GATE 2024 CY – Question 62
The wave function of a particle in a cubic box (of side $L$) is given by $$\psi(x,y,z)=\sqrt{32/L^3}\sin\frac{\pi x}{L}\cos\frac{\pi x}{L}\sin\frac{2\pi y}{L}\sin\frac{\pi z}{L}.$$ The ratio of the energy of the state corresponding to the above wave function to the ground state energy is ___. (rounded off to the nearest integer)
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Correct answer: 3
Explanation
**Wave function.** Use $2\sin\theta\cos\theta=\sin2\theta$:
$$\sin\frac{\pi x}{L}\cos\frac{\pi x}{L}=\tfrac12\sin\frac{2\pi x}{L}.$$
So $\psi\propto\sin\dfrac{2\pi x}{L}\,\sin\dfrac{2\pi y}{L}\,\sin\dfrac{\pi z}{L}$, i.e. the quantum numbers are $(n_x,n_y,n_z)=(2,2,1)$.
**Energy of a particle in a cubic box:**
$$E=\frac{h^2}{8mL^2}\left(n_x^2+n_y^2+n_z^2\right).$$
The state has $4+4+1=9$ in units of $h^2/8mL^2$, and the ground state $(1,1,1)$ has $1+1+1=3$:
$$\frac{E}{E_0}=\frac93=\mathbf{3}.$$