Three non-interacting bosonic particles, each of mass $m$, are in a one-dimensional infinite potential well of width $a$. The energy of the third excited state of the system is $x\hbar^2\pi^2/(ma^2)$. Find $x$ as an integer.
Single-particle energies are $E_n=n^2\hbar^2\pi^2/(2ma^2)$, with bosons allowed to share a level. The first distinct three-particle sums of $n_i^2$ are 3 for $(1,1,1)$, 6 for $(1,1,2)$, 9 for $(1,2,2)$, and 11 for $(1,1,3)$. Thus the third excited energy is $11\hbar^2\pi^2/(2ma^2)$ and $x=5.5$. The printed integer request is ambiguous for a half-integer result; the official key accepts 5–6, which is preserved here.
More One dimensional potentials: step potential, finite rectangular well, tunnelling from a potential barrier, particle in 1,2,3-dimensional box, particle in single and double delta function potentials, 1,2,3 dimensional harmonic oscillator: concept of degeneracy questions
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