Wave function collapse

For the constraint-solving algorithm, see Wave function collapse (algorithm).

In quantum mechanics, wave function collapse, also called reduction of the state vector,^[1] occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an observation, and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation.^[2]

Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation throughout this apparent collapse.^[3] More importantly, this is not enough to explain actual wave function collapse, as decoherence does not reduce it to a single eigenstate.^[4][5]

Historically, Werner Heisenberg was the first to use the idea of wave function reduction to explain quantum measurement.^{[6] [citation needed]}

Mathematical description

For an explanation of the notation used, see Bra–ket notation. For details on this formalism, see Quantum state.

In quantum mechanics each measurable physical quantity of a quantum system is called an observable which, for example, could be the position ${\displaystyle r}$  and the momentum ${\displaystyle p}$  but also energy ${\displaystyle E}$ , ${\displaystyle z}$  components of spin (${\displaystyle s_{z}}$ ), and so on. The observable acts as a linear function on the states of the system; its eigenvectors correspond to the quantum state (i.e. eigenstate) and the eigenvalues to the possible values of the observable. The collection of eigenstates/eigenvalue pairs represent all possible values of the observable. Writing ${\displaystyle \phi _{i}}$  for an eigenstate and ${\displaystyle c_{i}}$  for the corresponding observed value, any arbitrary state of the quantum system can be expressed as a vector using bra–ket notation:

$${\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle .}$$

 

The kets ${\displaystyle \{|\phi _{i}\rangle \}}$  specify the different available quantum "alternatives", i.e., particular quantum states.

The wave function is a specific representation of a quantum state. Wave functions can therefore always be expressed as eigenstates of an observable though the converse is not necessarily true.

Collapse

To account for the experimental result that repeated measurements of a quantum system give the same results, the theory postulates a "collapse" or "reduction of the state vector" upon observation,^[7]: 566 abruptly converting an arbitrary state into a single component eigenstate of the observable:

${\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle \rightarrow |\psi '\rangle =|\phi _{i}\rangle .}$ 

where the arrow represents a measurement of the observable corresponding to the ${\displaystyle \phi }$  basis.^[8] For any single event, only one eigenvalue is measured, chosen randomly from among the possible values.

Meaning of the expansion coefficients

The complex coefficients ${\displaystyle \{c_{i}\}}$  in the expansion of a quantum state in terms of eigenstates ${\displaystyle \{|\phi _{i}\rangle \}}$ ,

$${\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle .}$$

 

can be written as an (complex) overlap of the corresponding eigenstate and the quantum state:

$${\displaystyle c_{i}=\langle \phi _{i}|\psi \rangle .}$$

 

They are called the probability amplitudes. The square modulus ${\displaystyle |c_{i}|^{2}}$  is the probability that a measurement of the observable yields the eigenstate ${\displaystyle |\phi _{i}\rangle }$ . The sum of the probability over all possible outcomes must be one:^[9]

${\displaystyle \langle \psi |\psi \rangle =\sum _{i}|c_{i}|^{2}=1.}$ 

As examples, individual counts in a double slit experiment with electrons appear at random locations on the detector; after many counts are summed the distribution shows a wave interference pattern.^[10] In a Stern-Gerlach experiment with silver atoms, each particle appears in one of two areas unpredictably, but the final conclusion has equal numbers of events in each area.

This statistical aspect of quantum measurements differs fundamentally from classical mechanics. In quantum mechanics the only information we have about a system is its wave function and measurements of the wavefunction can only give statistical information.^[7]: 17

Terminology

The two terms "reduction of the state vector" (or "state reduction" for short) and "wave function collapse" are used to describe the same concept. A quantum state is a mathematical description of a quantum system; a quantum state vector uses Hilbert space vectors for the description.^[11]: 159 Reduction of the state vector replaces the full state vector with a single eigenstate of the observable.

