GATE 2026 PH – Question 41

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

The energy E and degeneracy d of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency ω are

$E=7\hbar\omega/2,\ d=6$
$E=7\hbar\omega/2,\ d=3$
$E=5\hbar\omega/2,\ d=3$
$E=5\hbar\omega/2,\ d=6$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $E=7\hbar\omega/2,\ d=6$

Explanation

The energies of a 3-D isotropic harmonic oscillator are
$$E=\hbar\omega\left(n_x+n_y+n_z+\tfrac32\right)=\hbar\omega\left(N+\tfrac32\right),\qquad N=0,1,2,\dots$$

Ground state: $N=0$.
First excited state: $N=1$.
**Second excited state: $N=2$**, with energy $E=\left(2+\tfrac32\right)\hbar\omega=\dfrac72\hbar\omega$.

**Degeneracy:** the number of non-negative integer triples $(n_x,n_y,n_z)$ with sum $N=2$ is
$$\binom{N+2}{2}=\binom42=6 :\quad(2,0,0),(0,2,0),(0,0,2),(1,1,0),(1,0,1),(0,1,1).$$

Answer: $E=\tfrac72\hbar\omega$ and $d=6$ (option A).