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A computational introduction to quantum statistics using harmonically trapped particles Martin Ligare∗ Department of Physics & Astronomy, Bucknell University, Lewisburg, PA 17837 (Dated: March 15, 2016) Abstract In a 1997 paper Moore and Schroeder argued that the development of student understanding of thermal physics could be enhanced by computational exercises that highlight the link between the statistical definition of entropy and the second law of thermodynamics [Am. J. Phys. 65, 26 (1997)]. I introduce examples of similar computational exercises for systems in which the quantum statistics of identical particles plays an important role. I treat isolated systems of small numbers of particles confined in a common harmonic potential, and use a computer to enumerate all possible occupation-number configurations and multiplicities. The examples illustrate the effect of quantum statistics on the sharing of energy between weakly interacting subsystems, as well as the distribution of energy within subsystems. The examples also highlight the onset of Bose-Einstein condensation in small systems. PACS numbers: 1 I. INTRODUCTION In a 1997 paper in this journal Moore and Schroeder argued that the development of student understanding of thermal physics could be enhanced by computational exercises that highlight the link between the statistical definition of entropy and the second law of thermodynamics.1 The first key to their approach was the use of a simple model, the Ein- stein solid, for which it is straightforward to develop an exact formula for the number of microstates of an isolated system. The second key was the use of a computer, rather than analytical approximations, to evaluate these formulas. This approach, using the micro- canonical ensemble appropriate for an isolated system, highlights the role of probability in determining the thermal properties of a system. (The use of the Einstein solid as a pedagogical tool was developed further in an introductory text by Moore2 and a more advanced text by Schroeder.3) In this paper I adopt the approach of Moore and Schroeder, and apply it to simple systems in which the quantum statistics of identical particles plays an important role. The examples are designed to amplify that connection between the statistical definition of entropy and temperature, as developed by Moore and Schroeder, while illustrating the profound effect of the statistics of identical particles. They also provide students with a connection to the vibrant field of trapped ultra-cold atoms. (For an overview of the field, see, for example, Ref. 4 and Ref. 5.) The examples considered in this paper demonstrate the breakdown of classical equiparti- tion, the different occupation-number distributions for bosons, fermions, and distinguishable particles, and the onset of Bose-Einstein condensation. The goal of the approach used in this paper is not the derivation of analytical results using sophisticated mathematics. Rather, it is the use of an unsophisticated technique — the explicit counting of the states of an isolated system with the aid of a computer — to help students gain insight into the effects of quantum statistics on thermal properties, thereby extending the pedagogical treatment of Moore and Schroeder. I first consider a gas with a fixed number of weakly-interacting particles trapped together in a one-dimensional harmonic potential. This potential has the advantage that the multiplicity for a collection of distinguishable classical particles is identical to that of the Einstein solid considered by Moore and Schroeder. The deviations from the thermal properties ex- plored in Ref. 1 are thus directly traceable to the quantum statistics of identical particles. 2 In order to highlight the unequal sharing of energy between fermions and bosons at a common temperature, as well as the different energy distributions for the two kinds of particles, the one-dimensional systems I consider consist of mixtures of both bosons and fermions in a common potential. I then consider a gas of particles trapped in a three-dimensional isotropic harmonic trap, and explore how the increased density of states (due to the degeneracy of the energy levels) leads to the onset of Bose-Einstein condensation. II. TRAPPED PARTICLES IN ONE DIMENSION A. States and Multiplicities The energy levels and multiplicities of the Einstein solid considered by Moore and Schroeder are identical to those of N non-interacting distinguishable particles of mass m confined together in a one-dimensional harmonic oscillator potential characterized by the angular frequency ω. The energy of any individual particle is Ei = mihω¯ ≡ ǫmi, (1) where the index i labels the particle, and mi is a non-negative integer. The total energy of the gas of N trapped particles is thus N U = miǫ ≡ qǫ, (2) Xi=0 where q is another non-negative integer . (I have ignored the zero-point energy, which plays no role in the statistical mechanics of the system.) As in the case of the Einstein solid discussed in Ref. 1, the multiplicity for N trapped distinguishable particles with q units of energy is q + N − 1 (q + N − 1)! ΩD)(N,q)= = . (3) q q!