The term "wave function" is typically used for a different mathematical representation of the quantum state, one that uses spatial coordinates also called the "position representation".^[11]: 324 When the wave function representation is used, the "reduction" is called "wave function collapse".

The measurement problem

The Schrodinger equation describes quantum systems but does not describe their measurement. Solution to the equations include all possible observable values for measurements, but measurements only result in one definite outcome. This difference is called the measurement problem of quantum mechanics. To predict measurement outcomes from quantum solutions, the orthodox interpretation of quantum theory postulates wave function collapse and uses the Born rule to compute the probable outcomes.^[12] Despite the widespread quantitative success of these postulates scientists remain dissatisfied and have sought more detailed physical models. Rather than suspending the Schrodinger equation during the process of measurement, the measurement apparatus should be included and governed by the laws of quantum mechanics.^[13]: 127

Physical approaches to collapse

Quantum theory offers no dynamical description of the "collapse" of the wave function. Viewed as a statistical theory, no description is expected. As Fuchs and Peres put it, "collapse is something that happens in our description of the system, not to the system itself".^[14]

Various interpretations of quantum mechanics attempt to provide a physical model for collapse.^[15]: 816 Three treatments of collapse can be found among the common interpretations. The first group includes hidden variable theories like de Broglie–Bohm theory; here random outcomes only result from unknown values of hidden variables. Results from tests of Bell's theorem shows that these variables would need to be non-local. The second group models measurement as quantum entanglement between the quantum state and the measurement apparatus. This results in a simulation of classical statistics called quantum decoherence. This group includes the many worlds interpretation and consistent histories models. The third group postulates additional, but as yet undetected, physical basis for the randomness; this group includes for example the objective collapse interpretations. While models in all groups have contributed to better understanding of quantum theory, no alternative explanation for individual events has emerged as more useful than collapse followed by statistical prediction with the Born rule.^[15]: 819

The significance ascribed to the wave function varies from interpretation to interpretation, and varies even within an interpretation (such as the Copenhagen Interpretation). If the wave function merely encodes an observer's knowledge of the universe then the wave function collapse corresponds to the receipt of new information. This is somewhat analogous to the situation in classical physics, except that the classical "wave function" does not necessarily obey a wave equation. If the wave function is physically real, in some sense and to some extent, then the collapse of the wave function is also seen as a real process, to the same extent.^{[citation needed]}

Quantum decoherence

Main article: Quantum decoherence

Quantum decoherence explains why a system interacting with an environment transitions from being a pure state, exhibiting superpositions, to a mixed state, an incoherent combination of classical alternatives.^[5] This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible in the same sense as in the second law of thermodynamics: the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Decoherence is thus very important for explaining the classical limit of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them.^[4][16][5]

History

The concept of wavefunction collapse was introduced by Werner Heisenberg in his 1927 paper on the uncertainty principle, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", and incorporated into the mathematical formulation of quantum mechanics by John von Neumann, in his 1932 treatise Mathematische Grundlagen der Quantenmechanik.^[17] Heisenberg did not try to specify exactly what the collapse of the wavefunction meant. However, he emphasized that it should not be understood as a physical process.^[18] Niels Bohr also repeatedly cautioned that we must give up a "pictorial representation", and perhaps also interpreted collapse as a formal, not physical, process.^[19]

The "Copenhagen" model espoused by Heisenberg and Bohr separated the quantum system from the classical measurement apparatus. In 1932 von Neumann took a more formal approach, developing "ideal" measurement scheme^{[20][21]: 1270} that postulated that there were two processes of wave function change:

1. The probabilistic, non-unitary, non-local, discontinuous change brought about by observation and measurement (state reduction or collapse).
2. The deterministic, unitary, continuous time evolution of an isolated system that obeys the Schrödinger equation (or a relativistic equivalent, i.e. the Dirac equation).

