List explicitly all the ways to arrange 2 quanta among 4 one-dimensional oscillators.

In this problem, the concept of a total number of microstates will be used to determine the list explicitly all the ways to arrange quanta among different dimensional oscillators.

The expression to calculate the total number of microstates of 2 quanta in 4 dimension oscillators is expressed as follows:

$N=\frac{\left(Numberofquanta+Numberofoscillator‐1\right)}{Numberofquanta!\left(numberofoscillator‐1\right)}$

Here, $N$ represents the total number of microstates of 2 quanta in 4 dimension oscillators.

Substitute the values in the above expression.

role="math" localid="1655489155238" $\mathit{\text{N}}\text{=}\frac{(2+4-1)}{2!(4-1)}$

role="math" localid="1655489167047" $=\mathit{}10$

Thus, the total numbers of microstates are 10.

The list of the ways to arrange 2 quanta among 4 one dimensional oscillators can be expressed as follows:

$$\begin{gathered} 1100,1100, \\ 1010,1010, \\ 1001,1001, \\ 0110,0110, \\ 0101,0101, \\ 0011,0011, \\ 2000,2000, \\ 0200,0200, \\ 0020,0020, \\ 0002,0002, \end{gathered}$$

Thus, the list of the ways to arrange 2 quanta among 4 one dimensional oscillators is

$$\begin{gathered} 1100,1100, \\ 1010,1010, \\ 1001,1001, \\ 0110,0110, \\ 0101,0101, \\ 0011,0011, \\ 2000,2000, \\ 0200,0200, \\ 0020,0020, \\ 0002,0002, \end{gathered}$$