In a large system of distinguishable harmonic oscillator how high does the temperature have to be for the probable number of particles occupying the ground state to be less than 1 ?

Like the notation used in the book, we also assume that the potential energy is shifted by$-\frac{1}{2}\hslash {\omega }_0$so that the allowed energy of a harmonic oscillator is given by:$E_n=n\hslash {\omega }_0\text{where}n⩾0.......\left(1\right)$

and its ground state energy is$E_0=0$.

The key idea here lies from the fact that we are now dealing with an enormous amount of distinguishable harmonic oscillators the probability at which an oscillator is in an energy level is now given by the Boltzmann probability distribution:

$P\left(E_n\right)=A{e}^{-E_n/k_{B}T}$ …… (2)

where $k_B$is Boltzmann's constant, and T is temperature.

Before we could utilize eq. (2), first must find the value of the normalization constant A . To do so, we plug in eq. (1) into (2), set the whole expression to l, and evaluate the integral over an interval from 0 to $\infty$. This translates to:

$1={\int }_0^{\infty }A{e}^{-n\hslash {\omega }_0/k_{B}T}dn$

Evaluating the above integral, and solving for A we obtain:

$\begin{array}{rcl}1& =& A{\int }_0^{\infty }{e}^{-n\hslash {\omega }_0/k_{B}T}\\ & =& A{\left[-\frac{k_{B}T}{\hslash {\omega }_0}{e}^{-n\hslash {\omega }_0/k_{B}T}\right]}_0^{\infty }\\ & =& A\left[-\frac{k_{B}T}{\hslash {\omega }_0}(0-1)\right]\\ & =& A\frac{k_{B}T}{\hslash {\omega }_0}\\ & & \end{array}$

Above equation can be written as,

$A=\frac{\hslash {\omega }_0}{k_{B}T}$

Eq. (2) now becomes

$P\left(n\right)=\frac{\hslash {\omega }_0}{k_{B}T}{e}^{-n\hslash {\omega }_0/k_{B}T}$ …… (3)

However, since we will be dealing with the probable number of particles in the ground state, then the probability becomes an occupation number N

$\mathcal{N}\left(n\right)=N\frac{\hslash {\omega }_0}{k_{B}T}{e}^{-n\hslash {\omega }_0/k_{B}T}$ …… (4)

Here N is the total number of particles.

To find the temperature T where the probable number of particles in the ground state is less than1, we set$n=0$ in eq. (4) and transform the expression into an inequality:

$N\left(1\right)>N\frac{\hslash {\omega }_0}{k_{B}T}{e}^{-\left(0\right)\hslash {\omega }_0/k_{B}T}$

Solving for T we finally get:

$$\begin{gathered} 1>\frac{\hslash {\omega }_0}{k_{B}T} \\ T>\frac{\hslash {\omega }_0}{k_B} \end{gathered}$$

Thus, the temperature is$T>\frac{\hslash {\omega }_0}{k_B}$ .