# 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. Course: [Advanced physics methods and spacetime](https://studybrix.com/en/courses/advanced-physics-methods/index.md) Level: advanced Prerequisites: Quantum expectation values and optimization. ## 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. ## Practice questions ### 1. A valid trial-state ground-energy estimate is normally what type of bound? A. An upper bound B. A lower bound always C. Exact for any trial state D. Unrelated to energy Answer: An upper bound Explanation: 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. Answer: 4 Explanation: Divide 12 by 3. ### 3. What improves the best variational bound when enlarging a nested trial family? A. The Hamiltonian disappears B. The minimum cannot increase C. The minimum must always become exact D. The minimum must increase Answer: The minimum cannot increase Explanation: 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. Answer: 3 Explanation: 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 / marking guidance: Both numerator and denominator scale by the squared magnitude of that constant, so the factor cancels. Explanation: 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 / 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. Explanation: 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. Answer: 2 Explanation: 0.8×1+0.2×6=2. ### 8. Which condition is essential before claiming the simple variational ground-energy bound? A. A complex-valued energy by design B. Ignoring boundary conditions C. A valid trial state in the operator domain D. A negative normalization Answer: A valid trial state in the operator domain Explanation: The expectation and boundary/domain conditions must be well-defined. ## Further reading - https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/ - https://openstax.org/details/books/university-physics-volume-3 Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.