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Time-energy coordinates for the classical linear potential. (a) Values of the time function, t(z); (b) Density plots of an energy-time Gaussian probability density in energy-time space; (c) Density plots of an energy-time Gaussian probability density in phase-space. Here, E 0 = 1.5, σ E = 0.5 and σ T = 1 in dimensionless units.
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In a previous paper, we introduced a way to generate a time coordinate system for classical and quantum systems when the potential function has extremal points. In this paper, we deal with the case in which the potential function has no extremal points at all, and we illustrate the method with the harmonic and linear potentials.
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Citations
... There is also a method of time propagation of the coordinate eigenstate δ(q) multiplied by p/m, generating that way the required time states [2]. This method is appropriate for the free particle and it was suggested to use it also for any potential function [3][4][5][6]. ...
The usual method for obtaining the eigenstates of an operator is to solve the corresponding eigenvalue equation. This procedure cannot be applied when the operator of interest is not known at all. We develop a method which generates the eigenstates of an operator, and the operator itself, which will be conjugate to a given known operator. This is particularly useful for the case of the time operator in Quantum Mechanics. We illustrate the method by obtaining time eigenstates for the free particle.
We give a short review of known exact inequalities that can be interpreted as
"energy-time" and "frequency-time" uncertainty relations. In particular we
discuss a precise form of signals minimizing the physical frequency-time
uncertainty product. Also, we calculate the "stationarity time" for mixed
Gaussian states of a quantum harmonic oscillator, showing explicitly that pure
quantum states are "more fragile" than mixed ones with the same value of the
energy dispersion. The problems of quantum evolution speed limits, time
operators and measurements of energy and time are briefly discussed, too.
