Back to Bohr: Quantum Jumps in Schrödinger’s Wave Mechanics

Abstract

The measurement problem of quantum mechanics concerns the question as to under which circumstances coherent wave evolution becomes disrupted to produce eigenstates of observables, instead of evolving superpositions of eigenstates. The problem already needs to be addressed within wave mechanics, before second quantization, because low-energy interactions can be dominated by particle-preserving potential interactions. We discuss a scattering array of harmonic oscillators, which can detect particles penetrating the array through interaction with a short-range potential. Evolution of the wave function of scattered particles, combined with Heisenberg’s assertion that quantum jumps persist in wave mechanics, indicates that the wave function will collapse around single oscillator sites if the scattering is inelastic, while it will not collapse around single sites for elastic scattering. The Born rule for position observation is then equivalent to the statement that the wave function for inelastic scattering amounts to an epistemic superposition of possible scattering states, in the sense that it describes a sum of probability amplitudes for inelastic scattering off different scattering centers, whereas, at most, one inelastic scattering event can happen at any moment in time. Within this epistemic interpretation of the wave function, the actual underlying inelastic scattering event corresponds to a quantum jump, whereas the continuously evolving wave function only describes the continuous evolution of probability amplitudes for scattering off different sites. Quantum jumps then yield definite position observations, as defined by the spatial resolution of the oscillator array.

1. Introduction

The problem of definite measurement outcomes from observations of quantum systems, which initially exist in a superposition of many possible eigenstates of the observed variable, is best illustrated and discussed in terms of basic examples. Unfortunately, this comes at the price of some redundancy with previous discussions, and at the risk of annoying knowledgeable readers. My excuses are twofold: On the one hand, the measurement problem is too important from a principal perspective to only be left to the initiated. On the other hand, staying close to actual manifestations of the measurement problem improves the clarity and efficiency of the discussion. Too much abstraction, on the other hand, can easily lead to elaborate proposals, which instantly fall apart when gauged against actual observations.

Position observations for electrons provide particularly neat illustrations of the measurement problem. For example, in the single-electron diffraction observations by Bach et al., single electrons from a 2-micrometer-wide collimated beam diffract off two slits (each 62 nm wide and 272 nm apart) to create a more than 4 mm wide well-resolved two-slit interference pattern at the location of a microchannel plate (MCP) detector, after accumulation of many single-electron signals [1]. The authors give $238.2\pm 6.6$ mm for the position resolution of the imaging system. The standard interpretation of quantum mechanics of this experiment, to which we adhere, is that the single-electron position probability density ${\left|\psi (\mathit{x},t)\right|}^2$, at the location of the detector, has a lateral width of more than 4 mm, with seven clearly identifiable peaks, after traversing the double-slit and traveling all the way to the detector. However, when an electron described by that wave function hits the detector, it creates a particle-like signal in the sense that the signal is confined to within the spatial resolution of the detector, which is much smaller than the inherent width of the wave function. This is exactly how we define a pointlike particle signal [2]: The spatial width of the signal is determined by detector resolution, but not by the width of the incoming wave function. Stated differently, the impact of the electron on the MCP creates a pointer state that corresponds to excitation of one channel (or, at most, a few channels within the spatial resolution of the imaging system), but not to excitation of several distinct macroscopically separated channels. The latter outcome would sometimes also be denoted as a superposition of pointer states, which would correspond to simultaneous detection of several positions of the electron.

The measurement problem for position observation then concerns the following question: The electron wave function coherently passes through the double-slit to evolve into the macroscopically wide wave of a double-slit interference pattern, and yet, when that wave hits the detector, it triggers a pointlike signal with a much smaller width. Why does the wave not light up an interference pattern across the MCP upon every impact? And how come that accumulation of many consecutive pointlike impacts creates that very interference pattern?

