the quantum origins of the classical". Rev Mod Phys vol. 75: pp. 715 ff.
- 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.