Variational Principle

The variational principle turns the ground-state problem into an optimization problem. Instead of solving the eigenvalue equation exactly, choose a family of trial states and minimize the energy expectation value.

For the chapter-wide map of ansatz, optimization, and evaluation errors, begin with Variational and Bound Methods. Variational Many-Body States compares determinant, Jastrow, projected, tensor-network, and neural state families. For symptom-based checks of suspicious calculations, use Common Variational Pitfalls.

For a Hamiltonian $H$ bounded below, the exact ground-state energy $E_0$ satisfies

$$E[\psi] = \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle} \ge E_0$$

for every nonzero trial state $\lvert\psi\rangle$ in the domain of $H$. Thus every admissible trial state gives an upper bound on the true ground-state energy.

This page is the canonical home for the ground-state variational bound. Finite-dimensional implementations are treated in the Rayleigh-Ritz method.

Statement and Assumptions

Let $H$ be a self-adjoint Hamiltonian with a lowest spectral value $E_0$. If the bottom of the spectrum is an isolated eigenvalue with normalized eigenstates $\lvert n\rangle$, the familiar discrete statement is especially simple:

$$H\lvert n\rangle=E_n\lvert n\rangle, \qquad E_0\le E_1\le E_2\le\cdots.$$

For any normalized state

$$\lvert\psi\rangle = \sum_n c_n\lvert n\rangle, \qquad \sum_n |c_n|^2=1,$$

the energy expectation value obeys $E[\psi]\ge E_0$.

The same idea extends to Hamiltonians with continuous spectrum, but then one must state domain assumptions more carefully. In applications, the trial wavefunction must be normalizable, satisfy the boundary conditions of the problem, and have finite kinetic and potential energy expectations.

Proof in an Eigenbasis

Using the spectral expansion,

$$\begin{aligned} E[\psi] &= \langle\psi|H|\psi\rangle \\ &= \sum_n |c_n|^2E_n. \end{aligned}$$

Since each $E_n\ge E_0$,

$$\sum_n |c_n|^2E_n \ge E_0\sum_n |c_n|^2 = E_0.$$

For an unnormalized trial state the same result follows by dividing by $\langle\psi|\psi\rangle$.

Equality holds only when $\lvert\psi\rangle$ has support entirely inside the ground-state eigenspace. If the ground state is nondegenerate, equality means the trial state is the exact ground state up to an overall phase.

Variational Families

A practical calculation chooses a family of normalized or normalizable trial states $\lvert\psi(\alpha)\rangle$ depending on parameters $\alpha$. The variational estimate is

$$E_{\mathrm{var}} = \min_\alpha \frac{ \langle\psi(\alpha)|H|\psi(\alpha)\rangle }{ \langle\psi(\alpha)|\psi(\alpha)\rangle }.$$

The result still satisfies

$$E_{\mathrm{var}}\ge E_0.$$

Adding more parameters or enlarging the trial family cannot make the best variational energy worse, because the older trial states remain available. This is the basic reason variational calculations can be systematically improved, though systematic improvement of the energy does not automatically imply equally good improvement of every observable.

The derivative and tangent-space conditions used to find that optimum are derived in Variational Parameters.

Physical Interpretation

A trial state has some overlap with the true ground state and some overlap with excited states. Excited-state contamination raises the average energy because excited components carry energies larger than $E_0$:

$$E[\psi]-E_0 = \sum_n |c_n|^2(E_n-E_0).$$

The variational method therefore tries to suppress high-energy features. In wave mechanics this often means balancing localization against gradients: squeezing a wavefunction may reduce potential energy, but it increases kinetic energy.

The method is especially effective when the trial family builds in known qualitative information:

• boundary conditions,
• symmetry and parity,
• regular behavior near singular points,
• correct decay at large distance,
• cusp or short-distance structure when known,
• limiting behavior in exactly solvable regimes.

