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Variational principle
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Upper bound on ground-state energy.
The expectation value of H in any trial state is ≥ the true ground-state energy; adjusting parameters to minimise it gives a good approximation.
Helium ground state: −77.5 eV from a simple trial function vs −79.0 measured.
Always an upper bound.
In more detail
The variational principle states that the energy estimated from any trial wavefunction is always greater than or equal to the true ground state energy. Better trial functions give better estimates.
Real-world uses
- Quantum chemists use variational methods for molecular ground states.
- Variational quantum algorithms run on quantum computers.
- Quantum chemists estimate the ground-state energy of molecules using trial wavefunctions and the variational principle.
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Frequently asked questions
Why is the estimate an upper bound?
Because the true ground state is the lowest possible energy, so any trial energy is at least that value.
How do I improve the estimate?
Use a trial function with more adjustable parameters.
Why is the variational estimate an upper bound?
Any trial state gives an energy at least as high as the true ground state, so the lowest estimate is closest to the truth.
Related topics in Quantum Mechanics
Zeeman effect · Stark effect · Perturbation theory · Dirac notation · Superposition · Measurement and collapse