At this point we might just sit back and relax and remind ourselves of the Born interpretation of the wave function. This is exactly what the wave function is supposed to do: “If we ask the electron where it is, it responds with a pointlike signal with a probability density distribution given by ${\left|\psi (\mathit{x},t)\right|}^2$”. Alas, it is not that simple. Both the passage through the double slit and the registration in the MCP detector are ultimately scattering events. How would the electron (or its wave function) “know” that scattering off the double-slit is not a detection event, and therefore scattering occurs coherently off both slits in a wavelike manner, but the scattering in the MCP is a detection event, and therefore scattering and the subsequent triggering of an electron avalanche occurs in one channel in a particle-like manner, instead of coherently exciting many channels over a 4 mm wide area?

The question is even more striking if we replace the double slit with elastic scattering off a metal surface, as in a low-energy electron diffraction (LEED) event. Electron scattering off the metal sample produces a macroscopic wave function that can (upon accumulation of many single-electron signals) create a macroscopic interference pattern once the electrons scatter in a fluorescent screen. Again, the electron wave function after scattering off the metal surface is a coherent superposition of scattering off different scattering centers, but the final signal is produced by scattering off only a single scattering center in the fluorescent screen. We were focusing on the double-slit example, because it is so well-documented as a single-electron diffraction experiment. To my knowledge, LEED has not be performed as a deliberate single-electron diffraction experiment, although this would be very nice, too. Of course, one would again expect to see gradual accumulation of the interference pattern from coherent elastic scattering off a metal surface, but the demonstration of initial elastic wave scattering followed by inelastic particle scattering would be even more striking.

What the two different kinds of scattering events separate in either case is that the first scattering event through the double-slit in Ref. [1], or off the metal sample in the LEED apparatus, does not change the energy of the scattered electron, whereas detection through an electron avalanche in the MCP [1], or through fluorescence or bremsstrahlung emission from the fluorescent screen in LEED, requires energy loss of the detected electron. We denote the scattering as elastic if the electron’s energy does not change, and as inelastic otherwise.

We note, in particular, that the first elastic scattering in each case leads to interference effects, and can therefore be denoted as wavelike, whereas the second inelastic scattering in each case provides a particle-like position signal. The energy transfer between the particle and the detector apparently breaks the coherent smooth evolution of the wave function.

It has been shown in the framework of decoherence theory that repeated and continuing elastic scattering off photons, or off atoms or molecules, can localize scattered particles [3,4,5] (see also [6,7] for excellent discussions of decoherence theory). However, the observation of interference patterns in the double-slit or LEED experiments shows that localization through elastic scattering cannot apply in those cases. Furthermore, it likely that it also cannot apply at the final detection stage through inelastic scattering, because the elastically scattered wave function emerging from the double-slit, or from the metal sample in LEED, covers a macroscopic area on the detector. Environmental monitoring through ongoing collisions in the detector cannot explain why the macroscopically extended incoming wave function will never produce two spatially separated position signals when hitting the detector.

On the other hand, it has been shown that inelastic potential scattering off the short-range potential of localized scattering centers already leads to a contraction of the scattered wave function after a single scattering event [8]. This does not require the creation of a collision term from ongoing scattering in the evolution equation of the density matrix of the scattered system, but uses the fact that the scattering matrix for scattering off localized short-range scattering centers imprints the length scale of the scattering centers onto the scattered wave function. However, that by itself does not necessarily imply a breakdown of smooth evolution of the wave function during inelastic scattering. On the other hand, the breakdown of smooth evolution is required to explain the absence of superposition effects from inelastic scattering during the position observation: the many-peaked probability density ${\left|\psi (\mathit{x},t)\right|}^2$ from interference effects after the initial elastic scattering does not generate more than one position signal at a time upon inelastic scattering in the detector. The scattered wave function contracts only here or there, but it never contracts both here and there. This absence of superposition effects after inelastic scattering is a clear indication of the disruption of smooth wave function evolution during an energy exchange between one quantum system and another quantum system, or equivalently, between two components of a quantum system.

A break in the smooth evolution of a system due to energy exchange is a quantum jump, and it can be argued in the framework of second quantization that the second-quantized time-dependent Schrödinger equation does not describe the smooth evolution of quantum states, but instead describes the smooth evolution of scattering matrix elements as probability amplitudes for quantum jumps between different sectors in Fock space [2,9].

