Answer to Question #164724 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) Is Helium at atmospheric pressure quantum mechanical? What about Hydrogen atoms in outer space (interatomic distance is 1 cm and temperature is 3 K)?

Answer 1

a). Equation de-Broglie wavelength

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

We know $eq^n, E=\frac{p^2}{2m}+V$

$\implies p^2=2m(E-V)$

$\implies p=\sqrt{2m(E-V)}$

Free particle , v=0

$\implies p=\sqrt{2mE}$

But we know , $E=\frac{3}{2}k_BT$

Hence $\lambda=\frac{h}{\sqrt{2m\times \frac{3}{2}k_BT}}$

$\implies \lambda=\frac{h}{\sqrt{3mk_BT}}$

Here k is the Boltzmann constant $=1.4 \times10^{-23}J/k$

Mass of an electron , m= $9.11\times10^{-31} kg$

Planck's constant, h= $6.626 \times10^{-34}Js$

The lattice spacing in a typical solid is d= 0.3 nm=  $0.3 \times10^{-9}m$

b). $\lambda >d$

$\frac{h}{\sqrt{3mk_BT}}>d$

$T<\frac{h^2}{{3mk_Bd^2}}$

$\frac{h^2}{{3mk_Bd^2}}={\frac{(6.626 \times10^{-34}J.s)^{2}}{{(3(9.11 \times10^{-31})}(1.4 \times10^{-23})(0.3 \times10^{-9})^2}}$

$T<1.3\times10^5K$

c). For helium

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

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

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

$\implies T< 2.8K$

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.