# Wave-particle duality and temperature

Today it was told me that wave properties of a particle increase if the temperature decreases. I'm surprised because I have never listened a similar thing, but I think that it's very interesting.

Could you explain me why it happens?

• Could you please say were did you hear this. In a lesson? – Constantine Black May 7 '15 at 18:09
• Please don't use the outdated language of the wave-particle duality. Is your question why many quantum effects become more relevant at low temperatures? – ACuriousMind May 7 '15 at 18:10
• @ConstantineBlack it wasn't a lesson... it was a.. small talk. He said "Can anyone answer to this question?" – sunrise May 7 '15 at 18:16
• @ACuriousMind yes, exactly – sunrise May 7 '15 at 18:16
• @danimal It should be useful, but as actually taught it is usually counterproductive. Student come away thinking that these ideas represent a mystery to be solved rather than two useful limits of a deeper theory. – dmckee May 7 '15 at 19:02

I think that your teacher (?) asked you about thermal de Broglie wavelength, where $$\lambda_T \propto\frac{1}{\sqrt{T}}.$$ You get this expression when you express the momentum in $\lambda=h/p$ in in terms of kinetic energy and the kinetic energy itself in terms of the energy due to temperature. (The derivation is also in the wikipedia article...) Indeed, when you drop the temperature the thermal de Broglie wavelength increases.
As far as I understand it, the thermal de Broglie wavelength indicitates roughly when you have to treat a gas clasically or quantum mechanically. In the sense that when $\lambda_T$ is much smaller than the particle separation then the particles "have to do with each other" and if the wavelength is approximately the same as the particle separation then the particles "have to do with each other". Depending on how they "have to do with each other" you then have to treat the gas as a Bose- or a Fermi gas.