However, this observation is not sufficient to resolve the measurement problem, because low-energy interactions, including low-energy interactions between particles and detectors, are dominated by particle-preserving potential interactions, and do not inherently require second quantization nor transitions between different sectors in Fock space. (For example, for electron scattering with electron kinetic energies $K_e\ll m_e{c}^2$, the ratio of the scattering matrix elements for electron scattering $\mathit{p}\to {\mathit{p}}^{\prime }$ off atoms or other electrons, from photon exchange versus Coulomb scattering in a Coulomb gauge, is $|S_{fi}^{\left(\gamma \right)}/S_{fi}^{\left(C\right)}|\lesssim |\mathit{p}+{\mathit{p}}^{\prime }{|}^2/4m_e^2{c}^2$ in the low-energy limit. See, e.g., Secs. 23.2, especially Equations 23.52 and 23.4 in Ref. [10]). Transitions between sectors of Fock space are helpful as clear indicators of quantum jumps, but we also need quantum jumps within the confines of Schrödinger’s wave mechanics to resolve the measurement problem. Indeed, within the framework of wave mechanics, Heisenberg had asked, soon after the inception of the Schrödinger equation, whether it only disguises quantum jumps through a smooth statistical evolution [11]. He repeated that suspicion in his lectures at the University of Chicago (see Appendix 5 in [12]). Heisenberg’s suspicion leads to a tempting resolution of the measurement problem: every observation necessitates an energy transfer between a detector and an observed system, and if this energy transfer disrupts the smooth wavelike evolution through a quantum jump, the annoying problem of the superposition of different pointer states as a consequence of coherent Schrödinger evolution disappears.

To explain this, we introduce an oscillator array as a model system for low-energy particle scattering in Section 2. The oscillator array serves both as a particle diffractor through elastic scattering, and as a particle detector through inelastic scattering. Interpreting the wave function after inelastic scattering as an ontic quantum state would inevitably predict the superposition of position signals from different locations, in contradiction to observation. On the other hand, we know that the wave function from a single elastic scattering event, as a coherent superposition of elastic scattering off different scattering centers, should have an ontic interpretation, because accumulation of position signals through single-electron inelastic scattering reveals the interference pattern predicted by coherent scattering off different centers. The difference in the scattering events is energy transfer, and we can break the dichotomy in quantum evolution between the different scattering events if we postulate that energy transfer evolves discontinuously through quantum jumps. Wavelike behavior then emerges from elastic scattering, whereas particle-like behavior emerges from inelastic scattering.

Our conclusions are summarized in Section 3. The apparent discrepancy of observation and coherent wavelike evolution of wave functions has led to epistemic interpretations of the wave function as a tool to predict possible outcomes of quantum evolution for given initial conditions [13,14,15,16]. Epistemic interpretations assume that quantum mechanics only provides means to describe our best possible knowledge about outcomes of measurements of a system in a purely operational sense, i.e., without implying that the quantum states describe the underlying physical reality of the system in an ontological sense. This is opposed to ontic interpretations, which assume that quantum states provide a complete description of the objective physical reality of the system. To my knowledge, Primas was the first to use both phrases together in a paper on quantum ontology, to contrast those different points of view [17]. Our results suggest that wave functions can describe the ontic evolution of quantum states as long as energy transfer within multi-partite systems can be neglected, while they describe epistemic superpositions of possible ontic outcomes as soon as energy transfer becomes relevant.

