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.

Formulation

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, 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,

$(\psi _{j},\ \psi )=\int _{\Omega }\ dx\ \psi _{j}^{*}(x)\psi (x)\ ,$

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. Is this uncertainty about us (the observers), or the cat? Can opening a box decide life or death?