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Physical Review

ψ Superposition and Quantum Periodicity

Alfred Landé

Published Nov 1, 1957• 9 citations
Publisher

Abstract

The superposition law for probability amplitudes, usually considered as an empirical rule and introduced into the quantum theory as an ad hoc assumption, is shown to be the mathematical solution of almost self-evident requirements for the structural metric of the probabilities in general. When combining the general $\ensuremath{\psi}$ metric of superposition with the postulate of a constant probability density in $q$-space for given $p$, and of a constant probability density in $p$-space for given $q$, one arrives at the mathematical result that $\ensuremath{\psi}(q, p)$ must necessarily have complex periodic form, $\mathrm{exp}(\frac{\mathrm{iqp}}{\mathrm{const}})$. Various quantum principles can thus be reduced to a nonquantal basis.

DOI
10.1103/physrev.108.891

Related papers

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•Sep 16, 2026•0 citations
The superposition law for probability amplitudes, usually considered as an empirical rule and introduced into the quantum theory as an ad hoc assumption, is shown to be the mathematical solution of almost self-evident requirements for the structural metric of the probabilities in general. When combining the general \ensuremathψ\ensuremath{\psi}\ensuremathψ metric of superposition with the postulate of a constant probability density in qqq-space for given ppp, and of a constant probability density in ppp-space for given qqq, one arrives at the mathematical result that \ensuremathψ(q,p)\ensuremath{\psi}(q, p)\ensuremathψ(q,p) must necessarily have complex periodic form, exp(iqpconst)\mathrm{exp}(\frac{\mathrm{iqp}}{\mathrm{const}})exp(constiqp​). Various quantum principles can thus be reduced to a nonquantal basis.
The superposition rule for probability amplitudes ψ is not a separate ad hoc assumption of a physical character but is a formal consequence of the arrangement of the states of a system in sets of mutually orthogonal states linked by probabilities of transition. Nor do the quantum prescriptions of Born and Schrödinger have to be introduced as ad hoc assumptions; they can be deduced as consequences of (a) the superposition rule, and (b) the definition of conjugates q and p as leading to a constant probability density in qp-space, a quality known from classical statistical mechanics. Quantum theory can thus be deduced from a nonquantal background.
The basic concepts and rules of quantum mechanics are shown to be immediate consequences, on the basis of simple reasoning, of a postulate of continuity in the domain of macroscopic thermodynamics. The continuity principle leads first to the concept of a fractional likeness between two states in general, as opposed to the classical alternative of either like or unlike. Then all possible states of a system can be arranged in sets of mutually unlike or “orthogonal” states, with members A and B of different sets interconnected by fractional likeness factors q(A, B) between zero and unity. These factors q are identical with the relative intensities in a splitting effect of a state into components, ruled by probability relations and by the basic principle of quantum statistics of not counting permutations of individual particles. The problem of relation between the various fractional likenesses, or probabilities of transition under analysis, is a purely mathematical problem and is solved by the introduction of probability amplitudes necessarily subject to a matrix law of multiplication, i.e., to the principle of superposition. Thus, starting from the continuity postulate of thermodynamics, quantum theory in its general outline is obtained by simple reasoning. All further details, such as the duality of waves and particles, the uncertainty relation, and the mechanics of special systems, need only one more bit of empirical information, namely, the symmetric and periodic form of the amplitude ψ of fractional likeness between a state of given coordinate and a state of given momentum.
Clarifying the nature of the quantum state |Ψ⟩ is at the root of the problems with insight into counter-intuitive quantum postulates. We provide a direct—and math-axiom free—empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure—quantum superposition—is based on a set-theoretic creation of ensemble formations and invokes the following three principia: (I) quantum statics, (II) doctrine of the number in the physical theory, and (III) mathematization of matching the two observations with each other (quantum covariance). All of the constructs rest upon a formalization of the minimal experimental entity—the registered micro-event, detector click. This is sufficient for producing the ℂ-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the spatio-temporal concepts. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.