2. Particle Diffraction and Detection

We model particle diffraction and detection through the interaction of a particle (with coordinates $\mathit{x}$ and momentum $\mathit{p}$) with an array of N scattering centers at positions ${\mathit{a}}_I$,

$$\begin{array}{cc}\hfill H& =H_0+V\hfill \\ \hfill & ={\displaystyle \frac{{\mathit{p}}^2}{2m}}+\sum _{I=1}^N\left({\displaystyle \frac{{\mathit{P}}_I^2}{2M}}+U({\mathit{y}}_I-{\mathit{a}}_I)\right)+\sum _{I=1}^{N}V(\mathit{x}-{\mathit{y}}_I),\hfill \end{array}$$

(1)

The potentials $U({\mathit{y}}_I-{\mathit{a}}_I)$ set the internal energy levels of the scattering centers, while the particle is scattered through the potential $V={\sum }_{I=1}^{N}V(\mathit{x}-{\mathit{y}}_I)$.

The Schrödinger equation predicts evolution of the corresponding $(N+1)$-particle wave function $\langle \mathit{x},{\mathit{y}}_1,\dots ,{\mathit{y}}_N|\mathsf{\Psi }\left(t\right)\rangle$ according to

$$\langle \mathit{x},{\mathit{y}}_1,\dots ,{\mathit{y}}_N|\mathsf{\Psi }\left(t\right)\rangle =\langle \mathit{x},{\mathit{y}}_1,\dots ,{\mathit{y}}_N|exp[-\mathrm{i}H(t-t^{\prime })/\hslash ]|\mathsf{\Psi }\left(t^{\prime }\right)\rangle .$$

(2)

We assume that the scattering centers are harmonic oscillators with equilibrium positions ${\mathit{a}}_I$,

$$U({\mathit{y}}_I-{\mathit{a}}_I)={\displaystyle \frac{1}{2}}M{\mathsf{\Omega }}^2{({\mathit{y}}_I-{\mathit{a}}_I)}^2.$$

(3)

The detector therefore consists of N “elementary detectors” or “detection units”, each of which can record the position of the incoming particle through a change in energy. The question is: why would several of those detection units not trigger simultaneously over the full width of the incoming particle wave function, despite the detector-plus-particle wave function $\langle \mathit{x},{\mathit{y}}_1,\dots ,{\mathit{y}}_N|\mathsf{\Psi }\left(t\right)\rangle$ already containing a superposition of several excited detector units in first order of the interaction potential V?

Please note that it does not matter at this point whether any of the “triggered” oscillators end up in an energy eigenstate or in a superposition of eigenstates for the detection event by the individual oscillator. The problem is that Equation (2) predicts a superposition of different excited oscillators over the width of the incoming wave function; see also Equation (7). This is exactly the problem that integration of the Schrödinger equation predicts for entanglement of detected particle and detector, resulting in a superposition of different pointer states, whereas the experiment yields unique pointer states, in contradiction to the prediction from the Schrödinger equation. It appears that the energy transfer during the scattering event that leads to particle detection must interrupt the wavelike evolution predicted by the Schrödinger equation.

We assume that the detector was in its ground state before scattering of the incoming particle,

$$\langle \mathit{x},{\mathit{y}}_1,\dots ,{\mathit{y}}_N|\mathsf{\Psi }\left(t^{\prime }\right)\rangle =\langle \mathit{x}|{\psi }_i\left(t^{\prime }\right)\rangle exp(-\mathrm{i}N{E}_{\mathbf{0}}{t}^{\prime }/\hslash )\prod _{I=1}^N{\varphi }_{\mathbf{0}}({\mathit{y}}_I-{\mathit{a}}_I),$$

(4)

where ${\varphi }_{\mathbf{0}}({\mathit{y}}_I-{\mathit{a}}_I)$ is the ground state wave function of the oscillator centered at ${\mathit{a}}_I$.

