GATE 2025 CY – Question 56
Wavefunctions and energies for a particle confined in a cubic box are $\psi_{n_x,n_y,n_z}$ and $E_{n_x,n_y,n_z}$, respectively. The functions $\phi_1,\phi_2,\phi_3$, and $\phi_4$ are written as linear combinations of $\psi_{n_x,n_y,n_z}$. Among these functions, the eigenfunction(s) of the Hamiltonian operator for this particle is/are $$\phi_1=(\psi_{1,4,1}-\psi_{2,2,3})/\sqrt2$$ $$\phi_2=(\psi_{1,5,1}+\psi_{3,3,3})/\sqrt2$$ $$\phi_3=(\psi_{1,3,8}+\psi_{3,8,1})/\sqrt2$$ $$\phi_4=\tfrac12\psi_{3,3,1}+\tfrac{\sqrt3}2\psi_{2,4,1}$$
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Correct answer: (A) $\phi_2$; (C) $\phi_3$
Explanation
The energy of a particle in a cubic box is
$$E_{n_x,n_y,n_z}=\frac{h^2}{8mL^2}\left(n_x^2+n_y^2+n_z^2\right).$$
A linear combination of eigenfunctions is itself an eigenfunction of $\hat H$ **only if all the functions in it have the same energy** (degenerate states). So compare the value of $n_x^2+n_y^2+n_z^2$ for the two terms in each function.
| Function | Terms | $n_x^2+n_y^2+n_z^2$ | Same? |
|---|---|---|---|
| $\phi_1$ | $\psi_{1,4,1}$, $\psi_{2,2,3}$ | $18$ and $17$ | No |
| $\phi_2$ | $\psi_{1,5,1}$, $\psi_{3,3,3}$ | $27$ and $27$ | **Yes** |
| $\phi_3$ | $\psi_{1,3,8}$, $\psi_{3,8,1}$ | $74$ and $74$ | **Yes** |
| $\phi_4$ | $\psi_{3,3,1}$, $\psi_{2,4,1}$ | $19$ and $21$ | No |
The eigenfunctions of the Hamiltonian are **$\phi_2$ and $\phi_3$** (options A and C).