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Physics · lesson 3 of 4

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.

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Prerequisites: Quantum expectation values and optimization.

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

Example 1

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

Example 2

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

Divide 12 by 3.

4

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

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An upper bound

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

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

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4

Divide 12 by 3.

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

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The minimum cannot increase

The old trial states remain available in the enlarged family.

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

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3

Complete the square: (a−2)²+3.

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

Sample answer and marking guidance

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

Credit complex magnitude as well as real scaling.

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

Sample answer and marking guidance

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.

7 A normalized trial state has probabilities 0.8 and 0.2 on eigenstates of energies 1 and 6. Find its mean energy.

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2

0.8×1+0.2×6=2.

8 Which condition is essential before claiming the simple variational ground-energy bound?

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A valid trial state in the operator domain

The expectation and boundary/domain conditions must be well-defined.

Further reading and review references

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