postulates

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Measurement postulate

An additional postulate, known as the Born rule[1] is often presented in the following form:

Any measurement of an observable a\, corresponding to some operator A\, will yield a measurement equal to one of the eigenvalues \lambda\, of A\,, which correspond to one or more eigenstates | \lambda\rangle_i\,. If the measurement is repeatedly performed on identically prepared states, the fraction of measurements yielding the value \lambda\, tends to
P(\lambda) = \sum_i \Vert \langle \Psi | \lambda\rangle_i \Vert^2\,.

The postulate is sometimes presented along with the requirement that

subsequently, the state of the system is given by the projection of the state onto the subspace spanned by \left\{\, |\lambda\rangle_i \,\right\}\,,
\sum_i | \lambda\rangle_i \langle \lambda |_i | \Psi \rangle\,,

though strictly this is only true of filtering experiments.

This postulate is distinguished from the others in that the act of measurement seems to play a preferred role. Since there is no a priori reason why the measurement process should be different from any other quantum mechanical process, this point deserves special clarification. Suppose the spin of a particle is prepared to be in the initial state \left|\Psi\right\rangle = \alpha \left|\uparrow\right\rangle + \beta \left|\downarrow\right\rangle\,, and a measurement of s_z\, is made via the Stern-Gerlach apparatus. There are two possibilities:

  • Firstly, the interpretation of a single measurement can only occur when when the particle is either in the final state \left|\uparrow\right\rangle\, or in the final state \left|\downarrow\right\rangle\,, but not in a more general superposition. It is conceivable that by interpreting the measurement, we are projecting |\Psi\rangle\, onto one or the other possibilities, and by some unknown (perhaps fundamental) mechanism, the projection is non-unitary and irreversible, and occurs with a certain probability. This process is termed wavefunction collapse.
  • Secondly, the interpretation of the measurement is necessarily a macroscopic process. It has been demonstrated that a pure state such as \left|\Psi\right\rangle\,, when coupled to a large number of other degrees of freedom (e.g. other spins), may evolve unitarily in such a way that the density matrix resembles that of an ensemble of systems, each with a definite spin state, with a certain fraction being in the state \left|\uparrow\right\rangle\, and another fraction being in the state \left|\downarrow\right\rangle\,. This process is termed environmentally-induced superselection, or einselection[2], or more generally, decoherence.

In either case, the predicted probabilities seem to be in agreement with the postulate. The second case is sometimes favoured because it makes fewer falsifiable assumptions.

back to density matrix
on to position representation

See also


References

[1]

  1. ↑ 1.0 1.1 Max Born (1926). "Zur Quantenmechanik der Stoßvorgänge (On the Quantum Mechanics of Collision)". Zeitschrift für Physik 37 (12): 863-867. DOI:10.1007/BF01397477.  Notes: English translation in Quantum theory and measurement, section I.2, J. A. Wheeler and W. H. Zurek, eds., Princeton, NJ: Princeton University Press, 1983. 
  2. ↑ Wojciech Hubert Zurek (2003). "Decoherence, einselection, and the quantum origins of the classical". Rev. Mod. Phys. 75 (3). arXiv:quant-ph/0105127. DOI:10.1103/RevModPhys.75.715. 
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