Question

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

Short answer

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

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

Step-by-step solution

Understanding the oscillator

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.

Determination of the total number of microstates of 2 quanta in 4 one 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.

Determination of the list of the ways to arrange 2 quanta among 4 one dimensional oscillators.

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

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