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 \(\underline{\hspace{2cm}}\).
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 \(\underline{\hspace{2cm}}\).
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In a one-dimensional infinite potential well, the energy levels are proportional to the square of the quantum number \( n \).
Updated On: Dec 29, 2025
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Correct Answer: 5
Solution and Explanation
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 \( \boxed{16} \).
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