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Entropy

Phase-Amplitude Relations for a Particle with a Superposition of Two Energy Levels in a Double Potential Well

Ofir Flom, Asher Yahalom A, Jacob Levitan, Haggai Zilberberg

Published Feb 22, 2022• 1 citation
Publisher

Abstract

We study the connection between the phase and the amplitude of the wave function and the conditions under which this relationship exists. For this we use the model of particle in a box. We have shown that the amplitude can be calculated from the phase and vice versa if the log analytical uncertainty relations are satisfied.
DOI
10.3390/e24030312

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Two distinct measures of information, connected respectively to the amplitude and phase of the wave function of a particle, are proposed. There are relations between the time derivatives of these two measures and their gradients on the configuration space, which are equivalent to the wave equation. The information related to the amplitude measures the strength of the potential coupling of the particle (which is itself aspatial) with each volume of its configuration space, i.e. its tendency to participate in an interaction localized in a region of ordinary physical space corresponding to that volume. The information connected to the phase is that required to obtain the time evolution of the particle as a persistent entity starting from a random succession of bits. It can be considered as the information provided by conservation principles. The meaning of the so-called "quantum potential" in this context is briefly discussed.
We investigate general differential relations connecting the respective behaviors of the phase and modulus of probability amplitudes of the form 〈ψf|ψ〉, where |ψf〉 is a fixed state in Hilbert space and |ψ〉 is a variable state, treated as a section of a U(1) bundle over a complex subspace of the corresponding ray space R=CPn. Amplitude functions on such holomorphic line bundles, while not strictly holomorphic, nevertheless satisfy generalized Cauchy–Riemann conditions involving the U(1) Berry–Simon connection on the parameter space. These conditions entail invertible relations between the gradients of the phase and modulus, therefore allowing for the reconstruction of the phase from the modulus (or vice versa) and other conditions on the behavior of either polar component of the amplitude. As a special case, we consider amplitude functions valued on the space of pure states, the ray space R=CPn, where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with the Fubini–Study metric. In conjunction with the generalized Cauchy–Riemann conditions, this geodesic interpretation leads to additional relations, in particular, a novel connection between the modulus of the amplitude and the phase gradient, somewhat reminiscent of the WKB formula. Finally, a connection with geometric phases is established.
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