The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. This simple model can be used to describe the classical ideal gas as well as the various quantum ideal gases such as the ideal massive Fermi gas, the ideal massive Bose gas as well as black body radiation which may be treated as a massless Bose gas. In physics, the particle in a box (or the square well) is a simple idealized system that can be completely solved within quantum mechanics. ... An ideal gas (also called a perfect gas) is a hypothetical fluid consisting of particles that are identical to each other, occupy negligible volume and undergo perfect elastic collisions with each other, with no intermolecular forces and no intramolecular storage of energy, as opposed to a real gas, a gas... A Fermi gas is a collection of non-interacting fermions. ... An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ...

Using the results from either Maxwell-Boltzmann statistics, Bose-Einstein statistics or Fermi-Dirac statistics we use the Thomas-Fermi approximation and go to the limit of a very large box, and express the degeneracy of the energy states as a differential, and summations over states as integrals. We will then be in a position to calculate the thermodynamic properties of the gas using the partition function or the grand partition function. These results will be carried out for both massive and massless particles. More complete calculations will be left to separate articles, but some simple examples will be given in this article. The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... For other topics related to Einstein see Einstein (disambig) In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ... In number theory, see Partition function (number theory) In statistical mechanics, see Partition function (statistical mechanics) In quantum field theory, see Partition function (quantum field theory) In game theory, see Partition function (game theory) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share...

Thomas-Fermi approximation for the degeneracy of states

For both massive and massless particles in a box, the states of a particle are enumerated by a set of quantum numbers [n_x, n_y,  n_z]. The absolute value of the momentum is given by:

where h is Planck's constant and L is the length of a side of the box. We can think of each possible state of a particle as a point on a 3-dimensional grid of positive integers. The distance from the origin to any point will be Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ...

Suppose each set of quantum numbers specify f  states where f  is the number of internal degrees of freedom of the particle that can be altered by collision. For example, a spin 1/2 particle would have f=2, one for each spin state. The Thomas-Fermi approximation assumes that the quantum numbers are so large that they may be considered to be a continuum. For large values of n , we can estimate the number of states with absolute value of momentum less than or equal to p from the above equation as

which is just f  times the volume of a sphere of radius n  divided by eight since we only consider the octant with positive n_i . The number of states with absolute value of momentum between p  and p+dp  is therefore

where V=L³  is the volume of the box. Notice that in using this continuum approximation, we have lost the ability to characterize the low-energy states including the ground state where n_i =1. For most cases this will not be a problem, but when considering Bose-Einstein condensation, in which a large portion of the gas is in or near the ground state, we will need to recover the ability to deal with low energy states.

Without using the continuum approximation, the number of particles with energy ε_i is given by

where

	for particles obeying Maxwell-Boltzmann statistics
	for particles obeying Bose-Einstein statistics
	for particles obeying Fermi-Dirac statistics

with β = 1/kT  with k  being Boltzmann's constant, T  being temperature, and μ being the chemical potential. Using the continuum approximation, the number of particles dN  with energy between E  and E+dE  is now written: The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... For other topics related to Einstein see Einstein (disambig) In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ...

The energy distribution function

We are now in a position to determine some distribution functions for the "gas in a box". The distribution function for any variable A is P_AdA and is equal to the fraction of particles which have values for A  between A  and A+dA

It follows that:

The distribution function for the absolute value of the momentum is:

and the distribution function for the energy E  is:

For a particle in a box, the relationship between energy E  and momentum p  is different for massive and massless particles. For massive particles, we have

while for massless particles:

where m  is the mass of the particle and c  is the speed of light. Using these relationships we have:

• For massive particles
  where Λ is the thermal wavelength of the gas.
  This is an important quantity, since when Λ is on the order of the interparticle distance (V/N)^1/3 , quantum effects begin to dominate and the gas can no longer be considered to be a Maxwell-Boltzmann gas.
• For massless particles
  where Λ is now the thermal wavelength for massless particles.

  Specific examples

  The following sections give an example of results for some specific cases. In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as: where h is Plancks constant m is the mass of a gas particle k is Boltzmanns constant T is the Temperature of the gas The thermal de Broglie... In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as: where h is Plancks constant m is the mass of a gas particle k is Boltzmanns constant T is the Temperature of the gas The thermal de Broglie...