(N − 1)! The difference between the gas of trapped particles and the Einstein solid arises when the particles are identical, i.e., indistinguishable. In the case of the Einstein solid the particles are distinguished by their positions in the lattice (with the assumption that the interparticle spacing is large compared to the spatial extent of a single-particle wavefunction); in the case of trapped particles, all particles occupy the same space near the minimum of the potential, and the quantum statistics of identical particles must be considered. 3 4ǫ 3ǫ 2ǫ ǫ Single particle energy 0 Distinguishable: 3 6 3 3 15 states Bosons: 11 1 1 4 states Fermions: 0 1 0 0 1 state Multiplicity FIG. 1: Possible occupation-number configurations of 3 particles with total energy 4ǫ in a one- dimensional harmonic oscillator potential. The number of distinct states for each configuration is given for distinguishable particles, bosons, and fermions. (For simplicity I have assumed that all of the fermions have the same spin orientation, so that at most one fermion may occupy a state.) For low-energy closed systems with small numbers of particles it is straightforward to demonstrate the effect of quantum statistics on occupation numbers by writing down all possible ways the units of energy can be distributed between the available states. A partic- ular distribution, with n0 units of energy in the ground state, n1 units in the first excited state, etc., is known as an occupation-number configuration, and can be written as the set of integers {n0, n1, n2, ... }. The enumeration of configurations “by hand” is the basis for problems in several undergraduate texts. For example, Griffiths6 presents problems based on small numbers of particles in square well potentials, and Schroeder7 leads students to in- vestigate particles in the one-dimensional harmonic potential considered in this paper. The strategy used in this paper is identical to that used to solve these textbook problems, so students who have completed exercises like these should be ready to appreciate how a computer can be used to extend this kind of investigation to larger systems with more energy. Before turning to larger systems, I first review the results of a typical textbook exercise by considering the case of 3 particles in a harmonic potential with 4 units of energy. The possible occupation-number configurations for this situation are illustrated in Fig. 1. The multiplicity for distinguishable particles is Ω(D) = 15, consistent with Eq. (3). The multiplicity for bosons, Ω(B) = 4, is less than the multiplicity for distinguishable particles, because without labels on the particles there are fewer distinct ways to arrange them, and the multiplicity 4 for fermions, Ω(F) = 1, is smaller still, because of the prohibition on multiply occupied states articulated in the Fermi exclusion principle. (To keep things simple, I am assuming that all of the fermions have the same spin orientation, so that only one fermion can occupy any single-particle state.) The number of energy units per particle in the example of Fig. 1 is only 4/3, which means that a large fraction of the configurations for distinguishable particles and bosons have multiply-occupied states. In the limit in which the number of energy units per particle is large, the fraction of configurations with multiply occupied levels goes to zero, and the multiplicity for distinguishable particles is larger than the multiplicity for identical particles by the factor N!. In this limit the entropy of a system of distinguishable particles differs from the entropy of a system of identical particles by the constant additive term ln(N!), and the thermodynamics of bosons, fermions, and distinguishable particles are identical. The quantum statistics of identical particles will only reveal itself at energies for which classical statistics predicts a significant probability of multiple occupation. The enumeration of all possible states “by hand” that is possible for small systems quickly becomes a daunting task as the number of particles grows. In the following paragraphs I discuss algorithms for determining the multiplicities Ω(B) and Ω(F) for arbitrary numbers of of bosons and fermions respectively. These algorithms are well-suited to implementation on a computer, so that students can go beyond standard textbook exercises and gain further insight into the thermodynamic properties of identical particles in larger systems. For identical bosons there is one state for each of the possible occupation-number configurations, {n0,n1,n2,... }. Each of the possible configurations with q units of energy must satisfy the condition ∞ q =0 × n0 +1 × n1 +2 × n2 + ··· = i × ni, (4) Xi=0 subject to the constraint that the total number of particles is fixed, i.e., ni = N.

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