In 1957 Hugh Everett III proposed a model of quantum mechanics that dropped von Neumann's first postulate. Everett observed that the measurement apparatus was also a quantum system and its quantum interaction with the system under observation should determine the results. He proposed that the discontinuous change is instead a splitting of a wave function representing the universe.^[21]: 1288 While Everett's approach rekindle interest in foundational quantum mechanics, it left core issues unresolved. Two key issues relate to origin of the observed classical results: what causes quantum systems to appear classical and to resolve with the observed probabilities of the Born rule.^{[21]: 1290 [20]: 5}

Beginning in 1970 H. Dieter Zeh sought a detailed quantum decoherence model for the discontinuous change without postulating collapse. Further work by Wojciech H. Zurek in 1980 lead eventually to a large number of papers on many aspects of the concept.^[22] Decoherence assumes that every quantum system interacts quantum mechanically with its environment and such interaction is not separable from the system, a concept called an "open system".^[21]: 1273 Decoherence has been shown to work very quickly and within a minimal environment, but as yet it has not succeeded in a providing a detailed model replacing the collapse postulate of orthodox quantum mechanics.^[21]: 1302

By explicitly dealing with the interaction of object and measuring instrument, von Neumann^[2] described a quantum mechanical measurement scheme consistent with wave function collapse. However, he did not prove the necessity of such a collapse. Although von Neumann's projection postulate is often presented as a normative description of quantum measurement, it was conceived by taking into account experimental evidence available during the 1930s (in particular Compton scattering was paradigmatic). Later work discussed so-called measurements of the second kind.^[23][24][25]

See also

• Arrow of time
• Interpretations of quantum mechanics
• Quantum decoherence
• Quantum interference
• Quantum Zeno effect
• Schrödinger's cat
• Stern–Gerlach experiment
• Wave function collapse (algorithm)

Notes

References

1. Penrose, Roger (May 1996). "On Gravity's role in Quantum State Reduction". General Relativity and Gravitation. 28 (5): 581–600. doi:10.1007/BF02105068. ISSN 0001-7701.
2. J. von Neumann (1932). Mathematische Grundlagen der Quantenmechanik (in German). Berlin: Springer.
   