  Massive Maxwell-Boltzmann particles

  For this case:

  Integrating the energy distribution function and solving for N gives

  Substituting into the original energy distribution function gives

  which are the same results obtained classically for the Maxwell-Boltzmann distribution. Further results can be found in the article on the classical ideal gas. The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ...

  Massive Bose-Einstein particles

  For this case:

  where z is defined as

  Integrating the energy distribution function and solving for N gives the particle number

  Where Li_s(z) is the polylogarithm function and Λ is the thermal wavelength. The polylogarithm term must always be positive and real, which means its value will go from 0 to ζ(3/2) as z  goes from 0 to 1. As the temperature drops towards zero, Λ will become larger and larger, until finally Λ will reach a critical value Λ_c where z=1 and The polylogarithm (also known as Jonquiéres function) is a special function and may be defined for all s and |z|<1 by: Both the parameter s and the argument z are taken to be complex numbers. ... In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as: where h is Plancks constant m is the mass of a gas particle k is Boltzmanns constant T is the Temperature of the gas The thermal de Broglie...

  The temperature at which Λ=Λ_c is the critical temperature. For temperatures below this critical temperature, the above equation for the particle number has no solution. The critical temperature is the temperature at which a Bose-Einstein condensate begins to form. The problem is, as mentioned above, that the ground state has been ignored in the continuum approximation. It turns out, however, that the above equation for particle number expresses the number of bosons in excited states rather well, and so we may write:

  where the added term is the number of particles in the ground state. (The ground state energy has been ignored.) This equation will hold down to zero temperature. Further results can be found in the article on the ideal Bose gas. An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. ...

  Massless Bose-Einstein particles (e.g. black body radiation)

  The most common massless bose gas is a gas of photons in a black body. Taking the "box" to be a black body cavity, the photons are continually being absorbed and re-emitted by the walls. The number of photons is not a conserved quantity, since we are actually in a relativistic regime. In the derivation of Bose-Einstein statistics, when the restraint on the number of particles is removed, this is effectively the same as setting the chemical potential (μ) to zero. Therefore: In physics, the photon (from Greek φοτος, meaning light) is a quantum of excitation of the quantised electromagnetic field and is one of the elementary particles studied by quantum electrodynamics (QED) which is the oldest part of the Standard Model of particle physics. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... For other topics related to Einstein see Einstein (disambig) In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ...

  The spectral energy density (energy per unit volume per unit frequency) is

  where E=hν has been used, and since the radiation is the same in all directions, and propagates at the speed of light (c), the spectral intensity (energy/time/area/solid angle/frequency) is

  which yields

  Since photons have two spin states, we have f=2 which yields Planck's law of black body radiation. Note that if we had carried out this procedure for massless Maxwell-Boltzmann particles, we would recover Wien's distribution which approximates a Planck's distribution for high temperature or low density.

  Massive Fermi-Dirac particles (e.g. electrons in a metal)

  For this case:

  Integrating the energy distribution function gives

  Where again, Li_s(z) is the polylogarithm function and Λ is the thermal de Broglie wavelength. Further results can be found in the article on the ideal Fermi gas. The polylogarithm (also known as Jonquiéres function) is a special function and may be defined for all s and |z|<1 by: Both the parameter s and the argument z are taken to be complex numbers. ... In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as: where h is Plancks constant m is the mass of a gas particle k is Boltzmanns constant T is the Temperature of the gas The thermal de Broglie... A Fermi gas is a collection of non-interacting fermions. ...

  References

  • Huang, Kerson, "Statistical Mechanics", John Wiley & Sons, New York, 1967.
  • A. Isihara, "Statistical Physics", Academic Press, New York, 1971.
  • L. D. Landau and E. M. Lifshitz, "Statistical Physics, 3rd Edition Part 1", Butterworth-Heinemann, Oxford, 1996.
  • Zijun Yan, "General thermal wavelength and its applications", Eur. J. Phys. 21 (2000) 625-631. http://www.iop.org/EJ/article/0143-0807/21/6/314/ej0614.pdf