91Ó°ÊÓ

We are given that the level A lies below $\mathrm{μ}$ by the same amount that level B lies above $\mathrm{μ}$.

We have to prove thatthe probability of level B being occupied is the same as the probability of level A being unoccupied.

Using the Fermi-Dirac distribution to calculate the probability of state B being occupied as follows:

$P\left(B\text{occupied}\right)=\frac{1}{e^{\frac{4\left({\mathrm{Î\mu }}_0-\mathrm{μ}\right)}{UT}}+1}$

Here, ${\mathrm{Î\mu }}_B$is the energy for the occupled state of $B,\mathrm{μ}$ is the chemical potential, k is the Boltzmann's constant, and T is the temperature.

The energy for the occupied state of B is,

${\mathrm{Î\mu }}_B=\mathrm{μ}+x$

Substitute $\mathrm{μ}+x$for ${\mathrm{Î\mu }}_B$ in the equation and simplifying, we get

$$\begin{gathered} P\left(B\text{occupied}\right)=\frac{1}{e^{\frac{t(\mathrm{μ}+x-\mathrm{μ})}{kT}}+1} \\ =\frac{1}{e^{\frac{x}{kT}}+1} \end{gathered}$$

The probability of state A being unoccupied is as follows,

$P\left(A\text{Unoccupied}\right)=1-P\left(A\text{occupied}\right)$

Using the Fermi-Dirac distribution, the probability of state A being occupied is as follows,

$P\left(A\text{occupied}\right)=\frac{1}{e^{\frac{+\left({\mathrm{Î\mu }}_A-\mathrm{μ}\right)}{NT}}+1}$

Where ${\mathrm{Î\mu }}_A$ is the energy for the occupied state of A.

Substituting the values, we get

$P\left(A\text{Unoccupied}\right)=1-\frac{1}{e^{\frac{+\left({\mathrm{Î\mu }}_A-\mathrm{μ}\right)}{kT}}}$

The energy for the occupied state of A is,

${\mathrm{Î\mu }}_A=\mathrm{μ}-x$

Substituting the values, we get

$$\begin{gathered} P\left(A\text{Unoccupied}\right)=1-\frac{1}{e^{\frac{+(\mathrm{μ}-x-\mathrm{μ})}{kT}}+1} \\ =1-\frac{1}{e^{\frac{-x}{kT}}+1} \\ =\frac{e^{\frac{-x}{kT}}}{e^{\frac{-x}{kT}}+1} \\ =\frac{1}{1+e^{\frac{x}{kT}}} \end{gathered}$$

Now, substituting $P\left(A\text{unoccupied}\right)=\frac{1}{1+e^{\frac{x}{kT}}}$in the equation,

$P\left(B\text{occupied}\right)=\frac{1}{e^{\frac{x}{kT}}+1}$, we get

$P\left(B\text{occupied}\right)=P\left(A\text{unoccupied}\right)$

Hence, the probability of level B being occupied is same as the probability of level A being unoccupied or the Fermi-Dirac distribution being symmetrical about the point where $\mathrm{Î\mu }=\mathrm{μ}$.