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Physical Review A

Temporal Evolution and the Approach to Equilibrium of a Quantum Particle in a Suddenly Expanded Box

Conley Stutz, D. W. Schlitt

Published Sep 1, 1970• 7 citations
Publisher

Abstract

The time evolution and its implications with regard to the approach to equilibrium of a single-particle system in a suddenly expanded box with perfectly reflecting walls is investigated using density-matrix techniques. It is shown that the time-dependent quantities exhibit a periodic behavior which is independent of the initial density matrix as long as the dimensions of the system are finite. The period is found to be $\frac{2\ensuremath{\pi}\ensuremath{\hbar}}{{E}_{1}}$ where ${E}_{1}$ is the ground-state energy of the expanded system. The continuum approximation is employed to determine the explicit time dependence of the particle density of a system initially in thermal equilibrium. It is found that the density for an expansion into a finite volume decays essentially as $\frac{1}{t}$ with small oscillations superposed on this behavior for times ranging from shortly after expansion up to one-fourth the characteristic period.

DOI
10.1103/physreva.2.897

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Temporal Evolution and the Approach to Equilibrium of a System of Noninteracting Quantum Particles

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A study is made of the temporal evolution and its implications in regards to the approach to equilibrium of a system of noninteracting particles that has undergone a sudden expansion. It is assumed that the system is initially in thermal equilibrium. It is shown that the particle density approaches an equilibrium density essentially as $\frac{1}{t}$ for times shortly after expansion up to one-half the characteristic period. This period is the same as for a single-particle system. An expression is derived for the particle density that is valid for Bose, Fermi, and Boltzmann particles. The density for Bose and Fermi particles is compared with that found for Boltzmann particles.

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I study the time evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density |Ψ( x , t )| 2 shows up quite a surprising behaviour: for definite times, plateaux appear for which |Ψ( x , t )| 2 is constant on finite intervals for x . Elements of theoretical explanation are given by analysing the singular component of the second derivative ∂ xx Ψ( x , t ). Analytical closed expressions are obtained for some specific times, which easily allow us to show that, at these times, the density organizes itself into regular patterns provided the size of the box is large enough; more, above some critical size depending on the specific time, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.

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I study the time-evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density |Ψ(x, t) | 2 shows up quite a surprising behaviour: for definite times, plateaux appear for which |Ψ(x, t) | 2 is constant on finite intervals for x. Elements of theoretical explanation are given by analyzing the singular component of the second derivative ∂xxΨ(x, t). Analytical closed expressions are obtained for some specific times, which easily allow to show that, at these times, the density organizes itself into regular patterns provided the size of the box in large enough; more, above some critical time-dependent size, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.

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Quantum Thermodynamics of the Spontaneous Approach to the Equilibrium State – Irreversible Exponential Decay of a Discrete Quantum State of Entropy Production into a Continuum

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We consider the temporal evolution of the entropy S of an isolated system from an initial non-equilibrium state 2· of entropy S = S i to the equilibrium state of maximum entropy S = S max > S i. The application of usual density matrix theory of quantum mechanics leads us to the surprising result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, because the entropy of the isolated system could not spontaneously increase from S i to S max by entropy production P > 0. This contradicts the second law of thermodynamics. We avoid this discrepancy by changing from quantum mechanics to quantum thermodynamics. Now the entropy production P = dS/dt and the time t are the relevant observables. In order to treat the temporal development of S and P, we at first prepare an initial state i of entropy production P i at time t < 0. Subsequently at ϑ t ≥ 0 the initial state i interacts with a continuum of final states f, which transfers the entropy production from i to f. The temporal evolution of the entropy production P(t) of our system is treated by means of the time-dependent perturbation theory. For that reason we split the operator of entropy production P = P 0 + W into two parts. The operator P 0 describes the unperturbed basic vectors of entropy production | i > and | f >, while the perturbation operator W determines the interaction between | i > and | f >. We expand the time-dependent state function < t | ψ > of the isolated system into the complete set of stationary eigenvectors | i > and | f > and calculate the time-dependent expansion coefficients c i(t) and c(f, t) by means of the dynamic equation of quantum thermodynamics − i k (d/dt) < t | ψ > = < t | P | ψ >. Here k means the Boltzmann constant, which is characteristic of quantum thermodynamics and represents the atomic entropy unit. The expansion coefficients are given by a system of coupled differential equations, from which we deduce an integral-differential equation for the coefficient c i(t) of the initial state i. Unfortunately it has been impossible to obtain an analytic solution of the complicated equation. However approximations are available. Accordingly, in the vicinity of the equilibrium state, the probability of still finding the initial state i of entropy production P i at time t is given by ω i(t) = | c i(t) |2 = exp (−Lt). The rate constant L ≥ σ depends on the density ρ(P f) of final states f of entropy production P f near by P i, and on the squared modulus of the perturbation matrix elements W if = < i | W | f >, which determines the interaction between | i > and | f >. The entropy production P(t) = P i ω i(t) = P i exp (−ϑLt) of the isolated system decreases exponentially and irreversibly with proceeding time t. Its rate-determining step is the transfer of entropy production from i to f. The integral over the entropy production of the final states f vanishes. Therefore the continuum states f do not contribute to the total entropy production P(t). The temporal evolution of the entropy S(t) can easily be calculated by integration of P(t). In conclusion, the entropy production of a non-equilibrium state is finite P(t) > 0 and only vanishes in the case of the equilibrium state P(∞) = 0. Thus, quantum thermodynamics agrees with the second law of thermodynamics.

