Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
Question Statement:
The non-relativistic Hamiltonian for a single electron atom is
H0 = p2
2m - V(r)
where V(r) is the Coulomb potential and m is the mass of the electron. Considering
the spin-orbit interaction term
H′ = 1
2m2c2
1
r
dV
dr L⃗ ∙ S
added to H0, which of the following statement is/are true?
Options:
(A) H′ commutes with L2
(B) H′ commutes with Lz and Sz
(C) For a given value of principal quantum number n and orbital angular momentum
quantum number l, there are 2(2l + 1) degenerate eigenstates of H0
(D) H0, L2, S2, Lz and Sz have a set of simultaneous eigenstates
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.48)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Classical Mechanics** within *Section 2: Classical Mechanics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Classical Mechanics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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