In physics, the thermal de Broglie wavelength (<math>\lambda_\text{th}</math>, sometimes also denoted by <math>\Lambda</math>) is a measure of the uncertainty in location of a particle of thermodynamic average momentum in an ideal gas. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source.
Massive particles
For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Assuming a 1-dimensional box of length , the partition function (using the energy states of the 1D particle in a box) is
<math display="block"> Z = \sum_n \exp{\left(-\frac{E_n}{k_\text{B}T}\right)} = \sum_n \exp{\left(-\frac{h^2 n^2}{8 m L^2 k_\text{B} T}\right)} .</math>
Since the energy levels are extremely close together, we can approximate this sum as an integral:
<math display="block"> Z = \int_0^\infty \exp{\left(-\frac{h^2 n^2}{8m L^2 k_\text{B}T}\right)} dn = \sqrt{\frac{2\pi m k_\text{B} T}{h^2 L \equiv \frac{L}{\lambda_\text{th .</math>
Hence,
<math display="block"> \lambda_\text{th} = \frac{h}{\sqrt{2\pi m k_\text{B} T ,</math>
where <math> h </math> is the Planck constant, is the mass of a gas particle, <math>k_\text{B}</math> is the Boltzmann constant, and is the temperature of the gas. If is the number of dimensions, and the relationship between energy () and momentum () is given by
<math display="block">E=ap^s</math>
(with and being constants), then the thermal wavelength is defined as
<math display="block">
\lambda_\text{th}=\frac{h}{\sqrt{\pi\left(\frac{a}{k_\text{B}T}\right)^{1/s}
\left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} ,
</math>
where is the Gamma function. This definition retains the following simple form for the chemical potential in the dilute (classical ideal gas) limit: