Question

Three non-interacting bosonic particles of mass \( m \) each, are in a one-dimensional infinite potential well of width \( a \). The energy of the third excited state of the system is \( x \times \frac{h^2 \pi^2}{ma^2} \). The value of \( x \) (in integer) is \(\underline{\hspace{2cm}}\).

Hint:

In a one-dimensional infinite potential well, the energy levels are proportional to the square of the quantum number \( n \).

Answer 1

Correct Answer: 5

For a system of non-interacting bosons in a one-dimensional infinite potential well, the energy levels are given by: \[ E_n = n^2 \times \frac{h^2 \pi^2}{2ma^2} \] For the third excited state, \( n = 4 \), so the energy is: \[ E_4 = 4^2 \times \frac{h^2 \pi^2}{2ma^2} = 16 \times \frac{h^2 \pi^2}{2ma^2} \] Thus, \( x = 16 \). The value of \( x \) is \( \boxed{16} \).