Questions: The Variational Principle in Quantum Mechanics

Question 1

A physicist uses a trial wavefunction to estimate the ground state energy of helium and obtains E_trial = −77.5 eV. The true ground state energy is E₀ = −79.0 eV. What does the variational principle say about this result?

• A. The result is invalid because it differs from the true answer by more than 1%.
• B. E_trial > E₀, which is consistent with the variational principle — the trial wavefunction gives an upper bound, not the exact answer.
• C. The variational principle has been violated because the true energy is lower than the computed value.
• D. The trial wavefunction is close enough that further optimization is unnecessary.

Answer: (B)

The variational principle states ⟨ψ|Ĥ|ψ⟩ ≥ E₀ for any normalized ψ. Here E_trial = −77.5 eV > −79.0 eV = E₀, which is exactly what the principle requires. The result is a valid upper bound — the true ground state energy is lower (more negative) than the trial estimate. This is not a violation; it is the expected behavior. The physicist should try to improve the trial wavefunction to bring E_trial closer to E₀ from above.

Question 2

Why is the variational method particularly powerful for many-electron systems like molecules, where exact solutions are impossible?

• A. It provides exact analytic solutions to the Schrödinger equation for any system with more than two electrons.
• B. It replaces the eigenvalue problem (finding E₀ exactly, which is often intractable) with an optimization problem that can be systematically improved by enriching the trial wavefunction family.
• C. It works only for ground states and is not applicable to excited states or molecular systems.
• D. It requires knowing the true ground state energy E₀ first, using it as a reference for the optimization.

Answer: (B)

The Schrödinger equation for many-electron systems cannot be solved exactly because of electron-electron interactions. The variational method bypasses this: instead of solving an eigenvalue problem, parameterize trial wavefunctions and minimize ⟨Ĥ⟩ over the parameter space. The resulting minimum is a rigorous upper bound on E₀. By choosing richer trial families (more parameters, more flexible forms), you get tighter bounds — systematically improvable without ever solving the full problem exactly.

Question 3

For any normalized quantum state |ψ⟩ that is not the true ground state, the expectation value ⟨ψ|Ĥ|ψ⟩ is strictly greater than the ground state energy E₀.

• T. True
• F. False

Answer: True

The proof shows ⟨ψ|Ĥ|ψ⟩ = Σ_n |c_n|² E_n ≥ Σ_n |c_n|² E₀ = E₀, with equality only when all weight is on the ground state (c_n = 0 for all n ≠ 0). If |ψ⟩ is not the ground state, it has nonzero weight on at least one excited state with E_n > E₀, which raises the expectation value strictly above E₀. This strict inequality is what makes the variational method useful: you know the minimum you find is above E₀, and equality means you've found it exactly.

Question 4

A variational calculation that produces a lower energy than a previous trial wavefunction has necessarily found a better approximation to the ground state.

• T. True
• F. False

Answer: False

Lower is better — up to a point. A lower E_trial is a tighter upper bound on E₀, which means the new trial wavefunction has higher overlap with the true ground state. However, if E_trial somehow came out *lower* than the true E₀, that would indicate a normalization error or a bug in the calculation, since the variational principle guarantees E_trial ≥ E₀. So 'lower' is only better within the allowed range — any result below E₀ is physically impossible and signals an error, not an improvement. The statement is false because it ignores this crucial floor.

Question 5

Why does the variational principle guarantee an upper bound — rather than a lower bound — on the ground state energy?

Model answer: Because the expectation value ⟨ψ|Ĥ|ψ⟩ is a weighted average of energy eigenvalues, with weights |c_n|² summing to 1. Since E₀ is the *minimum* energy eigenvalue, every term in the sum is ≥ E₀, and the weighted average is therefore ≥ E₀. You can never accidentally compute a weighted average that falls below the smallest value being averaged. Equality holds only when all weight is on the ground state eigenfunction itself.

The upper-bound property is what makes the method constructive: you can minimize over trial states knowing your minimum is above the truth. This lets you rank trial wavefunctions by quality (lower energy = better approximation) and systematically improve. A lower-bound principle would require knowing the true answer first, which defeats the purpose. The variational principle's one-sidedness — you can only approach E₀ from above — is both a constraint and a feature.