Projection of Equation (2) into the single-particle sector for the scattered particle yields in leading order of V (and after expressing initial and final states at fiducial time $t_0=0$, as usual)

$$\begin{array}{cc}\hfill |{\psi }_f\rangle =& |{\psi }_i\rangle -\sum _{I=1}^N\sum _{\mathit{n}\ge 0}{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{t^{\prime }}^{t}d\tau exp\left(\mathrm{i}{\omega }_{\mathit{n},\mathbf{0}}\tau \right)\int d^3\mathit{y}exp\left(\mathrm{i}{\displaystyle \frac{{\mathit{p}}^2\tau }{2m\hslash }}\right)V(\mathit{x}-\mathit{y})\hfill \\ \hfill & \times {\varphi }_{\mathit{n}}^{+}(\mathit{y}-{\mathit{a}}_I){\varphi }_{\mathbf{0}}(\mathit{y}-{\mathit{a}}_I)exp\left(-\mathrm{i}{\displaystyle \frac{{\mathit{p}}^2\tau }{2m\hslash }}\right)|{\psi }_i\rangle .\hfill \end{array}$$

(5)

Here, $\mathit{p}$ and $\mathit{x}$ are operators for the scattered particle. The transition frequency ${\omega }_{\mathit{n},\mathbf{0}}$ corresponds to the energy transfer from the particle to the scattering center,

$${\omega }_{\mathit{n},\mathbf{0}}=(E_{\mathit{n}}-E_{\mathbf{0}})/\hslash .$$

(6)

Projection of Equation (5) into position representation for the scattered particle yields, for scattered states (${\psi }_f\left(\mathit{x}\right)\ne {\psi }_i\left(\mathit{x}\right)$),

$$\begin{array}{cc}\hfill {\psi }_f\left(\mathit{x}\right)=& {\displaystyle \frac{1}{{\left(2\pi \right)}^3\mathrm{i}\hslash }}\sum _{I=1}^N\sum _{\mathit{n}\ge 0}\int d^3\mathit{y}\int d^3\mathit{z}\int d^3{\mathit{x}}^{\prime }\int d^3\mathit{k}{\int }_0^{\infty }d{k}^{\prime }{k}^{\prime }exp[\mathrm{i}\mathit{k}·(\mathit{x}-\mathit{z})]\hfill \\ \hfill & \times {\displaystyle \frac{sin\left(k^{\prime }|\mathit{z}-{\mathit{x}}^{\prime }|\right)}{\pi |\mathit{z}-{\mathit{x}}^{\prime }|}}V(\mathit{z}-\mathit{y}){\varphi }_{\mathit{n}}^{+}(\mathit{y}-{\mathit{a}}_I){\varphi }_{\mathbf{0}}(\mathit{y}-{\mathit{a}}_I){\psi }_i\left({\mathit{x}}^{\prime }\right)\hfill \\ \hfill & \times {\displaystyle \frac{sin\left([2m{\omega }_{\mathit{n},\mathbf{0}}+\hslash (k^2-k^{\prime 2})](t-t^{\prime })/4m\right)}{\pi [2m{\omega }_{\mathit{n},\mathbf{0}}+\hslash (k^2-k^{\prime 2})]/2m}}\hfill \\ \hfill & \times exp\left(\mathrm{i}[2m{\omega }_{\mathit{n},\mathbf{0}}+\hslash (k^2-k^{\prime 2})](t+t^{\prime })/4m\right).\hfill \end{array}$$

(7)

Equation (7) expresses the fact that the wave function after a single scattering event comprises a coherent superposition of contributions from each scattering center, as confirmed for elastic scattering through the observation of interference effects.

The factors in the two last lines of Equation (7) yield $\delta \left([2m{\omega }_{\mathit{n},\mathbf{0}}+\hslash (k^2-k^{\prime 2})]/2m\right)$ if

$$t-t^{\prime }\gg 2m/[2m{\omega }_{\mathit{n},\mathbf{0}}+\hslash (k^2-k^{\prime 2})].$$

(8)

