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European Journal of Physics

A note on the zero point energy of an ideal boson gas

Francis A. Deeney, J P O'Leary

Published Aug 2, 2011• 2 citations
Publisher

Abstract

We have recently demonstrated that, when investigating the internal energy and pressure of an ideal boson gas at temperatures below the Bose–Einstein temperature, it is necessary to include the contribution from the kinetic energy of the particles in the condensate. Here we express the same requirement in terms of the constraints of the Heisenberg uncertainty principle applied to the particles confined in the condensate. The problem may be used by undergraduate students as an additional example of the connection between the ground state energy of a system and the Heisenberg uncertainty principle.

DOI
10.1088/0143-0807/32/5/l01

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We have recently demonstrated that, when investigating the internal energy and pressure of an ideal boson gas at temperatures below the Bose–Einstein temperature, it is necessary to include the contribution from the kinetic energy of the particles in the condensate. Here we express the same requirement in terms of the constraints of the Heisenberg uncertainty principle applied to the particles confined in the condensate. The problem may be used by undergraduate students as an additional example of the connection between the ground state energy of a system and the Heisenberg uncertainty principle.
The ideal Bose gas has two major shortcomings: at zero temperature, all the particles 'condense' at zero energy or momentum, thus violating the Heisenberg principle; the second is that the pressure below the critical point is independent of density resulting in zero incompressibility (or infinite isothermal compressibility) which is unphysical. We propose a modification of the ideal Bose gas to take into account the Heisenberg principle. This modification results in a finite (in)compressibility at all temperatures and densities. The main properties of the ideal Bose gas are preserved, i.e. the relation between the critical temperature and density, but the specific heat has a maximum at the critical temperature instead of a discontinuity. Of course interactions are crucial for both cases in order to describe actual physical systems.
From a certain temperature on, the molecules “condense” without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there also some truth to it? Albert Einstein, Letter to Ehrenfest (Dec. 1924). Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein , Oxford University Press, New York (1982) Introduction For over 50 years the low-temperature liquid state of uncharged, spinless He 4 was the only system in which a Bose–Einstein (BE) condensation was considered experimentally realized. In that case, cold (<2.19K) liquid He 4 passes into an extraordinary phase of matter called a superfluid , in which the liquid's viscosity and entropy become zero. With advances in atomic cooling (to ≈ 10 –9 K) the number of Bose systems which demonstrably pass into a condensate has considerably increased. These include several isotopes of alkali gas atoms as well as fermionic atoms that pair into integerspin (boson) composites. Although an ideal Bose gas does exhibit a low-temperature critical instability, the ideal BE gas theory is not, on its own, able to describe the BE condensate wave state. In order for a theory of bosons to account for a condensate wave state, interactions between the bosons must be included. Nevertheless, considerable interesting physics is contained in the ideal Bose gas model.
Schrödinger's paper (1926) on Bose–Einstein condensation was submitted for publication on 15 December, 1925, immediately preceding the first of his papers on wave mechanics. Historians of science (Hanle, 1977; Klein, 1964) have seen ‘an organic connection’ between the two. It came between the three papers of Einstein (1924, 1925 a, b ) on the boson gas and Uhlenbeck's doctoral thesis (Uhlenbeck, 1927), and did not receive the attention it deserved, although, with hindsight, it is seen to contain the gem of the idea which was later to resolve the conflict between Einstein and Uhlenbeck. Einstein based his prediction (Einstein, 1925 a ) of condensation in the free boson gas on a combination of Bose–Einstein statistics and the classical density of states in phase-space; this leads to an expression for the mean particle number density as a function of kinetic energy which saturates as the total density increases so that there is a maximum possible density of particles having nonzero kinetic energy. Einstein claimed that, if the particle density is increased above this maximum, the excess goes into a state with zero kinetic energy. This is the phenomenon of condensation. Uhlenbeck (1927) objected that the result holds only ‘when the quantization of translational motion is neglected’. The situation remained unclear until the Amsterdam meeting to celebrate the centenary (23 November 1937) of the birth of Johannes van der Waals; here Kramers pointed out the importance of the thermodynamic limit for the sharp manifestatioon of phase transitions.
The pressure in the thermodynamic limit of a non-ideal Boson gas whose Hamiltonian includes only diagonal and pairing terms can be expressed as the infimum of a functional depending on two measures on momentum space: a positive measure describing the particle density and a complex measure describing the pair density. In this paper we examine this variational problem with the object of determining when the model exhibits Bose-Einstein condensation. In addition we show that if the pairing term in the Hamiltonian is positive then it has no effect
The nature of the Einstein-Bose condensation in an imperfect boson gas is discussed. It is argued that no matter how low the temperature, provided it is not zero, macroscopic second-sound waves should be weakly excited as thermal fluctuations about equilibrium. This implies that the single-particle momentum distribution must be smeared so that the state of zero momentum cannot contain a finite fraction of the particles. A microscopic mechanism which independently leads to a momentum spread is discussed. It is suggested that the actual Einstein-Bose condensation of an imperfect gas takes place in energy rather than momentum but that even in terms of this variable it becomes perfectly sharp only at zero temperature. This hypothesis is formulated analytically in terms of correlation functions of the second-quantized boson field.