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European Journal of Physics

Thermodynamics of a particle in square well potential

Satadal Bhattacharyya, Joydip Mitra

Published Oct 7, 2022• 2 citations
Publisher

Abstract

Abstract We have presented the thermodynamic properties of a particle moving freely both in an infinite and finite square well in contact with a heat reservoir at constant temperature. In both cases, we have computed the temperature dependence of the partition function, entropy, average energy and specific heat of the particle in an exact way. The exact results for various thermodynamic parameters deviate considerably from its known theoretical values and reveal the importance of such calculations. The effect of the width of the finite well on the different thermodynamic quantities have also been examined critically.

DOI
10.1088/1361-6404/ac986c

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Abstract We have presented the thermodynamic properties of a particle moving freely both in an infinite and finite square well in contact with a heat reservoir at constant temperature. In both cases, we have computed the temperature dependence of the partition function, entropy, average energy and specific heat of the particle in an exact way. The exact results for various thermodynamic parameters deviate considerably from its known theoretical values and reveal the importance of such calculations. The effect of the width of the finite well on the different thermodynamic quantities have also been examined critically.
Abstract In order to specify the uncertainty relation for a particle confined in a finite square well potential we derived the standard deviation of momentum Δ p and that of position Δ x . It is shown that Δ p has a peak as a function of the well width and both Δ x and Δ p ⋅ Δ x have a dip in contrast to the case of the particle confined in an infinite square well potential. We clarified that Δ p 0 ⋅ Δ x 0 of the particle occupying the ground state exists in the finite range as a function of the well width and the potential-barrier height, but a particle confined in an infinite square well potential has a constant Δ p 0 ⋅ Δ x 0 . In addition, we found that the values of Δ p n ⋅ Δ x n for excited states diverge in a certain range of well widths, where the energy level of the confined particle exceeds the potential barrier.
The partition function of a three-dimensional Ising model was calculated by a high-speed computer for a cubic array of twenty-seven points. The periodicity condition was imposed. The specific heat of the 3×3×3 lattice was calculated as a function of temperature. The critical temperature estimated from the specific heat curve seems slightly lower that obtained from the Padé approximation. The zeros of the partition function of the complex temperature plane were also obtained. The critical temperature estimated from the zeros near the positive real axis seems slightly higher than both of the above values. Similar calculations were carried out for the case of the two-dimensional Ising model. Several results are compared with those obtained from Onsager's exact solution.
At zero-temperature and finite-temperature, the thermodynamic properties of finite unitary Fermi gas in a three-dimensional harmonic trap are investigated by using fractional exclusion statistics, and the results are compared with those of the system which satisfies the thermodynamic limit. At zero-temperature, Fermi energy and average energy of per particle increase with the increase of the number of particles for finite unitary Fermi gas, and their limits are the corresponding parameters of the system which satisfy thermodynamic limits. Fermi energy and average energy of per particle each have a maximum value changing with the boundary of the potential well. For the finite-temperature trapped unitary Fermi system, when the number of particles is certain the average energy of per particle, average entropy of per particle, average heat capacity of per particle each have a characteristic temperature, respectively, when the temperature is equal to the characteristic temperature of the physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the temperature is lower (or higher) than the characteristic temperature of parameter, the physical parameter of the finite system will be greater (or less) than the corresponding parameter of the thermodynamic limit system. The characteristic temperature has particle number effect and boundary effect. When the temperature is determined, the average energy of per particle, average entropy of per particle and average heat capacity of per particle each have a characteristic number of particles, respectively, when the number of particles is equal to the characteristic number of particles for physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the number of particles is less (or more) than the characteristic number of particles for corresponding parameter, the corresponding parameter of the finite system will be less (or larger) than the thermodynamic limit of system.
Following the binary collision approach to the many-body problem in Quantum Statistical Mechanics, as developed recently by Lee and Yang, the equation of state, to the order a2/λ2, is derived for a system of particles interacting through a square-well potential and obeying Bose-Einstein or Fermi-Dirac statistics. Here, λ is the mean thermal wavelength of the particles and a represents the range of the two-body interaction. Expressions are obtained for the second and the third virial coefficients of the system and also for its various thermodynamical properties, such as energy, specific heat and entropy, at finite but low temperatures.
In this paper the detailed investigation of one of the most interesting models in the non relativistic quantum mechanics of one massive particle i.e., introduced by G. Poeschl and E. Teller in 1933 is continued; the starting point of dynamic analysis for potentials, wave functions and energy spectrum was carried on by the same authors in the paper. The generalization of these results on the case of nonzero temperature includes the approximate calculation of the partition function of the model and then also the Massieu Planck thermodynamic potential and its most important derivatives, i.e. internal energy, heat capacity and pressure. The analysis of the results is carried on separately for low and high temperature regions because of the lack of exact unifying expressions of the potential at all temperatures.
Abstract Explicit expressions for the critical amplitudes of the specific heat and other thermodynamic functions of the three‐dimensional Ising model with extracted universal part and non‐universal factor are obtained applying the non‐Gaussian measure density of fluctuation distribution for the calculation of the partition function. The curves for thermodynamic characteristics as functions of temperature and microscopic parameters of the Hamiltonian near the phase transition point are presented.