For inelastic scattering, $\mathit{n}\ne \mathbf{0}$, this condition is fulfilled for all times that satisfy $t-t^{\prime }\gg 1/\mathsf{\Omega }$, i.e., in this case, we find for the inelastic scattering contribution

$$\begin{array}{cc}\hfill {\psi }_f^{(\mathrm{inel}.)}\left(\mathit{x}\right)=& {\displaystyle \frac{m}{{\left(2\pi \right)}^3\mathrm{i}{\hslash }^2}}\sum _{I=1}^N\sum _{\mathit{n}\ne \mathbf{0}}\int d^3\mathit{y}\int d^3\mathit{z}\int d^3{\mathit{x}}^{\prime }\int d^3\mathit{k}exp[\mathrm{i}\mathit{k}·(\mathit{x}-\mathit{z})]\hfill \\ \hfill & \times {\displaystyle \frac{sin\left(\sqrt{k^2+(2m{\omega }_{\mathit{n},\mathbf{0}}/\hslash )}|\mathit{z}-{\mathit{x}}^{\prime }|\right)}{\pi |\mathit{z}-{\mathit{x}}^{\prime }|}}V(\mathit{z}-\mathit{y})\hfill \\ \hfill & \times {\varphi }_{\mathit{n}}^{+}(\mathit{y}-{\mathit{a}}_I){\varphi }_{\mathbf{0}}(\mathit{y}-{\mathit{a}}_I){\psi }_i\left({\mathit{x}}^{\prime }\right).\hfill \end{array}$$

(9)

Indeed, the emergence of the energy-preserving $\delta$-function follows both from the sinc function in the second-last line of Equation (7), and also from the fact that fast oscillation of the exponential factor in the last line would also erase the scattered wave function unless the energy is conserved.

Since $m=511\mathrm{keV}/c^2$ and ${\omega }_{\mathit{n},\mathit{0}}\ge \mathsf{\Omega }$, the normalized sinc function in the second line has, at most, an Ångström-scale width if $\hslash \mathsf{\Omega }>1$ eV. Furthermore, we also assume that the scattering potential V has a short range. Approximating those two short-range factors by the $\delta$-functions then yields

$${\psi }_f^{(\mathrm{inel}.)}\left(\mathit{x}\right)\propto \sum _{I=1}^N\sum _{\mathit{n}\ne \mathbf{0}}{\varphi }_{\mathit{n}}^{+}(\mathit{x}-{\mathit{a}}_I){\varphi }_{\mathbf{0}}(\mathit{x}-{\mathit{a}}_I){\psi }_i\left(\mathit{x}\right),$$

(10)

i.e., the inelastically scattered component of the outgoing wave function consists of N terms, which are cut out from the incoming wave function through the weight factors ${\varphi }_{\mathit{n}}^{+}(\mathit{x}-{\mathit{a}}_I){\varphi }_{\mathbf{0}}(\mathit{x}-{\mathit{a}}_I)$ and the short-range sinc function and scattering potential. These N terms correspond to N parts, which are centered around the detector units at locations ${\mathit{a}}_I$, and the widths of those parts are of order $\sqrt{\hslash /M\mathsf{\Omega }}$.

If we now argue with Heisenberg that wave function evolution is only a smooth probabilistic approximation to quantum jumps in the detector units, then we cannot assume continuation of wavelike superposition at this stage, and it appears reasonable to assume that only one quantum jump related to an energy transfer actually happens: the inelastically scattered part of the wave function collapses around a single detector site, thus yielding only a single position signal. The relative magnitude of the N terms in Equation (10) determines the relative probability amplitudes, and the detector unit and energy transition with the largest overlap factor

$$P_{iI\mathit{n}}=\int d^3\mathit{x}{\left|{\varphi }_{\mathit{n}}^{+}(\mathit{x}-{\mathit{a}}_I){\varphi }_{\mathit{0}}(\mathit{x}-{\mathit{a}}_I){\psi }_i\left(\mathit{x}\right)\right|}^2,\mathit{n}\ne \mathbf{0},$$

(11)

has the highest probability to yield the position signal.

