Subtopic: Wave Mechanics, Operators & Perturbation Theory
Question Statement:
Following trial wavefunctions
š1 = e-Zā²(r1+r2)
and
š2 = e-Zā²(r1+r2)(1 + g|r 1 - r 2|)
are used to get a variational estimate of the ground state energy of the helium atom.
Zā² and g are the variational parameters, r 1and r 2 are the position vectors of the
electrons. Let E0 be the exact ground state energy of the helium atom. E1 and E2 are
the variational estimates of the ground state energy of the helium atom
corresponding to š1 and š2, respectively. Which one of the fo llowing options is
true?
Options:
(A) E1 ā¤ E0 , E2 ā¤ E0, E1 ā„ E2
(B) E1 ā„ E0 , E2 ā¤ E0, E1 ā„ E2
(C) E1 ā¤ E0 , E2 ā„ E0, E1 ā¤ E2
(D) E1 ā„ E0 , E2 ā„ E0, E1 ā„ E2
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.28)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantum Mechanics** within *Section 4: Quantum Mechanics* (1 Mark, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Quantum 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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