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European Journal of Physics

-potential in the microcanonical ensemble

J C Martinez, E. Polatdemir

Published Dec 10, 2002• 0 citations
Publisher

Abstract

As an effort to present a new working example for an advanced class in statistical mechanics, we employ Lawson's path integral formalism to calculate the microcanonical density operator and apply it to a one-dimensional particle interacting with an attractive delta potential. As an offshoot we derive the ground-state energy for a system of bosons in one dimension interacting via the delta potential.

DOI
10.1088/0143-0807/24/1/305

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As an effort to present a new working example for an advanced class in statistical mechanics, we employ Lawson's path integral formalism to calculate the microcanonical density operator and apply it to a one-dimensional particle interacting with an attractive delta potential. As an offshoot we derive the ground-state energy for a system of bosons in one dimension interacting via the delta potential.
Path integral representations for the quantum microcanonical ensemble are presented. In the quantum microcanonical ensemble, two operators are of primary interest. First, $\stackrel{^}{\ensuremath{\rho}}=\ensuremath{\delta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})$ corresponds to the microcanonical density matrix and can be used to calculate expectation values. Second, $\mathrm{N\ifmmode \hat{}\else \^{}\fi{}}=\ensuremath{\Theta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})$ can give the number of states with energy En<E.{E}_{n}<E.En​<E. We consider position matrix elements of both of these operators $\ensuremath{\Omega}{(x,x}^{\ensuremath{'}},E)=〈{x}^{\ensuremath{'}}|\ensuremath{\delta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})|x〉$ and $\ensuremath{\Theta}{(x,x}^{\ensuremath{'}},E)=〈{x}^{\ensuremath{'}}|\ensuremath{\theta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})|x〉.$ A path integral formalism leads to exact integral representations for $\ensuremath{\Omega}{(x,x}^{\ensuremath{'}},E)$ and $\ensuremath{\Theta}{(x,x}^{\ensuremath{'}},E).$ We present both phase space and configuration space forms. For simple systems, such as the free particle and harmonic oscillator, exact solutions are possible. For more complicated systems, expansion schemes or numerical evaluations are required. A perturbative calculation and numerical integration results are presented for the quantum anharmonic oscillator.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.
An expansion method in the path-integral formulation of quantum mechanics, as proposed in a previous paper, is extended to account for states with odd parity. The method is tested on the case of a one-dimensional attractive delta-function potential and the well-known solutions for both bound and scattering states are obtained analytically by an exact summation of the expansion series. The applicability of the method is discussed and it is shown how the energy spectrum could be determined by means of Stieltjes theory of moments.
The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the qp and atoms are restricted to the sites of a one-dimensional lattice. A path integral formalism is developed within the context of the canonical ensemble in which the qp is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. For the case of a free particle, analytical expressions for the energy, its fluctuations, and the qp-qp correlation function are derived and compared with the Monte Carlo simulations. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp for a periodic interaction potential, forcing the qp to occupy extended states. We consider a striped potential in one dimension, where every other lattice site is occupied by an atom with potential ε, and every other lattice site is empty. This potential serves as a stress test for the path integral formalism because of its rapid site-to-site variation. An analytical solution was determined in this case by utilizing Bloch's theorem due to the periodicity of the potential. Comparisons of the potential energy, the total energy, the energy fluctuations, and the correlation function are made between the results of the Monte Carlo simulations and the analytical calculations.