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GATE 2020 PH – Question 14

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

For harmonic-oscillator lowering and raising operators $\hat a,\hat a^\dagger$, the energy eigenstate $|n\rangle$ is also an eigenstate of $(\hat a^\dagger)^2\hat a^2$. Its eigenvalue is

  1. $n(n-1)$
  2. $n(n+1)$
  3. $(n+1)^2$
  4. $n^2$

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Correct answer: (A) $n(n-1)$

Explanation

Apply the two lowering operators first: $\hat a^2|n\rangle=\sqrt{n(n-1)}|n-2\rangle$. The two raising operators return the state with another factor $\sqrt{n(n-1)}$. Their product gives $(\hat a^\dagger)^2\hat a^2|n\rangle=n(n-1)|n\rangle$, A. This also gives zero for n = 0 or 1.

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