StudyBrixFree worksheets, study guides and courses

Physics · Advanced university

Variational bounds and trial states

For a normalized trial state in the Hamiltonian domain, the expectation of a Hamiltonian bounded below is at least its ground energy.

Make a worksheet

Prerequisites

Quantum expectation values and optimization.

Study notes

Bound

For a normalized trial state in the Hamiltonian domain, the expectation of a Hamiltonian bounded below is at least its ground energy.

Parameters

Optimize a trial-state family to lower the upper bound. A restricted family need not contain the true ground state.

Normalization

A non-normalized trial function requires the quotient ⟨ψ∣H∣ψ⟩/⟨ψ∣ψ⟩. Omitting the denominator changes the problem.

Limits

A bound on energy is not automatically a pointwise guarantee about the approximate wavefunction. Excited states require additional conditions.

Worked examples

A valid trial-state ground-energy estimate is normally what type of bound?

The Rayleigh quotient is at least the true ground energy under the stated hypotheses.

An upper bound

For numerator 12 and trial norm squared 3, calculate the Rayleigh quotient.

Divide 12 by 3.

4

Check your understanding

  1. What improves the best variational bound when enlarging a nested trial family?

    1. The Hamiltonian disappears
    2. The minimum cannot increase
    3. The minimum must always become exact
    4. The minimum must increase

  2. A trial energy is E(a)=a²−4a+7 for real a. Find its minimum value.

  3. Why does multiplying a trial function by a nonzero constant not change its Rayleigh quotient?

  4. State why a very accurate variational energy does not alone prove every observable is equally accurate.

Answers and marking guidance

Answers and marking notes
  1. The minimum cannot increase

    The old trial states remain available in the enlarged family.

  2. 3

    Complete the square: (a−2)²+3.

  3. Both numerator and denominator scale by the squared magnitude of that constant, so the factor cancels.

    Credit complex magnitude as well as real scaling.

  4. The energy constrains an expectation value, while other observables can be sensitive to different state features. Further error analysis or convergence checks are needed.

    Credit the distinction between quantities being approximated.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.

AI-assisted preview; independent educational review required. Level labels describe prerequisites, not accreditation or official curriculum alignment.