Question

What are classical limits? Explain how quantum distribution laws are reduced to classical Maxwell-Boltzmann distribution?

Answer 1

Classical Limits:-The classical limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior.

The function f(
e) depends on whether or not the particles obey the Pauli exclusion principle.

We know that for the bosons distribution function $f_{BE}(\epsilon)=\dfrac{1}{e^{\alpha+\frac{\epsilon}{kT}}-1}$

For fermions, the distribution function is $f_{FE}(\epsilon)=\dfrac{1}{e^{\alpha+\frac{\epsilon}{kT}}+1}$

We know that max well distribution function $f(\epsilon)=Ae^{\frac{-\epsilon}{kT}}$

The number of particles having energy $\epsilon$ at temperature T,

$n(\epsilon)= Ag(\epsilon)e^{-\epsilon/kT}$

So, for atom in ground state,

$n(\epsilon_1)= Ag(\epsilon_1)e^{-\epsilon_1/kT}$ --------(i)

So, for atom in first excited state,

$n(\epsilon_2)= Ag(\epsilon_2)e^{-\epsilon_21/kT}$ -----(ii)

Now dividing equation (ii) by (i)

$\dfrac{n(\epsilon_2)}{n(\epsilon_1)}=\dfrac{g(\epsilon_2)}{g(\epsilon_1)}e^{-\frac{\epsilon_2-\epsilon_1}{kT}}$

=$Ae^{-\epsilon/kT}$