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GATE 2022 PH – Question 45

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

For the harmonic oscillator $a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$ and $x=\sqrt{\hbar/(2m\omega)}(a+a^\dagger)$. Which is the first three rows and columns of x?

  1. $\sqrt{\hbar/(2m\omega)}\begin{pmatrix}1&0&0\\0&\sqrt2&0\\0&0&\sqrt3\end{pmatrix}$
  2. $\sqrt{\hbar/(2m\omega)}\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}$
  3. $\sqrt{\hbar/(2m\omega)}\begin{pmatrix}0&1&0\\1&0&\sqrt2\\0&\sqrt2&0\end{pmatrix}$
  4. $\sqrt{\hbar/(2m\omega)}\begin{pmatrix}1&0&\sqrt3\\0&0&0\\\sqrt3&0&1\end{pmatrix}$

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Correct answer: (C) $\sqrt{\hbar/(2m\omega)}\begin{pmatrix}0&1&0\\1&0&\sqrt2\\0&\sqrt2&0\end{pmatrix}$

Explanation

The position operator is $\hat x=\sqrt{\dfrac{\hbar}{2m\omega}}\,(a+a^\dagger)$, with
$$a|n\rangle=\sqrt n\,|n-1\rangle,\qquad a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle .$$

**Matrix elements** $\langle m|\hat x|n\rangle$ are non-zero only when $m=n\pm1$:
- $\langle0|x|1\rangle=\sqrt{\tfrac{\hbar}{2m\omega}}\,\sqrt1$,
- $\langle1|x|2\rangle=\sqrt{\tfrac{\hbar}{2m\omega}}\,\sqrt2$.

(The matrix is symmetric because $x$ is Hermitian.)

**The first three rows and columns** (states $|0\rangle,|1\rangle,|2\rangle$):
$$x=\sqrt{\frac{\hbar}{2m\omega}}\begin{pmatrix}0&1&0\\1&0&\sqrt2\\0&\sqrt2&0\end{pmatrix}\quad(\text{option C}).$$