On the other hand, since the energy preservation also follows from the exponential factor in the last line of Equation (7), we find for the elastically scattered component of the wave function

$$\begin{array}{cc}\hfill {\psi }_f^{(\mathrm{el}.)}\left(\mathit{x}\right)=& {\displaystyle \frac{m}{{\left(2\pi \right)}^3\mathrm{i}{\hslash }^2}}\sum _{I=1}^N\int d^3\mathit{y}\int d^3\mathit{z}\int d^3{\mathit{x}}^{\prime }\int d^3\mathit{k}exp[\mathrm{i}\mathit{k}·(\mathit{x}-\mathit{z})]\hfill \\ \hfill & \times {\displaystyle \frac{sin\left(k|\mathit{z}-{\mathit{x}}^{\prime }|\right)}{\pi |\mathit{z}-{\mathit{x}}^{\prime }|}}V(\mathit{z}-\mathit{y}){\varphi }_{\mathbf{0}}^{+}(\mathit{y}-{\mathit{a}}_I){\varphi }_{\mathbf{0}}(\mathit{y}-{\mathit{a}}_I){\psi }_i\left({\mathit{x}}^{\prime }\right).\hfill \end{array}$$

(12)

Except for extremely low-energy electrons, the remaining sinc function still has, at most, an Ångström-scale width, and we can still approximate short-range factors with $\delta$-functions,

$${\psi }_f^{(\mathrm{el}.)}\left(\mathit{x}\right)\propto \sum _{I=1}^N{\varphi }_{\mathbf{0}}^{+}(\mathit{x}-{\mathit{a}}_I){\varphi }_{\mathbf{0}}(\mathit{x}-{\mathit{a}}_I){\psi }_i\left(\mathit{x}\right).$$

(13)

This would again cut N pieces of widths of order $\sqrt{\hslash /M\mathsf{\Omega }}$ out of the incoming wave function. However, this time, we cannot expect a quantum jump as a consequence of the energy transfer, and the superposition principle persists. Elastic scattering off the oscillator array therefore produces electron diffraction, and the coherent superposition of scattered waves from the N scattering centers will imprint an interference pattern on the elastically scattered wave function. Following-up with inelastic scattering in a second oscillator array for position detection, and accumulation of many single-particle signals, will reveal that interference pattern. Please note that a set of scattering centers can principally yield both elastic and inelastic scattering, in agreement with Equations (9) and (12). It depends on the relative magnitude of scattering matrix elements, whether elastic or inelastic scattering will dominate. For example, low-energy scattering of electrons off metal surfaces is dominated by elastic scattering because of a lack of inelastic channels within the range of the kinetic energy of the scattered electrons.

3. Conclusions

Heisenberg suggested that the continuous wave function evolution represents only a continuous evolution of probability amplitudes for quantum jumps if an energy transfer is involved. On the other hand, observation requires an energy transfer between the observed system and the detector, and quantum jumps would break continuous wave function evolution, thereby also breaking the superposition principle. However, breaking the superposition principle of wave function evolution is exactly what is needed to avoid prediction of mutually contradictory pointer states in low-energy quantum mechanics. Mutually contradictory pointer states would correspond, e.g., to macroscopically separated position signals in single-particle experiments.

The picture that emerges from this investigation favors an interpretation of the wave function as an epistemic superposition of possible ontic outcomes, where elastic scattering corresponds to one possible outcome, while every inelastic scattering channel corresponds to another possible outcome. Interference effects from elastic scattering then accounts for wavelike behavior, while inelastic scattering yields particle-like behavior in observations.

The time-dependent Schrödinger equation does not evolve a quantum system per se, but only evolves relative probability amplitudes for different channels of possible ontic outcomes of evolution of the system. The different channels are labeled by quantum jumps related to energy transfer within a multi-partite system (or between systems), or the absence thereof. By the same token, the time-dependent Schrödinger equation describes evolution of our best possible knowledge about evolution of a quantum system, until we update our knowledge through observation.

This proposal entails a critical dependence on the availability of elastic versus inelastic scattering channels, and on the magnitude of the corresponding scattering amplitudes, whether a scattering array will primarily work as a diffractor or as a detector. Dependence of these properties on the energy of incident particles should therefore open a window to experimental tests of the proposal. Furthermore, a “golden bullet” experiment would continuously vary properties of the scattering array to tune the array from being a diffractor to being a detector, for fixed incident particle energy.