In a recent article (quant-ph/9906015) Deutsch, using concepts of decision theory, provided a strikingly simple argument to explain why quantum-mechanical probabilities are squares of amplitudes. The decisiontheoretic reasoning was criticized by Barnum et al. (quant-ph/9907024). Taking a more conventional approach, we reinterpret a crucial substitution in Deutsch’s paper as the time evolution of an apparatus, capable of separating internal states of an “atom ” into a suitable number of “cavities”. Permutational symmetries play a central role in this analysis. PACS numbers: 03.65.BzIn a recent paper, Deutsch [1] invoked the non-probabilistic part of classical decision theory to deduce from only non-probabilistic axioms of quantum mechanics the Statistical Axiom: The probability for measuring xi in a superposition state |ψ 〉 = ∑ i ψi|xi 〉 is |ψi | 2. The paper has attracted some vehement criticism [2] relating to its decision-theoretic setting, but also to priorities. As pointed out by Ref. [2], in 1957 already Gleason [3] gave a mathematically rigorous proof for the redundance of the Statistical Axiom, showing that if normalized probabilities are to be assigned to the eigenvectors of each bounded hermitian operator, then the only solution is the usual quantum mechanical prescription. Unfortunately for a result of such fundamental importance to quantum physics, the existing proofs of Gleason’s theorem are not easily received by many physicists, although mathematically elementary [4], synoptical [5], and even textbook [6] elaborations have been presented. Ref. [1] has a strong point in this respect, suggesting that the mathematics behind amplitudes being square-roots of probabilities essentially is the Pythagorean theorem in Hilbert space. Consider a superposition state of the special form |ψ 〉 =
Recent results obtained in quantum measurements indicate that the fundamental relations between three physical properties of a system can be represented by complex conditional probabilities. Here, it is shown that these relations provide a fully deterministic and universally valid framework on which all of quantum mechanics can be based. Specifically, quantum mechanics can be derived by combining the rules of Bayesian probability theory with only a single additional law that explains the phases of complex probabilities. This law, which I introduce here as the law of quantum ergodicity, is based on the observation that the reality of physical properties cannot be separated from the dynamics by which they emerge in measurement interactions. The complex phases are an expression of this inseparability and represent the dynamical structure of transformations between the different properties. In its quantitative form, the law of quantum ergodicity describes a fundamental relation between the ergodic probabilities obtained by dynamical averaging and the deterministic relations between three properties expressed by the complex conditional probabilities. The complete formalism of quantum mechanics can be derived from this one relation, without any axiomatic mathematical assumptions about state vectors or superpositions. It is therefore possible to explain all quantum phenomena as the consequence of a single fundamental law of physics.
One of the most successful, yet most puzzling, structures of quantum mechanics is Born's rule:P(\mathbf{x},t)=|\Psi(\mathbf{x},t)|^2.It establishes, with extraordinarily high experimental accuracy, the connection between the quantum wave function and the statistical distribution of actual measurement outcomes. Yet a more fundamental question remains worth asking: Why is probability given precisely by the modulus squared of the complex wave amplitude? Standard quantum mechanics uses Born's rule as a fundamental rule. This paper instead poses an open research question: Could Born's rule be not an irreducible probabilistic axiom, but a statistical emergence resulting from the combined action of continuous wave-amplitude superposition, quantized energy exchange, and local interaction dynamics? We propose a candidate physical picture:wave-amplitude superposition->spatial interference structure->local interaction->quantized energy exchange->discrete events}->probability distribution The central mathematical question is \Gamma(\mathbf{x})\stackrel{?}{\propto}|\Psi(\mathbf{x})|^2 where Γ(x)\Gamma(\mathbf{x})Γ(x) denotes the occurrence rate of localized discrete detection events. This paper does not claim to have completed such a derivation. Instead, it explicitly formulates the problem as an \textbf{open, computable, testable, and falsifiable research challenge}. We particularly invite young physicists, mathematicians, and researchers skilled in AI and modern computational tools to explore this problem.