Contents
Introduction The Statement Examples Hartree-Fock DFT Variational Quantum Eigensolver Misconceptions FAQ Sources Introduction
The variational principle provides a powerful way to estimate ground-state energies in quantum systems where exact solutions are unavailable. Pick a trial wave function with adjustable parameters; the expectation value of the Hamiltonian gives an upper bound on the true ground-state energy. Minimize over the parameters to get the best bound. This idea underlies the Rayleigh-Ritz method, Hartree-Fock theory, density functional theory, quantum chemistry, and modern variational quantum algorithms.
The Statement
For any normalized trial wave function |ψtrial ⟩:
⟨ψtrial |Ĥ|ψtrial ⟩ ≥ E0
where E0 is the true ground-state energy. Equality holds only if |ψtrial ⟩ = |ψ0 ⟩, the true ground state [1 ].
Proof Sketch
Expand |ψtrial ⟩ = Σ cn |n⟩ in eigenstates of Ĥ. Then ⟨H⟩ = Σ |cn |²En ≥ E0 Σ|cn |² = E0 . The bound is achieved when all weight is on the ground state.
Excited States
If you restrict to wave functions orthogonal to the ground state, the variational principle gives an upper bound for the first excited state. Extends to higher states with appropriate orthogonality constraints.
Examples
Hydrogen Atom Test
Try ψtrial = (α³/π)^(1/2) exp(−αr). Computing ⟨H⟩ as a function of α and minimizing gives α = 1/a₀ and E = −13.6 eV — the exact ground state. The variational method gives the exact answer here because the trial function family contains the true solution.
Helium
Helium has two electrons and is not exactly solvable. A simple trial function ψ = ψ1s (r₁)ψ1s (r₂) with effective nuclear charge Zeff as a variational parameter gives Zeff = 27/16 ≈ 1.69 and ground-state energy ≈ −77.5 eV, compared to experimental −79.0 eV. Improved trial functions get within milli-eV of experiment [2 ].
Hartree-Fock Theory
For multi-electron atoms and molecules, the Hartree-Fock approximation uses a Slater determinant of single-particle orbitals as the trial wave function. The orbitals are optimized variationally. This is the foundation of much of quantum chemistry [3 ].
Hartree-Fock includes exchange exactly but neglects electron correlations beyond exchange. Post-HF methods (configuration interaction, coupled cluster, MP2) add correlations systematically.
Density Functional Theory
The Hohenberg-Kohn theorems (1964) established that the electron density determines all properties of an electronic ground state [4 ]. DFT uses this to reformulate the variational problem in terms of density instead of wave function.
The Kohn-Sham equations (1965) provide a practical implementation: solve a single-particle Schrödinger-like equation with an effective potential that includes exchange-correlation. DFT is now the workhorse of materials science and quantum chemistry, with millions of calculations done per year.
Variational Quantum Eigensolver
VQE is a hybrid quantum-classical algorithm: a parameterized quantum circuit prepares trial states; classical optimization adjusts parameters to minimize energy. Applied to molecular electronic-structure problems on near-term quantum hardware [5 ].
VQE is the leading example of quantum advantage in the "NISQ" era — using small quantum computers for tasks where their natural noise tolerance helps.
Common Misconceptions
"The variational bound is tight"
It's an upper bound; the gap depends on how good the trial function is. For poor trial functions, the bound can be far above the true energy.
"Variational methods give the ground state"
They give an upper bound on the ground-state energy. The corresponding trial function is the best approximation within the chosen family, not necessarily the true ground state.
"The variational principle applies to all properties"
Only to the energy directly. Other properties calculated from the trial function may not have the same upper-bound structure.
"DFT is exact"
In principle, exact (if you know the exchange-correlation functional). In practice, approximate functionals are used, and DFT's accuracy depends on the functional choice.
FAQ
How do I choose a trial function?
Use physical intuition and known limiting forms. For atomic problems, hydrogenic orbitals with adjustable Z. For molecules, linear combinations of atomic orbitals. For general systems, basis-set expansions.
What if my trial function gives a lower energy than the true ground state?
You've made an error. The variational bound is rigorous; the trial function calculation should always give an upper bound.
Can variational methods predict excited states?
Yes, with orthogonality constraints to lower states. Or via response-theory methods. Or by configuration interaction over a basis.
Is the variational principle related to the principle of least action?
Both involve stationary points of functionals. The quantum variational principle is about energy expectation values; the action principle is about classical trajectories. They are related but distinct.
Sources
Griffiths, D. J. (2018). Introduction to Quantum Mechanics , 3rd ed.
Bethe, H. A., Salpeter, E. E. (1957). Quantum Mechanics of One- and Two-Electron Atoms .
Szabo, A., Ostlund, N. S. (1996). Modern Quantum Chemistry . Dover.
Hohenberg, P., Kohn, W. (1964). "Inhomogeneous electron gas." Physical Review , 136(3B), B864.
Peruzzo, A., et al. (2014). "A variational eigenvalue solver on a photonic quantum processor." Nature Communications , 5, 4213.