We seek in this paper the fundamental form of the equation of quantum dynamics that causes systems to evolve according to the master equation. The direction of this investigation starts out with the master equation taken as an experimental fact and the state of a quantum system described as usual by a vector in Hilbert space. We first prove that the only time evolution operator that will lead to the master equation depends on the exponential of time to the half power. This time dependence suggests that it is due to a random walk. Random walks arise due to a random variable determining the direction taken by each step. On modeling the behavior of the system as a random walk in Hilbert space it is found that the resulting time evolution operator depended on the exponential of time to the half power. This is the same functional form as the one found earlier in the paper that led to the master equation. It is concluded that random time evolution operators describe the underlying process that cause systems to evolve by the master equation.
Keywords: Random time evolution operator; Master equation; Foundation of Statistical Mechanics: Quantum theory
Jan 02, 2025
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Examined is a consequence of using the probability to calculate both the entropy of a system and as a measure of the likelihood of the system being found in a specific microstate. Using three simple examples, it is found in each case that a contradiction exists. The contradiction arises if we require that the entropy of a microstate in which the system can be found must equal that of the systems entropy as calculated using quantum statistical mechanical theory. It is argued that the contradiction can be resolved by retaining the usual equations of statistical mechanics, but interpreting the meaning of the probabilities so that they no longer provide a measure of the system being found in a specific microstate.
Keywords: Quantum Statistical Mechanics, Probability, Contradiction, Entropy
Nov 18, 2022
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The equations of quantum mechanics are time reversal symmetric. Here we use a simple conceptual double slit experiment to show that under certain circumstances time reversal symmetry is violated. We send a single particle from a source through a double slit to a detecting screen. Irrespective of how we reverse the momentum of the particle at the screen, we expect that it is unlikely to return to the original source. We interpret this to mean that when the position at which the particle arrives is probabilistically determined, time reversal symmetry is violated. Looking at the experiment in quantum mechanical term, we find the need to use a random unitary matrix in the equation giving the time evolution of the state of the system. This is due to a selection process that takes place during a measurement, wherein a system that can be in any of a number of states randomly ends up in only one of those states. This equation is not time reversal symmetric.
Keywords: Time reversal symmetry; Quantum theory
PACS:
03.65.Ta Foundations of quantum mechanics; measurement theory;
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
2015
We explore the consequences of making the following assumptions for transitions in isolated, degenerate systems, where we describe the state of the system by a vector in a linear vector space: changes in the state are due to a series of random events where each event is given by a unitary operator that is equal to an exponential function of a random Hermitian matrix; the principal of superposition applied to all possible paths gives the path taken; the statistical properties of the random variable are a system property. For systems where the elements of the random Hermitian matrix are i.i.d. and uncorrelated, we find that on using the central limit theorem the above leads to a unitary time evolution operator that is a function of time increments to the half power. On applying this operator to an isolated thermodynamic system we show it leads to an equation consisting of two terms. The time derivative of the first term is the master equation; the second term is shown to represent fluctuations. The transition probabilities in the master equation satisfy the principle of detailed balance. The two terms taken together describe a system that evolves to a final state in which the system fluctuates about equilibrium. A single set of assumptions lead to the master equation, the principle of detailed balance and fluctuations.
Keywords: Fractional time evolution operator; Quantum theory; Master equation; Fluctuations; Foundation of Statistical Mechanics: Fractional master equation; Fractals
PACS:
03.65.Ta Foundations of quantum mechanics; measurement theory
05.30.-d Quantum statistical mechanics
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.45.-a Nonlinear dynamics and chaos
2010
E-mail address: bhrosof@alum.mit.edu E-mail address: b.rosof@rosof.com