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Measurement of the Schrödinger Wave of a Single Particle

Yakir Aharonov, Lev Vaidman A

Published Jan 1, 1993• 122 citations
OpenAlex

Abstract

We show that it is possible to measure the Schrödinger wave of a single quantum system. This provides a strong argument for associating physical reality with the quantum state of a single system, and challenges the usual assumption that the quantum state has physical meaning only for an ensemble of identical systems. While the Schrödinger wave is a basic element of quantum theory, it is generally believed that one cannot associate physical reality to the wave of a single particle. Rather, the Schrödinger wave is often viewed as a mathematical tool for calculating the probabilities of various outcomes of certain experiments when many such experiments are performed on an ensemble of identical systems, all in the same quantum state. Several arguments seem to support this point of view: i) We have never seen the quantum state of a single particle in a laboratory. Indeed, while a wave is typically spread over a region of space, we never see a particle simultaneously in several distinct locations. ii) If we could see a quantum state, we could presumably distinguish it from any other quantum state, but the unitary time evolution of states in quantum

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Based on an analysis of protective measurements, we show that the quantum state represents the physical state of a single quantum system. This result is more definite than the PBR theorem [Pusey, Barrett, and Rudolph, Nature Phys. 8, 475 (2012)]. The physical meaning of the quantum state is an important interpreta-tive problem of quantum mechanics. A long-standing question is whether a pure state relates only to an ensemble of identically prepared systems or di-rectly to the state of a single system. Recently, Pusey, Barrett and Rudolph (PBR) demonstrated that under an independence assumption, the quantum state is a representation of the physical state of a single quantum system [1]. This poses a further interesting question, namely whether ψ-ontology can be argued without resorting to nontrivial assumptions such as the in-dependence assumption (cf. Ref. [2-4]). In this Letter, we will show that protective measurements [5,6] already provide such an argument. The meaning of the quantum state is usually analyzed in the context of conventional impulsive measurements of an ensemble of identically prepared systems. However, it has been known that the quantum state of a single prepared system can be protectively measured [5-9]. During a protective measurement, the measured system is protected by an appropriate procedure (e.g. via the quantum Zeno effect) so that its quantum state neither changes nor becomes entangled with the quantum state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables from a single quantum system, even if the system is initially not in an eigenstate of the measured observable, and in particular, the quantum state of the system can also be measured as expectation values of certain observables. It is expected that protective measurements will be realized in the near future with the rapid development of quantum technologies.

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All alternative interpretations have at their roots some form of disgust at the Copenhagen–Gottingen philosophy, and all have a desire to seek an alternative truth for the understanding of quantum mechanics. The alternative view is of an intrinsic reality prior to the measurement, as discussed by A. Einstein, and by most of the founders of the wave theory of quantum mechanics. A key point of Neils Bohr’s arguments with Einstein, particularly with regard to the Einstein–B. Podolsky–N. Rosen type of experiments, was that one could not make multiple measurements on the same particle, or quantum system. In his view, the measurement itself defined the quantum system. With the arrival of the Schrodinger equation, quantum mechanics had a causal, deterministic equation of motion for the wave function. Perturbation theory, based upon the Schrodinger equation, provided transition details that undermined the assertions of quantum jumps.

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The concept of wave-particle duality, which is a key element of quantum theory, has been remarkably found to manifest itself in several experimental realizations as in the famous double-slit experiment. In this specific case, a single particle seems to travel through two separated slits simultaneously. Nevertheless, it is never possible to measure it in both slits, which naturally appears as a manifestation of the collapse postulate. In this respect, one could as well ask if it is possible to "perceive" the presence of the particle at the two slits simultaneously, once its collapse could be avoided. In this article, we use the recently proposed entanglement mediation protocol to provide a positive answer to this question. It is shown that a photon which behaves like a wave, i.e., which seems to be present in two distant locations at the same time, can modify two existing physical realities in these locations. Calculations of the "weak trace" left by such photon also enforce the validity of the present argumentation.

