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GATE 2025 PH – Question 13

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 · 1 mark · Multiple choice

A particle of mass $m$ is in a potential $V(x)=\frac12m\omega^2x^2$ for $x>0$ and $V(x)=\infty$ for $x\le0$, where $\omega$ is the angular frequency. The ratio of the energies corresponding to the lowest energy level to the next higher level is

  1. $3/7$
  2. $1/2$
  3. $1/3$
  4. $3/5$

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Correct answer: (A) $3/7$

Explanation

**The potential** is a half harmonic oscillator: $V=\tfrac12m\omega^2x^2$ for $x>0$ and an infinite wall for $x\leq0$.

**Boundary condition.** The wave function must vanish at $x=0$ (the infinite wall). The full harmonic oscillator has eigenfunctions of alternating parity. The **even** states ($n=0,2,4,\dots$) do not vanish at $x=0$, but the **odd** states ($n=1,3,5,\dots$) do. So only the odd states survive, with the energies of the full oscillator,
$$E_n=\left(n+\tfrac12\right)\hbar\omega,\qquad n=1,3,5,\dots$$

The lowest two levels are $n=1$ and $n=3$:
$$E_1=\tfrac32\hbar\omega,\qquad E_3=\tfrac72\hbar\omega .$$

**Ratio:**
$$\frac{E_{lowest}}{E_{next}}=\frac{3/2}{7/2}=\frac37\quad(\text{option A}).$$