Let |l, m> be the simultaneous eigenstates of L^2 and L_z. Here L is the angular momentum operator with Cartesian components (L_x, L_y, L_z), l is the angular momentum quantum number and m is the azimuthal quantum number. The value of <1, 0 | (L_x + i L_y) | 1, -1> is
Options:
(A) 0
(B) \hbar
(C) \sqrt{2}\hbar
(D) \sqrt{3}\hbar
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2016 Q.60 / PH Q.50)**
**1. Angular Momentum Raising Operator**:
- The ladder operator is defined as $L_+ = L_x + i L_y$.
- Its action on an eigenstate $|l, m\rangle$ is:
$L_+ |l, m\rangle = \hbar \sqrt{l(l+1) - m(m+1)} |l, m+1\rangle$
**2. Applying to $|1, -1\rangle$**:
- Here $l = 1$ and $m = -1$:
$L_+ |1, -1\rangle = \hbar \sqrt{1(2) - (-1)(0)} |1, 0\rangle = \hbar \sqrt{2 - 0} |1, 0\rangle = \sqrt{2}\hbar |1, 0\rangle$
**3. Inner Product Calculation**:
- Taking the matrix element:
$\langle 1, 0 | L_+ | 1, -1\rangle = \langle 1, 0 | \left(\sqrt{2}\hbar |1, 0\rangle\right) = \sqrt{2}\hbar \langle 1, 0 | 1, 0\rangle = \sqrt{2}\hbar$
- **Official Master Answer**: **C**
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