In physics, the thermal de Broglie wavelength ( λ th {\displaystyle \lambda _{\text{th}}} , sometimes also denoted by Λ {\displaystyle \Lambda } ) is a measure of the uncertainty in location of a particle of thermodynamic average momentum in an ideal gas. It is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature.
Quantum-classical boundary We can take the average interparticle spacing in the gas to be approximately (V/N)1/3 where V is the volume and N is the number of particles. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. On the other hand, when the thermal de Broglie wavelength is on the order of or larger than the interparticle distance, quantum effects will dominate and the gas must be treated as a Fermi gas or a Bose gas, depending on the nature of the gas particles. The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. That is, the quantum nature of the gas will be evident for
V N λ th 3 ≤ 1 , or ( V N ) 1 / 3 ≤ λ th {\displaystyle \displaystyle {\frac {V}{N\lambda _{\text{th}}^{3}}}\leq 1\ ,{\text{ or }}\left({\frac {V}{N}}\right)^{1/3}\leq \lambda _{\text{th}}}
i.e., when the interparticle distance is less than the thermal de Broglie wavelength; in this case the gas will obey Bose–Einstein statistics or Fermi–Dirac statistics, whichever is appropriate. This is for example the case for electrons in a typical metal at T = 300 K, where the electron gas obeys Fermi–Dirac statistics, or in a Bose–Einstein condensate. On the other hand, for
V N λ th 3 ≫ 1 , or ( V N ) 1 / 3 ≫ λ th {\displaystyle \displaystyle {\frac {V}{N\lambda _{\text{th}}^{3}}}\gg 1\ ,{\text{or}}\ \left({\frac {V}{N}}\right)^{1/3}\gg \lambda _{\text{th}}}
i.e., when the interparticle distance is much larger than the thermal de Broglie wavelength, the gas will obey Maxwell–Boltzmann statistics. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source.
Massive particles For non-interacting particles with mass, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Assuming a 1-dimensional box of length L, the partition function (using the energy states of the 1D particle in a box) is
Z = ∑ n exp ( − E n k B T ) = ∑ n exp ( − h 2 n 2 8 m L 2 k B T ) . {\displaystyle Z=\sum _{n}\exp {\left(-{\frac {E_{n}}{k_{\text{B}}T}}\right)}=\sum _{n}\exp {\left(-{\frac {h^{2}n^{2}}{8mL^{2}k_{\text{B}}T}}\right)}.}
Since the energy levels are extremely close together, we can approximate this sum as an integral:
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