arXiv:1608.00868•Scientific Reports•Jan 3, 2017•11 citations
We show that it is possible to measure the Schrödinger wave of a single quantum system. This provides a strong argument for associating physical reality with the quantum state of a single system, and challenges the usual assumption that the quantum state has physical meaning only for an ensemble of identical systems. While the Schrödinger wave is a basic element of quantum theory, it is generally believed that one cannot associate physical reality to the wave of a single particle. Rather, the Schrödinger wave is often viewed as a mathematical tool for calculating the probabilities of various outcomes of certain experiments when many such experiments are performed on an ensemble of identical systems, all in the same quantum state. Several arguments seem to support this point of view: i) We have never seen the quantum state of a single particle in a laboratory. Indeed, while a wave is typically spread over a region of space, we never see a particle simultaneously in several distinct locations. ii) If we could see a quantum state, we could presumably distinguish it from any other quantum state, but the unitary time evolution of states in quantum
Protective measurements yield properties of the quantum state of a single quantum system without affecting the quantum state. A protective measurement involves adiabatic coupling to the measuring device together with a procedure to protect the state from changing. For nondegenerate energy eigenstates the protection is provided by the system itself. In this case it is actually possible to measure the Schrödinger wave via measurements on a single system. This fact provides an argument in favor of associating physical reality with a quantum state of a single system, challenging the usual ensemble interpretation. We also believe that the complete description of a quantum system requires a two-state vector formalism involving (in addition to the usual one)a future quantum state evolving backwards in time. Protective measurements testing the two-state vector reality are constructed.
Although quantum mechanics is one of our most successful physical theories, there has been a long-standing debate about the interpretation of the wave function--the central object of the theory. Two prominent views are that (i) it corresponds to an element of reality, i.e., an objective attribute that exists before measurement, and (ii) it is a subjective state of knowledge about some underlying reality. A recent result [M. F. Pusey, J. Barrett, and T. Rudolph, arXiv:1111.3328] has placed the subjective interpretation into doubt, showing that it would contradict certain physically plausible assumptions, in particular, that multiple systems can be prepared such that their elements of reality are uncorrelated. Here we show, based only on the assumption that measurement settings can be chosen freely, that a system's wave function is in one-to-one correspondence with its elements of reality. This also eliminates the possibility that it can be interpreted subjectively.
Based on an analysis of protective measurements, we show that the quantum state represents the physical state of a single quantum system. This result is more definite than the PBR theorem [Pusey, Barrett, and Rudolph, Nature Phys. 8, 475 (2012)]. The physical meaning of the quantum state is an important interpreta-tive problem of quantum mechanics. A long-standing question is whether a pure state relates only to an ensemble of identically prepared systems or di-rectly to the state of a single system. Recently, Pusey, Barrett and Rudolph (PBR) demonstrated that under an independence assumption, the quantum state is a representation of the physical state of a single quantum system [1]. This poses a further interesting question, namely whether ψ-ontology can be argued without resorting to nontrivial assumptions such as the in-dependence assumption (cf. Ref. [2-4]). In this Letter, we will show that protective measurements [5,6] already provide such an argument. The meaning of the quantum state is usually analyzed in the context of conventional impulsive measurements of an ensemble of identically prepared systems. However, it has been known that the quantum state of a single prepared system can be protectively measured [5-9]. During a protective measurement, the measured system is protected by an appropriate procedure (e.g. via the quantum Zeno effect) so that its quantum state neither changes nor becomes entangled with the quantum state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables from a single quantum system, even if the system is initially not in an eigenstate of the measured observable, and in particular, the quantum state of the system can also be measured as expectation values of certain observables. It is expected that protective measurements will be realized in the near future with the rapid development of quantum technologies.
All alternative interpretations have at their roots some form of disgust at the Copenhagen–Gottingen philosophy, and all have a desire to seek an alternative truth for the understanding of quantum mechanics. The alternative view is of an intrinsic reality prior to the measurement, as discussed by A. Einstein, and by most of the founders of the wave theory of quantum mechanics. A key point of Neils Bohr’s arguments with Einstein, particularly with regard to the Einstein–B. Podolsky–N. Rosen type of experiments, was that one could not make multiple measurements on the same particle, or quantum system. In his view, the measurement itself defined the quantum system. With the arrival of the Schrodinger equation, quantum mechanics had a causal, deterministic equation of motion for the wave function. Perturbation theory, based upon the Schrodinger equation, provided transition details that undermined the assertions of quantum jumps.
The concept of wave-particle duality, which is a key element of quantum theory, has been remarkably found to manifest itself in several experimental realizations as in the famous double-slit experiment. In this specific case, a single particle seems to travel through two separated slits simultaneously. Nevertheless, it is never possible to measure it in both slits, which naturally appears as a manifestation of the collapse postulate. In this respect, one could as well ask if it is possible to "perceive" the presence of the particle at the two slits simultaneously, once its collapse could be avoided. In this article, we use the recently proposed entanglement mediation protocol to provide a positive answer to this question. It is shown that a photon which behaves like a wave, i.e., which seems to be present in two distant locations at the same time, can modify two existing physical realities in these locations. Calculations of the "weak trace" left by such photon also enforce the validity of the present argumentation.