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arXiv:2401.15110

Collapse of wave functions in Schroedinger's wave mechanics

Rainer Dick

Published Jan 25, 2024• 0 citations
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Abstract

We show that inelastic scattering leads to a collapse of the wave function within standard evolution through the Schroedinger equation, whereas elastic scattering will not collapse the wave function. Specifically, we find that the initial width of the emerging wave function in inelastic scattering is primarily determined by the size of the participating scattering center, but not by the width of the incoming wave function. This implies that dynamical collapse of the wave function through inelastic scattering, together with energy quantization in bound quantum systems, can explain the emergence of particle-like signals without the need to invoke the Born rule.

DOI
10.48550/arxiv.2401.15110

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We show that inelastic scattering leads to a collapse of the wave function within standard evolution through the Schroedinger equation, whereas elastic scattering will not collapse the wave function. Specifically, we find that the initial width of the emerging wave function in inelastic scattering is primarily determined by the size of the participating scattering center, but not by the width of the incoming wave function. This implies that dynamical collapse of the wave function through inelastic scattering, together with energy quantization in bound quantum systems, can explain the emergence of particle-like signals without the need to invoke the Born rule.
We show that inelastic scattering in a short-range potential leads to a collapse of the wave function within standard evolution through the Schrödinger equation, whereas elastic scattering will not collapse the wave function. Specifically, we find that the initial width of the emerging wave function in inelastic scattering is primarily determined by the size of the participating scattering center and the range of the scattering potential, but not by the width of the incoming wave function. This indicates that dynamical collapse of the wave function through inelastic scattering can explain the emergence of particle-like signals.
It is often claimed that the collapse of the wave function and Born's rule to interpret the square of the norm as a probability, have to be introduced as separate axioms in quantum mechanics besides the Schroedinger equation. Here we show that this is not true in certain models where quantum behavior can be attributed to underlying deterministic equations. It is argued that indeed the apparent spontaneous collapse of wave functions and Born's rule are features that strongly point towards determinism underlying quantum mechanics.
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The measurement problem of quantum mechanics concerns the question under which circumstances coherent wave evolution becomes disrupted to produce eigenstates of observables, instead of evolving superpositions of eigenstates. The problem needs to be addressed already 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.
The Born rule, P(x)=∣ψ(x)∣2P(\mathbf{x})=|\psi(\mathbf{x})|^2P(x)=∣ψ(x)∣2 is one of the most successful relations in modern physics. Yet its physical meaning remains conceptually distinctive. Schrödinger's equation describes the continuous evolution of a complex wave amplitude, whereas an actual detector records discrete and localized events. The Born rule connects these two descriptions with remarkable precision, but the physical mechanism responsible for the quadratic relation between wave amplitude and event frequency is not contained explicitly in the Schrödinger equation itself. This paper revisits the historical and physical relationship between Schrödinger's wave mechanics and Born's statistical interpretation of scattering. The purpose is not to deny the empirical validity of the Born rule, but to identify the physical gap between continuous wave evolution and discrete detection statistics. We distinguish three levels of description: the preparation of localized quantum excitations, their continuous propagation and scattering, and the final microscopic detector event. This leads to a central question: \boxedtext{Why should the rate of discrete detection events be proportional to ∣ψ∣2|\psi|^2∣ψ∣2?} The paper argues that the answer cannot be obtained merely by defining ∣ψ∣2|\psi|^2∣ψ∣2 to be a probability density. Instead, a physical theory must explain how a continuously distributed complex amplitude produces a localized event rate. We investigate the possibility that the quadratic dependence emerges from a local interaction functional, energy-transfer mechanism, or bilinear response of the detector to the incident quantum amplitude. A particularly important distinction is made between the wave amplitude itself and the experimentally recorded event frequency. A single detection event is localized, whereas the Born distribution emerges only through repeated preparations and statistical accumulation. The paper therefore proposes that the Born rule may be viewed as the macroscopic statistical image of a microscopic event-generation process. No completed derivation of Born's rule is claimed here. Rather, the paper formulates the precise dynamical problem that such a derivation must solve and identifies several mathematical and experimental constraints that any successful mechanism must satisfy.
The space quantization induced by a Stern-Gerlach experiment is normally explained by invoking the ``collapse of the wave function.'' This is a rather mysterious idea; it would be better to explain the Stern-Gerlach results without using it. We re-analyze the Stern-Gerlach experiment using path integrals. We find if we model explicitly the finite width of the beam, coherent interference within the beam itself provides the space quantization -- without need to invoke the collapse. If we insist on employing only wave functions with the space and spin parts kept forcibly disentangled, we recreate the need to invoke the collapse. The collapse-free approach makes more specific predictions about the shape and position of the scattered beams; if the interaction region has finite length, these may be testable. Pending experimental disambiguation, the chief arguments in favor of the collapse-free approach are that it is simpler and less mysterious, has no adjustable parameters, and requires the invocation of no new forces.