Physics · Advanced university
Variational bounds and trial states · revision notes
For a normalized trial state in the Hamiltonian domain, the expectation of a Hamiltonian bounded below is at least its ground energy.
Key ideas and conditions
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.
Explain without looking
- A normalized trial state has probabilities 0.8 and 0.2 on eigenstates of energies 1 and 6. Find its mean energy.
- Which condition is essential before claiming the simple variational ground-energy bound?
Further reading and review references
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