Abstract
Werner Heisenberg introduced the notion of quantum potentia in order to accommodate the indeterminism associated with quantum measurement. Potentia captures the capacity of the system to be found to possess a property upon a corresponding sharp measurement in which it is actualized. The specific potentiae of the individual system are represented formally by the complex amplitudes in the measurement bases of the eigenstate in which it is prepared. All predictions for future values of system properties can be made by an experimenter using the probabilities which are the squared moduli of these amplitudes that are the diagonal elements of the density matrix description of the pure ensemble to which the system, so prepared, belongs. Heisenberg considered the change of the ensemble attribution following quantum measurement to be analogous to the classical change in Gibbs’ thermodynamics when measurement of the canonical ensemble enables a microcanonical ensemble description. This analogy, presented by Heisenberg as operating at the epistemic level, is analyzed here. It has led some to claim not only that the change of the state in measurement is classical mechanical, bringing its quantum character into question, but also that Heisenberg held this to be the case. Here, these claims are shown to be incorrect, because the analogy concerns the change of ensemble attribution by the experimenter upon learning the result of the measurement, not the actualization of the potentia responsible for the change of the individual system state which—in Heisenberg’s interpretation of quantum mechanics—is objective in nature and independent of the experimenter’s knowledge.
Highlights
A quantum physical system may be seen upon measurement manifestly to possess a specific property that it was not certain to possess beforehand, even though its state had been fully specified by its preparation
At the completion of an exact measurement of its dynamical properties, the system is once again isolated from the measuring apparatus and its state has generally changed discontinuously from |ψi to a different Hilbert-state vector |φk i corresponding to its actual value; upon awareness of the actual value as that of the kth possible outcome, an experimenter assigns the system for predictive purposes to the pure ensemble described by the density matrix ρk = |φk ihφk | [6]
It has been shown here that the analogy made by Heisenberg under consideration here is one between, on the one hand, the change in quantum mechanics of assignment of a system from a mixed ensemble described by a diagonal density matrix to a pure quantum ensemble enabled by a sharp measurement, and on the other hand, the change in Gibbs’ thermodynamics from a classical system’s assignment to canonical ensemble to its assignment to a microcanonical ensemble enabled by a precise measurement
Summary
A quantum physical system may be seen upon measurement manifestly to possess a specific property that it was not certain to possess beforehand, even though its state had been fully specified by its preparation. At the completion of an exact measurement of its dynamical properties, the system is once again isolated from the measuring apparatus and its state has generally changed discontinuously from |ψi to a different Hilbert-state vector |φk i corresponding to its actual value; upon awareness of the actual value as that of the kth possible outcome, an experimenter assigns the system for predictive purposes to the pure ensemble described by the density matrix ρk = |φk ihφk | [6] Heisenberg viewed this change of ensemble assignment as “exactly analogous” to that occurring in classical statistical thermodynamics upon the measurement of a Gibbs ensemble with an outcome associated with a given microstate. It is shown that the actualization of potentia is distinctly quantum and occurs in the quantum state space, which is a complex Hilbert space, even though the two theories involve analogous changes—which correspond to the knowledge provided by measurements in each case—in the ensembles associated with the systems used for predictive purposes
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