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American Journal of Physics

ψ Superposition and Quantum Rules

Alfred Landé

Published Feb 1, 1956• 10 citations
Publisher

Abstract

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.

DOI
10.1119/1.1934142

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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 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 qqq-space for given ppp, and of a constant probability density in ppp-space for given qqq, one arrives at the mathematical result that $\ensuremath{\psi}(q, p)$ 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 Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.
The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.
In quantum mechanics, the physics of probability amplitudes, that exists in the imaginary and complex mathematical realm, is brought to the measurable realm by the iconic Born rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 that convert the superposition probability amplitude https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 into a measurable interferometric probability distribution. When Born first introduced his rule, in 1926, he did so in the absence of a physical derivation. In fact, Born never offered a derivation for his most crucial rule. In this chapter a transparent interferometric derivation of Born's rule is given. Furthermore, it is indicated that far from a ‘collapse’ the https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition, for interferometric configurations, clearly involves a mathematical expansion . The https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition for quantum entanglement is also discussed.
The basic notion of physical system states is different in classical statistical mechanics and in quantum mechanics. In classical mechanics, the particle system state is determined by its position and momentum; in the case of fluctuations, due to the motion in environment, it is determined by the probability density in the particle phase space. In quantum mechanics, the particle state is determined either by the wave function (state vector in the Hilbert space) or by the density operator. Recently, the tomographic-probability representation of quantum states was proposed, where the quantum system states were identified with fair probability distributions (tomograms). In view of the probability-distribution formalism of quantum mechanics, we formulate the superposition principle of wave functions as interference of qubit states expressed in terms of the nonlinear addition rule for the probabilities identified with the states. Additionally, we formulate the probability given by Born’s rule in terms of symplectic tomographic probability distribution determining the photon states.
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.