Example: Gaussian Width for the Harmonic Oscillator

Consider the one-dimensional harmonic oscillator

$$H = \frac{P^2}{2m} + \frac12m\omega^2X^2.$$

Use the normalized Gaussian trial state

$$\psi_a(x) = \left(\frac{a}{\pi}\right)^{1/4} \exp\left(-\frac{a x^2}{2}\right), \qquad a>0.$$

For this state,

$$\langle X^2\rangle=\frac{1}{2a}, \qquad \langle P^2\rangle=\frac{\hbar^2a}{2}.$$

The variational energy is

$$E(a) = \frac{\hbar^2a}{4m} + \frac{m\omega^2}{4a}.$$

Minimizing with respect to $a$ gives

$$\frac{dE}{da} = \frac{\hbar^2}{4m} - \frac{m\omega^2}{4a^2} = 0, \qquad a=\frac{m\omega}{\hbar}.$$

Substituting this value,

$$E_{\mathrm{var}} = \frac{\hbar\omega}{2}.$$

The Gaussian family contains the exact ground state, so the bound is saturated. For the anharmonic oscillator, the same Gaussian family gives a useful upper bound but not an exact answer.

Energy Accuracy Versus State Accuracy

The variational energy can be accurate even when the trial wavefunction misses some details. Suppose the exact ground state is nondegenerate and

$$\lvert\psi\rangle = c_0\lvert0\rangle + \sum_{n>0}c_n\lvert n\rangle.$$

Then

$$E[\psi]-E_0 = \sum_{n>0}|c_n|^2(E_n-E_0).$$

The energy error is second order in small excited-state amplitudes. By contrast, expectation values of other observables can have first-order errors if the observable connects the ground state to the excited components. A good variational energy is therefore not by itself proof of a uniformly good wavefunction.

Common Mistakes

• Treating a variational estimate as a two-sided error bar. The basic principle gives an upper bound on $E_0$, not a lower bound.
• Forgetting normalization. The quotient $E[\psi]$ is homogeneous in $\lvert\psi\rangle$, so one may either normalize first or use the denominator explicitly.
• Using a trial state outside the Hamiltonian domain. A wavefunction with the wrong boundary behavior can give misleading or divergent terms.
• Optimizing a family that violates a required symmetry. If the physical ground state is constrained to a sector, the trial states should lie in that sector.
• Assuming a good energy guarantees a good value for every observable.

Relation to Other Methods

Perturbation theory expands around a nearby exactly solved Hamiltonian. The variational method instead optimizes over a chosen set of states and can remain useful when no small perturbative parameter is available.

In numerical quantum mechanics, the variational principle appears as finite-basis diagonalization. In quantum algorithms, variational eigensolvers use the same expectation-value objective, with a parameterized quantum circuit replacing an analytic trial wavefunction.

Exercises

1. Show that multiplying a trial state by a nonzero constant does not change $E[\psi]$.

Solution

Let $\lvert\phi\rangle=c\lvert\psi\rangle$ with $c\ne0$. Then

$$\frac{\langle\phi|H|\phi\rangle}{\langle\phi|\phi\rangle} = \frac{|c|^2\langle\psi|H|\psi\rangle}{|c|^2\langle\psi|\psi\rangle} = \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}.$$

2. For the harmonic oscillator Gaussian trial state above, verify that the stationary point is a minimum.

Solution

The second derivative is

$$\frac{d^2E}{da^2} = \frac{m\omega^2}{2a^3}.$$

For $a>0$ this is positive, so the stationary point is a minimum.

3. Suppose a normalized trial state has $|c_0|^2=0.99$ and all remaining weight lies above a gap $\Delta=E_1-E_0$. Find a lower estimate for $E[\psi]-E_0$ in terms of $\Delta$.

Solution

The excited-state weight is $1-|c_0|^2=0.01$. Since each excited component has energy at least $E_0+\Delta$,

$$E[\psi]-E_0 = \sum_{n>0}|c_n|^2(E_n-E_0) \ge 0.01\,\Delta.$$

This is a lower estimate for the energy excess given the stated overlap information. In practice, the overlap is usually unknown; the variational principle itself gives the upper bound $E[\psi]\ge E_0$.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978.