   J. von Neumann (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press.

3. Zurek, Wojciech Hubert (2009). "Quantum Darwinism". Nature Physics. 5 (3): 181–188. arXiv:0903.5082. Bibcode:2009NatPh...5..181Z. doi:10.1038/nphys1202. S2CID 119205282.
4. Schlosshauer, Maximilian (2005). "Decoherence, the measurement problem, and interpretations of quantum mechanics". Rev. Mod. Phys. 76 (4): 1267–1305. arXiv:quant-ph/0312059. Bibcode:2004RvMP...76.1267S. doi:10.1103/RevModPhys.76.1267. S2CID 7295619.
5. Fine, Arthur (2020). "The Role of Decoherence in Quantum Mechanics". Stanford Encyclopedia of Philosophy. Center for the Study of Language and Information, Stanford University website. Retrieved 11 April 2021.
6. Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43: 172–198. Translation as 'The actual content of quantum theoretical kinematics and mechanics' here
7. Griffiths, David J.; Schroeter, Darrell F. (2018). Introduction to quantum mechanics (3 ed.). Cambridge ; New York, NY: Cambridge University Press. ISBN 978-1-107-18963-8.
8. Hall, Brian C. (2013). Quantum theory for mathematicians. Graduate texts in mathematics. New York: Springer. p. 68. ISBN 978-1-4614-7115-8.
9. Griffiths, David J. (2005). Introduction to Quantum Mechanics, 2e. Upper Saddle River, New Jersey: Pearson Prentice Hall. p. 107. ISBN 0131118927.
10. Bach, Roger; Pope, Damian; Liou, Sy-Hwang; Batelaan, Herman (2013-03-13). "Controlled double-slit electron diffraction". New Journal of Physics. 15 (3). IOP Publishing: 033018. arXiv:1210.6243. Bibcode:2013NJPh...15c3018B. doi:10.1088/1367-2630/15/3/033018. ISSN 1367-2630. S2CID 832961.
11. Messiah, Albert (1966). Quantum Mechanics. North Holland, John Wiley & Sons. ISBN 0486409244.
12. Zurek, Wojciech Hubert (2003-05-22). "Decoherence, einselection, and the quantum origins of the classical". Reviews of Modern Physics. 75 (3): 715–775. arXiv:quant-ph/0105127. doi:10.1103/RevModPhys.75.715. ISSN 0034-6861.
13. Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014). Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics]. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books. ISBN 978-0-465-06290-4.
14. Fuchs, Christopher A.; Peres, Asher (2000-03-01). "Quantum Theory Needs No 'Interpretation'". Physics Today. 53 (3): 70–71. doi:10.1063/1.883004. ISSN 0031-9228.
15. Stamatescu, Ion-Olimpiu (2009). Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.). Wave Function Collapse. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 813–822. doi:10.1007/978-3-540-70626-7_230. ISBN 978-3-540-70622-9.
16. Wojciech H. Zurek (2003). "Decoherence, einselection, and the quantum origins of the classical". Reviews of Modern Physics. 75 (3): 715. arXiv:quant-ph/0105127. Bibcode:2003RvMP...75..715Z. doi:10.1103/RevModPhys.75.715. S2CID 14759237.
17. C. Kiefer (2002). "On the interpretation of quantum theory—from Copenhagen to the present day". arXiv:quant-ph/0210152.
18. G. Jaeger (2017). ""Wave-Packet Reduction" and the Quantum Character of the Actualization of Potentia". Entropy. 19 (10): 13. Bibcode:2017Entrp..19..513J. doi:10.3390/e19100513.
19. Henrik Zinkernagel (2016). "Niels Bohr on the wave function and the classical/quantum divide". Studies in History and Philosophy of Modern Physics. 53: 9–19. arXiv:1603.00353. Bibcode:2016SHPMP..53....9Z. doi:10.1016/j.shpsb.2015.11.001. S2CID 18890207. We can thus say that, for Bohr, the collapse is not physical in the sense of a physical wave (or something else) collapsing at a point. But it is a description – in fact the best, or most complete, description – of something happening, namely the formation of a measurement record (e.g. a dot on a photographic plate).
20. Hartle, James B. "The quantum mechanics of cosmology." Notes from the lectures by the author at the 7th Jerusalem Winter School 1990 on Quantum Cosmology and Baby Universes. arXiv:1805.12246 (2018).
21. Schlosshauer, Maximilian (2005-02-23). Decoherence, the measurement problem, and interpretations of quantum mechanics. Vol. 76. pp. 1267–1305. doi:10.1103/RevModPhys.76.1267. ISSN 0034-6861.
22. Camilleri, Kristian (2009-12-01). "A history of entanglement: Decoherence and the interpretation problem". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. On The History Of The Quantum. 40 (4): 290–302. doi:10.1016/j.shpsb.2009.09.003. ISSN 1355-2198.
23. W. Pauli (1958). "Die allgemeinen Prinzipien der Wellenmechanik". In S. Flügge (ed.). Handbuch der Physik (in German). Vol. V. Berlin: Springer-Verlag. p. 73.
24. L. Landau & R. Peierls (1931). "Erweiterung des Unbestimmtheitsprinzips für die relativistische Quantentheorie". Zeitschrift für Physik (in German). 69 (1–2): 56–69. Bibcode:1931ZPhy...69...56L. doi:10.1007/BF01391513. S2CID 123160388.)
25. Discussions of measurements of the second kind can be found in most treatments on the foundations of quantum mechanics, for instance, J. M. Jauch (1968). Foundations of Quantum Mechanics. Addison-Wesley. p. 165.; B. d'Espagnat (1976). Conceptual Foundations of Quantum Mechanics. W. A. Benjamin. pp. 18, 159.; and W. M. de Muynck (2002). Foundations of Quantum Mechanics: An Empiricist Approach. Kluwer Academic Publishers. section 3.2.4.

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