GATE 2025 CY – Question 57
If a particle’s state function is an eigenfunction of the operator $\hat L^2$ with eigenvalue $30\hbar^2$, then the possible eigenvalue(s) of the operator $\hat L_z^2$ for the same state function is/are
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (B) $16\hbar^2$; (C) $25\hbar^2$; (D) 0
Explanation
For the angular momentum operators:
$$\hat L^2\psi=l(l+1)\hbar^2\psi,\qquad\hat L_z\psi=m\hbar\psi,\quad m=-l,\dots,+l .$$
**Find $l$.** Solve $l(l+1)=30$: $l=5$ (since $5\times6=30$).
**Allowed values of $m$:** $-5,-4,\dots,0,\dots,+5$, so $L_z^2=m^2\hbar^2$ can be
$$0,\ 1,\ 4,\ 9,\ 16,\ 25\quad(\text{in units of }\hbar^2).$$
Check the options:
- $10\hbar^2$: not a perfect square of an integer $m$ between 0 and 5. ✗
- $16\hbar^2$ ($m=\pm4$): possible. ✓
- $25\hbar^2$ ($m=\pm5$): possible. ✓
- $0$ ($m=0$): possible. ✓
Answer **B, C and D**.