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Approximations of time-dependent phenomena in quantum mechanics: adiabatic versus sudden processes

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By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.

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Approximations of time-dependent phenomena in quantum mechanics: adiabatic versus sudden processes

S V Melnichuk, W. van Dijk, Yukihisa Nogami

By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.

European Journal of Physics•Dec 15, 2004•6 citations
The time evolution and its implications with regard to the approach to equilibrium of a single-particle system in a suddenly expanded box with perfectly reflecting walls is investigated using density-matrix techniques. It is shown that the time-dependent quantities exhibit a periodic behavior which is independent of the initial density matrix as long as the dimensions of the system are finite. The period is found to be 2\ensuremathπ\ensuremathℏE1\frac{2\ensuremath{\pi}\ensuremath{\hbar}}{{E}_{1}}E1​2\ensuremathπ\ensuremathℏ​ where E1{E}_{1}E1​ is the ground-state energy of the expanded system. The continuum approximation is employed to determine the explicit time dependence of the particle density of a system initially in thermal equilibrium. It is found that the density for an expansion into a finite volume decays essentially as 1t\frac{1}{t}t1​ with small oscillations superposed on this behavior for times ranging from shortly after expansion up to one-fourth the characteristic period.
A study is made of the temporal evolution and its implications in regards to the approach to equilibrium of a system of noninteracting particles that has undergone a sudden expansion. It is assumed that the system is initially in thermal equilibrium. It is shown that the particle density approaches an equilibrium density essentially as 1t\frac{1}{t}t1​ for times shortly after expansion up to one-half the characteristic period. This period is the same as for a single-particle system. An expression is derived for the particle density that is valid for Bose, Fermi, and Boltzmann particles. The density for Bose and Fermi particles is compared with that found for Boltzmann particles.
I study the time evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density |Ψ( x , t )| 2 shows up quite a surprising behaviour: for definite times, plateaux appear for which |Ψ( x , t )| 2 is constant on finite intervals for x . Elements of theoretical explanation are given by analysing the singular component of the second derivative ∂ xx Ψ( x , t ). Analytical closed expressions are obtained for some specific times, which easily allow us to show that, at these times, the density organizes itself into regular patterns provided the size of the box is large enough; more, above some critical size depending on the specific time, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.
I study the time-evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density |Ψ(x, t) | 2 shows up quite a surprising behaviour: for definite times, plateaux appear for which |Ψ(x, t) | 2 is constant on finite intervals for x. Elements of theoretical explanation are given by analyzing the singular component of the second derivative ∂xxΨ(x, t). Analytical closed expressions are obtained for some specific times, which easily allow to show that, at these times, the density organizes itself into regular patterns provided the size of the box in large enough; more, above some critical time-dependent size, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.
We consider the temporal evolution of the entropy S of an isolated system from an initial non-equilibrium state 2· of entropy S = S i to the equilibrium state of maximum entropy S = S max > S i. The application of usual density matrix theory of quantum mechanics leads us to the surprising result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, because the entropy of the isolated system could not spontaneously increase from S i to S max by entropy production P > 0. This contradicts the second law of thermodynamics. We avoid this discrepancy by changing from quantum mechanics to quantum thermodynamics. Now the entropy production P = dS/dt and the time t are the relevant observables. In order to treat the temporal development of S and P, we at first prepare an initial state i of entropy production P i at time t < 0. Subsequently at ϑ t ≥ 0 the initial state i interacts with a continuum of final states f, which transfers the entropy production from i to f. The temporal evolution of the entropy production P(t) of our system is treated by means of the time-dependent perturbation theory. For that reason we split the operator of entropy production P = P 0 + W into two parts. The operator P 0 describes the unperturbed basic vectors of entropy production | i > and | f >, while the perturbation operator W determines the interaction between | i > and | f >. We expand the time-dependent state function < t | ψ > of the isolated system into the complete set of stationary eigenvectors | i > and | f > and calculate the time-dependent expansion coefficients c i(t) and c(f, t) by means of the dynamic equation of quantum thermodynamics − i k (d/dt) < t | ψ > = < t | P | ψ >. Here k means the Boltzmann constant, which is characteristic of quantum thermodynamics and represents the atomic entropy unit. The expansion coefficients are given by a system of coupled differential equations, from which we deduce an integral-differential equation for the coefficient c i(t) of the initial state i. Unfortunately it has been impossible to obtain an analytic solution of the complicated equation. However approximations are available. Accordingly, in the vicinity of the equilibrium state, the probability of still finding the initial state i of entropy production P i at time t is given by ω i(t) = | c i(t) |2 = exp (−Lt). The rate constant L ≥ σ depends on the density ρ(P f) of final states f of entropy production P f near by P i, and on the squared modulus of the perturbation matrix elements W if = < i | W | f >, which determines the interaction between | i > and | f >. The entropy production P(t) = P i ω i(t) = P i exp (−ϑLt) of the isolated system decreases exponentially and irreversibly with proceeding time t. Its rate-determining step is the transfer of entropy production from i to f. The integral over the entropy production of the final states f vanishes. Therefore the continuum states f do not contribute to the total entropy production P(t). The temporal evolution of the entropy S(t) can easily be calculated by integration of P(t). In conclusion, the entropy production of a non-equilibrium state is finite P(t) > 0 and only vanishes in the case of the equilibrium state P(∞) = 0. Thus, quantum thermodynamics agrees with the second law of thermodynamics.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.