Subtopic: Spin Angular Momentum & Pauli Spin Matrices
Question Statement:
H is the Hamiltonian, L⃗ the orbital angular momentum and Lz is the
z-component of L⃗ . The 1s state of the hydrogen atom in the non-relativistic
formalism is an eigen function of which one of the following sets of operators?
Options:
(A) H, L2 and Lz
(B) H, L⃗ , L2 and Lz
(C) L2 and Lz only
(D) H and Lz only
Official Answer Key:
MTA
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.15)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 4: Quantum Mechanics* | **Topic**: *Angular momentum & Hydrogen Atom* (`Spin Angular Momentum & Pauli Matrices`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Angular momentum & Hydrogen Atom** (`Spin Angular Momentum & Pauli Matrices`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `H, L⃗ , L2 and Lz`**.
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