Thermal wavelength

The thermal wavelength or thermal de Broglie wavelength is the mean de Broglie wavelength of a particle at a certain temperature . The thermal wavelength characterizes the spatial "expansion" of a particle and represents the link between classical and quantum statistics .

definition

According to the wave-particle dualism, a particle can have a wavelength of

${\displaystyle \lambda ={\frac {h}{p}}}$

With

• Planck's quantum of action ${\displaystyle h=2\pi \hbar }$
• relativistic impulse ${\displaystyle p}$

be assigned.

For the energy of the particle will

${\displaystyle E=\pi \cdot k_{\mathrm {B} }\cdot T={\frac {p^{2}}{2\cdot m}}}$

assumed with

• Boltzmann's constant ${\displaystyle k_{\mathrm {B} }}$
• absolute temperature ${\displaystyle T}$
• The factor comes from the sum of states , it appears there in the same power as the thermal energy and is therefore adopted as an energy factor . ${\displaystyle \pi }$ ${\displaystyle k_{\mathrm {B} }T}$

The result is the thermal wavelength

${\displaystyle \Rightarrow \lambda ={\frac {h}{\sqrt {2\pi \cdot m\cdot k_{\mathrm {B} }\cdot T}}}=\hbar {\sqrt {\frac {2\pi }{m\cdot k_{\mathrm {B} }\cdot T}}}}$

with the mass of the particle. ${\displaystyle m}$

motivation

To motivate the above definition, consider the wave vector that is given in a statistical ensemble by ${\displaystyle {\vec {k}}}$

${\displaystyle {\begin{aligned}k&={\frac {1}{\hbar }}\int _{-\infty }^{+\infty }\mathrm {d} p\;\exp \left(-\beta \,{\frac {p^{2}}{2m}}\right)\\&={\frac {1}{\hbar }}{\sqrt {\frac {2m\pi }{\beta }}}\\&={\frac {2\pi }{h}}{\sqrt {2m\pi k_{\mathrm {B} }T}}\end{aligned}}}$

with the energy normalization ${\displaystyle \beta :=(k_{\mathrm {B} }T)^{-1}.}$

The following also applies to the magnitude of the wave vector ( circular wave number ): ${\displaystyle k}$ ${\displaystyle k={\frac {2\pi }{\lambda }}.}$

Examples

Some examples of the thermal de Broglie wavelength at 298 K.

Particle	${\displaystyle m}$ (kg)	${\displaystyle \lambda _{\mathrm {th} }}$ (m)
H 2	3.3474E-27	7.1228E-11
N 2	4.6518E-26	1.91076E-11
O 2	5.31352E-26	1.78782E-11
F 2	6.30937E-26	1.64105E-11
Cl 2	1.1614E-25	1.2093E-11
HCl	5.97407E-26	1.68586E-11

meaning

The thermal wavelength is a simple means of estimating the quantum nature of a system. Quantum effects start to play a role when the thermal wavelength is combined with other characteristic lengths of the system - such as the mean free path of the particles or the mean particle distance , where the particle number density is - becomes comparable. In the case of a sharp phase transition between the classical and quantum system, the temperature at the transition is also called the transition temperature . ${\displaystyle n^{(-1/3)}}$ ${\displaystyle n}$

As can be seen directly from the above definition, the wavelength increases with decreasing temperature. The mean free path decreases with increasing pressure. As a result, a gas no longer behaves classically at low temperatures or high pressures. From such considerations one deduces, for example, that white dwarfs are stabilized by quantum effects due to the extremely high pressures inside.

Bose-Einstein condensates can arise when the thermal wavelength is in the range of the distance between two atoms. Therefore, to generate such condensates, the materials must be brought to extremely low temperatures.

literature

• Walter Grimus: Introduction to Statistical Physics and Thermodynamics: Fundamentals and Applications , Oldenbourg, 2010, ISBN 978-3-486-70205-7 , p. 75, limited preview in the Google book search.