The GATE Grind

GATE 2023 PH – Question 51

Quantum Mechanics · 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 · 2 marks · Multiple choice

A particle occupies a 2D infinite square well $0\le x,y\le L$. Its wavefunction vanishes only on $y=L/2$ apart from the boundaries. If its energy is E, what is the ground energy?

  1. $E/4$
  2. $2E/5$
  3. $3E/8$
  4. $E/2$

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Show answer and explanation

Correct answer: (B) $2E/5$

Explanation

**Energy of a particle in a 2-D square well:**
$$E_{n_x,n_y}=\frac{\pi^2\hbar^2}{2mL^2}\left(n_x^2+n_y^2\right).$$

**Nodal lines.** The wave function is $\sin\dfrac{n_x\pi x}{L}\sin\dfrac{n_y\pi y}{L}$. It has $n_x-1$ interior nodal lines parallel to the $y$-axis and $n_y-1$ interior nodal lines parallel to the $x$-axis.

The wave function vanishes **only** on the line $y=L/2$ (apart from the boundaries), so:
- there is no interior nodal line in $x$: $n_x=1$,
- there is exactly one interior nodal line in $y$, at $y=L/2$: $n_y=2$.

**Energy of this state:** $E=(1+4)\epsilon_0=5\epsilon_0$ with $\epsilon_0=\dfrac{\pi^2\hbar^2}{2mL^2}$.

**Ground state:** $(1,1)$ with $E_g=2\epsilon_0$, so
$$E_g=\frac25E\quad(\text{option B}).$$