Abstract
This article re-examines Schrödinger’s charge density hypothesis, according to which the charge of an electron is distributed in the whole space, and the charge density in each position is proportional to the modulus squared of the wave function of the electron there. It is shown that the charge distribution of a quantum system can be measured by protective measurements as expectation values of certain observables, and the results as predicted by quantum mechanics confirm Schrödinger’s original hypothesis. Moreover, the physical origin of the charge distribution is also investigated. It is argued that the charge distribution of a quantum system is effective, that is, it is formed by the ergodic motion of a localized particle with the charge of the system.
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Notes
Note that Born’s interpretation and Schrödinger’s charge density hypothesis can be reconciled (see later analysis and Gao 2017).
Another protection scheme is to introduce a protective potential such that the measured wave function of a quantum system is a nondegenerate energy eigenstate of the Hamiltonian of the system with finite gap to neighboring energy eigenstates (Aharonov and Vaidman 1993). By this scheme, the measurement of an observable is required to be weak and adiabatic.
Note that after the measurement the pointer wavepacket does not spread, and the width of the wavepacket is the same as the initial width. This ensures that the pointer shift can represent a valid measurement result.
Similarly, we can protectively measure another observable \(B ={\hbar \over {2mi}}(A\nabla + \nabla A)\). The measurements will give the electric flux density \(j_Q(x,t) ={\hbar Q\over {2mi}}(\psi ^* \nabla \psi - \psi \nabla \psi ^* )\) everywhere in space. Moreover, we can also protectively measure the charge density (and electric flux density) of a many-body system, and the density turns out to be the same as that given by Schrödinger. For example, a protective measurement of \(A_1+A_2\) in a two-particle state \(\psi (x_1,x_2,t)\) yields \(\langle A_1+A_2\rangle = Q_1\int _{V}\int _{-\infty }^{+\infty }|\psi (x_1,x_2,t)|^2 {\hbox {d}}x_2{\hbox {d}}x_1+Q_2\int _{V}\int _{-\infty }^{+\infty }|\psi (x_1,x_2,t)|^2 {\hbox {d}}x_1{\hbox {d}}x_2\). When \(v \rightarrow 0\) we can find the charge density is \(\rho _Q(x,t)=Q_1\int _{-\infty }^{+\infty }|\psi (x,x_2,t)|^2 {\hbox {d}}x_2+Q_2\int _{-\infty }^{+\infty }|\psi (x_1,x,t)|^2{\hbox {d}}x_1\).
Note that in Nelson’s stochastic mechanics, the particle, which is assumed to undergo Brownian motion, moves only within a region bounded by the nodes (Nelson 1966). Obviously this sort of motion is not ergodic and cannot generate the right charge distribution. This conclusion also holds true for the motion of particles in Bohm’s theory (Bohm 1952), as well as in some variants of stochastic mechanics and Bohm’s theory (Bell 1986; Vink 1993).
Moreover, these two parts will be also entangled and their wave function be defined in a six-dimensional configuration space.
It is a fundamental assumption in physics that a physical entity being at an instant has no interactions with itself being at another instant, while two distinct physical entities may have interactions with each other.
Note that this argument does not assume that real charges which exist at the same time are classical charges and they have classical interaction. By contrast, the Schrödinger–Newton equation, which was proposed by Diósi (1984) and Penrose (1998), treats the mass distribution of a quantum system as classical.
Besides, for normalized wave functions, the probability current density must also be equal to the formed charge flux density divided by the charge of the particle.
This result may have implications for the ontological meaning of the wave function. For further discussion see Gao (2017).
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I am grateful to two anonymous referees of this journal for their insightful comments, constructive criticisms and helpful suggestions.
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Gao, S. Is an Electron a Charge Cloud? A Reexamination of Schrödinger’s Charge Density Hypothesis. Found Sci 23, 145–157 (2018). https://doi.org/10.1007/s10699-017-9521-3
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DOI: https://doi.org/10.1007/s10699-017-9521-3