Answer to Question #164888 in Quantum Mechanics for Vinit Kumawat

Question

Quantum mechanics is relevant, when the de Broglie wavelength of the particle is 

greater than the distance between particles. The purpose of this problem is to determine 

which systems will have to be treated quantum mechanically and which can be 

described classically.

a) Show that the typical de Broglie wavelength of a particle in an ideal gas in 

equilibrium is 𝜆 =

ℎ/√3𝑚𝐾𝐵𝑇

b) Solids: The lattice spacing in a typical solid is d = 0.3 nm. Find the temperature 

below which the free electrons in a solid are quantum mechanical? (Hint: Refer the 

a) part of the question and treat free electrons as a gas and the lattice spacing as the 

typical distance between the electrons)

c) Gases: For what temperatures are the atoms in an ideal gas at pressure 𝑃 quantum 

mechanical? (Hint: Use the ideal gas law, to deduce the inter atomic distance)

Below what temperature, is Helium at atmospheric pressure quantum mechanical?

Below what temperature is Hydrogen atoms in outer space quantum mechanical? 

(interatomic distance is 1 cm and temperature is 3 K)

Answer 1

a) In thermal equilibrium at (Kelvin) temperature T, the average kinetic energy of a particle is

$$\frac{p^2}{2m}=\frac{3}{2}kT$$

so the typical de Broglie wavelength is

$$\lambda=\frac{h}{\sqrt{3mkT}}$$

b)

$$T=\frac{h^2}{3mkd^2}\\T=\frac{(6.6\cdot10^{-34})^2}{3(3.9\cdot10^{-26})(1.4\cdot10^{-23})(3\cdot10^{-10})^2}=3\ K$$

c)

$$Pd^3=kT\\T=\frac{1}{k}\left(\frac{h^2}{3m}\right)^\frac{3}{5}(P)^\frac{2}{5}$$

He:

$$T=\frac{1}{(1.4\cdot10^{-23})}\left(\frac{(6.6\cdot10^{-34})^2}{3(6.8\cdot10^{-27})}\right)^\frac{3}{5}(10^5)^\frac{2}{5}=2.8\ K$$

To treat the helium quantum mechanically we need it to be in a temperature less than 2.8 K.

H:

$$T=\frac{h^2}{3mkd^2}\\T=\frac{(6.6\cdot10^{-34})^2}{3(3.4\cdot10^{-27})(1.4\cdot10^{-23})(0.01)^2}=3.1\cdot10^{-14}\ K$$

In the outer space hydrogen shows a classical behaviour.