Subtopic: Variational Method & Ground State Estimation
Question Statement:
Following trial wavefunctions
\phi1 = e-Z'(r1+r2)
and
\phi2 = e-Z'(r1+r2)(1 + g|r 1 - r 2|)
are used to get a variational estimate of the ground state energy of the helium atom.
Z' 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 \phi1 and \phi2, respectively. Which one of the fo llowing options is
true?
Options:
(A) E1 ≤ E0 , E2 ≤ E0, E1 ≥ E2
(B) E1 ≥ E0 , E2 ≤ E0, E1 ≥ E2
(C) E1 ≤ E0 , E2 ≥ E0, E1 ≤ E2
(D) E1 ≥ E0 , E2 ≥ E0, E1 ≥ E2
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.28)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 4: Quantum Mechanics* | **Topic**: *Perturbation theory & Approximation Methods* (`Time-Independent Non-Degenerate & Degenerate Perturbation`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Perturbation theory & Approximation Methods** (`Time-Independent Non-Degenerate & Degenerate Perturbation`).
**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): `E1 ≥ E0 , E2 ≤ E0, E1 ≥ E2`**.
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