The Variational Method

When no small parameter exists to justify perturbation theory, the variational method provides a non-perturbative handle on the ground state. Its foundation is a simple but powerful inequality: the expectation of the Hamiltonian in any trial state is an upper bound on the true ground-state energy. Minimizing over a family of trial states squeezes that bound down, often to remarkable accuracy — it is the method behind much of quantum chemistry and, in operator form, the mean-field theories of many-body physics.

1. The Variational Principle

For any normalized state $|\psi ⟩$ (not necessarily an eigenstate),

$$\boxed{E[\psi ]\equiv ⟨\psi |\hat{H}|\psi ⟩\ge E_0,}$$

with equality iff $|\psi ⟩$ is the ground state. Proof: expand in energy eigenstates $|\psi ⟩={\sum }_n{c}_n|n⟩$; then $E[\psi ]={\sum }_n|c_n{|}^2{E}_n\ge E_0{\sum }_n|c_n{|}^2=E_0$, since every $E_n\ge E_0$. The bound is one-sided and monotone: lower trial energies are always better estimates.

A complementary fact — the functional $E[\psi ]$ is stationary at every eigenstate, not just the minimum — means small errors in the trial state give second-order-small errors in the energy. This is why crude trial functions still yield good energies.

2. The Rayleigh–Ritz Procedure

Choose a trial wavefunction $\psi (\mathbf{r};{\alpha }_1,\dots ,{\alpha }_k)$ depending on adjustable parameters, compute

$$E({\alpha }_1,\dots ,{\alpha }_k)=\frac{⟨\psi |\hat{H}|\psi ⟩}{⟨\psi |\psi ⟩},$$

and minimize over the parameters, $\partial E/\partial {\alpha }_i=0$. The minimum is the best ground-state estimate available within that family. Enlarging the family (more parameters) can only lower — never raise — the bound, giving a systematic route to improvement.

Linear (Ritz) version. If the trial state is a linear combination $|\psi ⟩={\sum }_i{c}_i|{\varphi }_i⟩$ of fixed basis functions, minimization reduces to the generalized eigenvalue problem $H\mathbf{c}=ES\mathbf{c}$, with $H_{ij}=⟨{\varphi }_i|\hat{H}|{\varphi }_j⟩$ and overlap $S_{ij}=⟨{\varphi }_i|{\varphi }_j⟩$. The lowest root bounds $E_0$; higher roots bound higher levels (the Hylleraas–Undheim / MacDonald theorem), so diagonalizing in a finite basis brackets several states at once. This is the computational core of quantum chemistry.

3. Worked Example: The Helium Ground State

Helium ($Z=2$, two electrons) is not exactly solvable because of the electron–electron repulsion $e^2/|{\mathbf{r}}_1-{\mathbf{r}}_2|$. Take a trial state of two hydrogenic $1s$ orbitals with an effective charge $Z_{\text{eff}}$ as the variational parameter — physically, each electron partially screens the nucleus from the other. Minimizing gives

$$Z_{\text{eff}}=Z-\frac{5}{16}=1.6875,E_0\le -77.5\text{eV},$$

within ~2% of the experimental $-79.0\text{eV}$ — far better than first-order perturbation theory, and with transparent physics (screening) built into a one-parameter ansatz.

4. Excited States and the Operator Form

Excited states can be bounded by restricting the trial space to be orthogonal to lower states (exactly, or by symmetry — e.g. the lowest state of each angular-momentum or parity sector is bounded by the variational energy within that sector). Promoting the variational principle from states to operators — minimizing an energy functional over single-particle orbitals or density matrices — yields the Hartree–Fock and density-functional mean-field methods that dominate practical many-electron computation, and, applied to a variational ground-state ansatz for a field, the mean-field approximations of many-body and field theory.

See also

• Time-independent perturbation theory — the complementary small-parameter method.
• WKB approximation — the semiclassical alternative.
• Hydrogen atom — the orbitals used as trial functions.