Free Particle Density Matrix and Partition Function

This is a methodical approach with the physics being the system of free particles, for example say, ideal gas. The result for the free particle may be applied to a large number of free particles. Applying the Cauchy problem in the coordinate q representation for the free particle we have:

This is a system of equations of a diffusive motion with standard solution for given initial conditions:

For the free particle transition amplitude (see details in [1] when U{q) = 0 in 4.39 of chapter 4), we expect a motion with constant velocity,

Then following the procedure from 4.47 to 4.79 of chapter 4, the transition amplitude and in terms of the classical action where the classical action

was derived in [1] via the classical equation of motion. The quantity 15.110 simply is a diffusive motion or spreading of a localized wave packet that started as a Dirac delta-function centered at q.

We apply statistical physics by doing the analytic continuation in 15.55, which transforms the transition amplitude К in quantum mechanics to the density matrix p{qi„q_aY-

where, the density matrix is p(q_b,q_a) [1,5,9]. So, all formulae obtained in path integration in quantum mechanics may be transformed into statistical physics formulae via the change of variables (analytical prolongation)

and the quantum-statistical density matrix for the free particles corresponds formally to or

This corresponds to the solution of 15.105 for the one-dimensional case and for/? = 0 we have If L is the length of the one-dimensional system then

For the case of the three-dimensional system with volume V then For the case of N particles then

This permits us to derive all the thermodynamic quantities.

Quantum Harmonic Oscillator Density Matrix and Partition Function

Detail derivation of the action functional and transition amplitude of the harmonic oscillator may be found in the reference [1]. The Lagrangian of a harmonic oscillator has the form:

and the action for the classical path of the harmonic oscillator is found to be

and the transition amplitude

with

From the procedure 15.74 to 15.77, then the quantum-statistical density matrix p(q_b,q_a>-ihP) for the harmonic oscillator is obtained:

with

We find the partition function from the quantum-statistical density matrix via the trace:

This involves setting q_b = q_a = q. This implies integrating over periodic paths starting and ending at q as in Figure 100. Integrating over q is equivalent to integrating over all periodic paths with period hfi. This yields the Gaussian integral

This agrees with the partition function for the one-dimensional oscillator evaluated from

This equation may be obtained via the Cauchy problem in the coordinate ^-representation for the harmonic oscillator:

The exact solution for this problem is written exactly in 15.124 and 15.125.

The free energy F of the harmonic oscillator is evaluated:

The average over the ensemble of the energy E in thermodynamics is also called internal energy and may be denoted by E:

where the average occupation of statistical oscillators (Bose-Einstein distribution n)

The model of real physical systems are not the oscillators themselves but the selection of infinite number of free or forced oscillators. Suppose we have an infinite number of selections of free oscillators with the spectrum {<«;}. From the property of additivity of the free energy F, and the internal energy E of the system, and in the absence of these expressions in the terms describing an interaction of oscillators then we have, respectively, the free energy F and the internal energy E:

Here, the average selection of occupation numbers n, is given by:

Except the term, the contribution to F of the i^lb oscillator is negligible when /ihto, » 1. For low

temperatures, the high-frequency modes are frozen out and so do not contribute to the specific heat. For the case of the Lagrangian for an oscillator subjected to a time-dependent force y(t) then

and the classical action is found to be with

If y(t) = 0 then we achieve the action of a free harmonic oscillator. The transition amplitude of a forced harmonic oscillator corresponds to a purely quantum mechanical result is found to be

From the procedure 15.74 to 15.77, then the quantum-statistical density matrix p(q_b,q_a,-ihf)) for the forced harmonic oscillator

Here, the classical action is found to be

Considering the standard Gaussian integral:

Then the partition function for the forced harmonic oscillator becomes where

Here, the driving force у is not yet defined. We can equally rewrite the partition function in terms of occupation numbers, say, for example, phonons via

But the mean occupation n(hcop) is and

So,

All these formulae will be useful in the evaluation of the polaron and bipolaron characteristics in the quasi-OD quantum dot and quasi-ID quantum wire.