Answer to Question #164500 in Quantum Mechanics for Nidhin S

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)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) For helium

$m=4mp=4(1.61 \times10^{-27})$

$1 atm=(1.0 \times10^5 N/m^2)$

$\implies T< {\frac{(6.6 \times10^{-34})^{6/5}}{(3(6.8 \times10^{-27}))^{3/5}}} \times \frac{(1.0 \times10^{5})^{2/5}}{1.4 \times10^{-23}}$

$\implies T< 2.8K$

b)For hydrogen

$m=2mp=2(1.67 \times10^{-27})=3.2 \times10^{-27}$

$width, d=1cm$

$\implies T< \frac{h^2}{3mK_Bd^2}$

$\implies T< {\frac{(6.6 \times10^{-34})^{2}}{{(3(3.4 \times10^{-27})}(1.4 \times10^{-23})(1 \times10^{-2})^2}}$

$\implies T< 3.1 \times10^{-14}$

So at 3K , it is classical region.