Measurement in quantum mechanics

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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, 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.

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

  1. W. Hubert Zurek (July, 2003). "Decoherence, einselection, and the quantum origins of the classical". Rev Mod Phys vol. 75: pp. 715 ff.
  2. 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. 
  3. 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.