You apply the variational method with a Gaussian trial function to the hydrogen atom and obtain an energy estimate of −11.5 eV. The true ground state energy is −13.6 eV. What does this tell you?
AYou made a computational error — a correctly applied variational method should give the exact ground state energy
BThe result is a valid upper bound: −11.5 eV > −13.6 eV, consistent with the variational theorem
CThe variational method underestimated the true energy, which is expected since it approaches from below
DThe Gaussian is a poor trial function and you should use a lower energy estimate from a different method
The variational theorem guarantees ⟨ψ|H|ψ⟩/⟨ψ|ψ⟩ ≥ E₀ for any trial state. −11.5 eV > −13.6 eV is exactly what the theorem predicts — an upper bound that is above (less negative than) the true ground state. Option C reverses the direction: the variational method always approaches the ground state from ABOVE. A more flexible trial function would give a tighter (more negative) upper bound, but it can never go below E₀.
Question 2 Multiple Choice
Why can't the variational method ever yield an energy estimate below the true ground state energy?
ABecause trial wavefunctions are always normalized, and normalization constrains the energy to be positive
BBecause any state expanded in energy eigenstates has all eigenvalue contributions ≥ E₀, so the expectation value — a weighted average of eigenvalues — must also be ≥ E₀
CBecause the minimization procedure always converges to the global minimum, which is E₀
DBecause the calculus of variations guarantees stationary points are minima, not maxima or saddle points
Write |ψ⟩ = Σcₙ|Eₙ⟩. Then ⟨H⟩ = Σ|cₙ|²Eₙ. Since each Eₙ ≥ E₀ and Σ|cₙ|² = 1, this is a weighted average of quantities each at least E₀ — so ⟨H⟩ ≥ E₀. The bound is saturated only when cₙ = 0 for all n > 0, i.e., when |ψ⟩ is exactly the ground state. No choice of parameters in the trial function can make this weighted average go below E₀.
Question 3 True / False
Adding more variational parameters to a trial function can only lower or preserve the variational energy estimate — it can never raise it.
TTrue
FFalse
Answer: True
True. A trial function with more parameters spans a larger family of states. The minimum over a larger family is always less than or equal to the minimum over a smaller subfamily (since the smaller family is a subset). So adding parameters gives access to states that were previously unavailable, potentially achieving a lower ⟨H⟩, but never forcing ⟨H⟩ to be higher than it was with fewer parameters. This is why the variational method is systematically improvable.
Question 4 True / False
The variational method can give an energy estimate below the true ground state energy if the trial wavefunction is chosen to have the correct symmetry and nodal structure.
TTrue
FFalse
Answer: False
False. The variational theorem holds for any trial state, regardless of its symmetry or nodal structure. ⟨H⟩ ≥ E₀ always. Symmetry and nodal structure matter for the quality of the upper bound (how close it is to E₀) and for targeting excited states, but they cannot cause the bound to be violated. No cleverness in choosing the trial function can push the estimate below the true ground state energy.
Question 5 Short Answer
Explain why minimizing the energy expectation value over variational parameters always gives an upper bound on the ground state energy. Why is this one-sided guarantee useful in practice?
Think about your answer, then reveal below.
Model answer: The proof follows from expanding the trial state in energy eigenstates: |ψ⟩ = Σcₙ|Eₙ⟩, so ⟨H⟩ = Σ|cₙ|²Eₙ. This is a weighted average of eigenvalues, all of which are ≥ E₀ (the ground state energy is by definition the minimum eigenvalue). A weighted average of quantities each ≥ E₀ is itself ≥ E₀. The bound is tight only when |ψ⟩ = |E₀⟩ exactly. The one-sided guarantee is useful because it makes the method systematically improvable: a better trial function gives a lower (tighter) upper bound, and you always know you are approaching the true answer from above. You can therefore compare competing trial functions by their variational energies — lower is always better — and you know you have not accidentally gone below the true answer.
The practical power is that for systems where exact solutions are impossible (most multi-electron atoms, all molecules beyond H₂⁺), the variational principle gives a controlled approximation: you know the sign of your error and can reduce it by increasing the flexibility of the trial function. This is the foundation of Hartree-Fock theory and all of modern quantum chemistry: larger basis sets lower the variational energy toward the exact answer.