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We report model calculations of the time-dependent internal energy and entropy for a single quasi-free massive quantum particle at a constant temperature. We show that the whole process started from a fully coherent quantum state to thermodynamic equilibrium can be understood, based on statistics of diffracted matter waves. As a result of thermal interaction between the particle and its surroundings, the motion of the particle shows new feature.

Recent experiments of absorption of a single photon by a single atom [

We consider a structureless massive quantum particle moving in a space at constant temperature T as shown in figure 1. Although there is no interaction with other particles, the particle is quasi-free because a circular aperture of radius a_{0} for its matter wave is considered. The quantum particle is assumed to be of kinetic energy E_{0} initially at the origin and is described by a wave-packet sharply peaked at the de Broglie wavelength , with m being its mass and h the Plank’s constant. The wave front propagates along x-axis at group velocity V_{g} = h/(mλ). For simplicity, we assume that the radius a_{0} of the matter wave aperture is large compared with the wavelength λ so that the shape and linear dimension of the forward-going wave-front remains unchanged as the wave propagates. It is important that every point at the edge of the wave-front generates out-going fully spherical waves and the kinetic energy associated with the forward-going wave-front follows the form [

where L is a temperature dependent parameter of dimension length and is expected to be infinitely large as the temperature tends to zero. This is just the energy for the source to generate out-going fully spherical waves. We define a probability density function for the particle

There exists a step length d_{0}, so that the energy for fully spherical waves generated in d_{0} is E_{0}P_{E}(nd_{0})d_{0} when the forward-going plane-wave front is at the position nd_{0}, where n is an integer. In principle, the particle may be in any of these diffracted states besides the forward-going plane-wave state, i.e., the particle itself constitutes automatically a thermodynamic system as a result of diffraction at the edge of its matter-wave front. The probability for different energy states is known here and, in general, all energy states are not equally likely. If one can distinguish energy states in more detail, a smaller step Δx should be used and the degeneracy replaced by d_{0}/Δx. Thermal interaction between the particle’s system and the surrounding space becomes possible and the space here acts as the heat reservoir at constant temperature T.

The partition function of the particle should take into account contributions from forward-going wave-front and all spherical waves diffracted at the edge. The partition function at a given time is then

where the notation β = 1/k_{B}T is used as usual with k_{B} being the Boltzmann constant. The constant Z_{0} is defined as the non-zero real solution of the transcend equation

With the help of the Lambert W function [_{0} = −W_{−1}(1/2e^{1/2}) − 1/2 and we use an approximate value Z_{0} = 1.25643 in our numerical calculations. Here we have used the correct form of the step length instead of the wavelength, and t_{r} = t/t_{c} is the time scaled with a temperature dependent characteristic time defined by

The expectation value for the energy of the particle or its internal energy is

The first term comes from all those out-going spherical waves diffracted at the edge, and the second term from the forward-going wave-front itself as a whole. The internal energy of the particle is written as

with

,

depending on the temperature and the particle’s initial energy in a complicated form.

In _{0} is less or more than k_{B}T/2. At the special point E_{0} = k_{B}T/2, our numerical calculations show that the curve moves downward initially and then upward continuously, indicating exchange of energy taking place between the particle and its surrounding space in the whole process. The limiting value

of the internal energy for the freedom in the x-direction is k_{B}T/2, regardless of its initial energy. The limit is already reached for t/t_{c} > 3 in our numerical calculations and the overall decay or increase in internal energy does not follow the simple exponential form. Physically, the process describes the evolution of the particle starting from a fully coherent initial state with well defined energy in quantum mechanics to a series of possible states. Although states representing the forward-going wavefront remain coherent, those states diffracted at the edge at different time are no longer coherent at finite temperatures as a result of heat exchange. The process breaks down if a collision occurs in a gas.

After a sufficient long time, the probability for the particle in the coherent state disappears and the internal energy reaches the universal thermodynamic energy in equilibrium with the surrounding environment. The time needed to reach the limit of internal energy could in some sense be viewed as the coherence decay time of a quantum particle in a space at a finite temperature. Therefore, the present work might be utilized in understanding the temperature dependence for the coherence decay time observed recently for individual molecules at room temperature [

We now calculate the average value of the particle’s position to obtain information of where it is at a given time. Although out-going spherical waves diffracted at the edge are no longer coherent at finite temperatures, the overall probability projected on x-axis in the range –V_{g}t < x < V_{g}t can be calculated without mathematical difficulties. The thermodynamic expectation of the particle’s position in the x-direction is written as

The first term represents contributions from all those out-going spherical waves and the second term from the forward-going wave-front. The explicit result is

with being the exponential integral function of argument z [

_{c} for various combinations of temperature and initial energy. It increases continuously with increasing time, approaching a limit within a time of several t_{c}. It is true that a particle with higher initial energy reaches the limit at a later time. The limit is

where C = 0.577216 is the Euler’s constant [

The entropy, measuring the amount of uncertainty of the particle, can also be calculated because the probability distribution is known. The uncertainty relation in quantum mechanics tells us that waves diffracted at the edge should be indistinguishable when the wave-front moves forward within about one wavelength. We choose λ as the step length and rewrite the probability in discrete form. The entropy is calculated using the definition, which is perfectly unambiguous for time-dependent system of any size. Under the condition, we find the timedependent entropy of the particle for the coordinate is

Here the constant term has been neglected. The last term can be understood as the result of expansion of the particle’s matter-wave. The entropy starts from zero and then increase monotonically with increaseing time. As and reach their limits, the entropy for each freedom converges to a universal value

with no regards to the temperature and the initial state of the particle. The result holds also for a free quantum particle. As a consequence of the wavelike nature of the particle, the feature is purely quantum mechanical without classical counterpart.

The possibility of experimental observation depends on developing single quantum particle experiment in which there is very little chance for two wave packets to overlap. It seems to have been demonstrated by past experiments on the diffractions of neutrons [

The motion of a quasi-free massive quantum particle in a space at a finite temperature shows interesting phenomena arising from thermal interaction. The important point is that a nonzero temperature of the environment inherently destroys the coherence of a quantum particle and the whole process can be modeled.