Measurement in quantum mechanics: Difference between revisions
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*any measurement of the property ''O'' results in an eigenvalue of ''O'' | *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 {{nowrap|<nowiki>|</nowiki>(ψ, ψ<sub>j</sub>)<nowiki>|</nowiki><sup>2</sup>,}} where ψ<sub>j</sub> corresponds to an eigenvector of ''O'' with the ''j''-th eigenvalue, and it is assumed that {{nowrap|<nowiki>|</nowiki>(ψ, ψ)<nowiki>|</nowiki><sup>2</sup> <nowiki>=</nowiki> 1}}. | *the probability that the measurement will result in the ''j''-th eigenvalue is {{nowrap|<nowiki>|</nowiki>(ψ, ψ<sub>j</sub>)<nowiki>|</nowiki><sup>2</sup>,}} where ψ<sub>j</sub> corresponds to an eigenvector of ''O'' with the ''j''-th eigenvalue, and it is assumed that {{nowrap|<nowiki>|</nowiki>(ψ, ψ)<nowiki>|</nowiki><sup>2</sup> <nowiki>=</nowiki> 1}}. | ||
*a repetition of the measurement results in the same eigenvalue provided the system is not further disturbed between measurements | *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<sub>j</sub>. | ||
Here (f, g) is shorthand for the [[scalar product]] of ''f'' and ''g''. For example, | Here (f, g) is shorthand for the [[scalar product]] of ''f'' and ''g''. For example, | ||
:<math>(\psi_j,\ \psi) = \int_\Omega \ dx \ \psi_j^*(x)\psi(x) \ , </math> | :<math>(\psi_j,\ \psi) = \int_\Omega \ dx \ \psi_j^*(x)\psi(x) \ , </math> | ||
for a single-particle wavefunction in one dimension, with ‘*’ denoting a complex conjugate, and Ω the region in which the particle is confined. | 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== | ==Paradox== | ||
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{{cite journal |title=The present situation in quantum mechanics; a translation of Schrödinger's "cat paradox paper" |author=Erwin Schrödinger (John D. Trimmer, translator) |journal=Proc American Phil Soc |volume=vol. 124 |pages=pp. 323-388 |url=http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html#sect5 |date=Original published in German in ''Naturwissenschaften'' 1935}} | {{cite journal |title=The present situation in quantum mechanics; a translation of Schrödinger's "cat paradox paper" |author=Erwin Schrödinger (John D. Trimmer, translator) |journal=Proc American Phil Soc |volume=vol. 124 |pages=pp. 323-388 |url=http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html#sect5 |date=Original published in German in ''Naturwissenschaften'' 1935}} | ||
</ref> Is this uncertainty about us (the observers), or | </ref> Is this uncertainty about us (the observers), or the cat? Can opening a box decide life or death? | ||
==Notes== | ==Notes== | ||
<references/> | <references/>[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:00, 17 September 2024
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
- ↑ 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.