The Variational Principle

The variational principle is the single most-used idea in computational quantum physics. Every self-consistent-field method (Hartree–Fock) minimizes $\langle H\rangle$ over antisymmetrized product states; density-functional theory recasts the minimization over densities via the Hohenberg–Kohn variational theorem; quantum Monte Carlo optimizes many-parameter trial wavefunctions with the bound as its quality certificate. In the last decade the same principle powers the variational quantum eigensolver, where a quantum processor prepares parametrized trial states and a classical optimizer turns the knobs — the ground-state search of molecules and lattice models on near-term quantum hardware is the $\langle H\rangle$-minimization of this article, run on qubits. Neural-network wavefunctions (variational Monte Carlo with deep ansätze) are the newest trial family, and they obey the same unbreakable rule: whatever they report, the truth is at or below it.

The whole method lives in one picture: energy as a landscape over the space of wavefunctions, with the eigenstates sitting at its critical points and the ground state at the lowest. Every normalizable state $\psi$ has an energy $\langle H\rangle_\psi$. Sweep over all of Hilbert space and this number has a floor — the ground-state energy $E_{\text{gs}}$ — reached only at the ground state itself. So if you can only afford to search a slice of Hilbert space (an exponential here, a Gaussian there, a knob to turn), the lowest energy you find on that slice is pinned above the true floor. You cannot accidentally dig below the ground state, because there is no state below it to find.

There is a second, subtler feature of this landscape, and it is the reason the method is used everywhere. Near the true ground state the surface is flat — the ground state is a stationary point, so the leading variation is quadratic, a smooth bowl rather than a slope. Sit a little off the bottom and you are barely above it: a wavefunction that is $10\%$ wrong in shape gives an energy that is only about $1\%$ wrong. A crude guess is rewarded far beyond its accuracy. That flatness is a theorem, proved below, not a lucky accident.

The claim is that $\langle H\rangle$ can never dip below $E_{\text{gs}}$. The proof needs nothing but a complete set of energy eigenstates — the resolution of the identity guaranteed by Completeness Relation and Resolution of Identity — and the ordering of the spectrum. Any trial state expands as $|\psi\rangle = \sum_n c_n |n\rangle$ in the exact eigenbasis $\hat H|n\rangle = E_n|n\rangle$ from Eigenvalues and Eigenstates, even though we do not know that basis explicitly. Sandwich $\hat H$ between the expansion and divide by the norm; because the $|n\rangle$ are orthonormal, the cross terms vanish and only the diagonal survives:

where: $c_n = \langle n|\psi\rangle$ is the (unknown) overlap of the trial state with the $n$-th exact eigenstate, and $E_n$ are the (unknown) exact energies ordered $E_{\text{gs}} = E_0 \le E_1 \le E_2 \le \cdots$. This is a weighted average of the exact energies with non-negative weights $|c_n|^2$. A weighted average of numbers each $\ge E_0$ is itself $\ge E_0$ — read off directly by subtracting the floor:

where: every factor $E_n - E_0 \ge 0$ and every weight $|c_n|^2 \ge 0$, so the whole fraction is non-negative. The average sits above the smallest member and can only equal it when all the weight lands on that member — that is, when $\psi$ is the ground state.

The bound alone would be a curiosity if trial functions had to be nearly perfect to be useful. They do not, and the reason is a theorem about the shape of the energy landscape near the ground state — the owned structural result of this article. Suppose the trial state is the true ground state contaminated by a small orthogonal admixture,

where: $\varepsilon$ measures how wrong the guess is in shape (a first-order error), and $|\delta\rangle$ is any normalized state orthogonal to the ground state. The claim is that the resulting energy error is $O(\varepsilon^2)$ — one order better than the shape error. Carrying the admixture through the energy functional, the numerator picks up $E_{\text{gs}} + \varepsilon^2\langle\delta|\hat H|\delta\rangle$ (the cross term $\langle\text{gs}|\hat H|\delta\rangle$ vanishes because $|\delta\rangle\perp|\text{gs}\rangle$ and $\hat H|\text{gs}\rangle\propto|\text{gs}\rangle$), and the denominator becomes $1+\varepsilon^2$:

where: the leading coefficient is the (positive) energy gap between the admixed direction and the ground state. There is no term linear in $\varepsilon$ — the tell-tale signature of a stationary point.

The absence of a linear term is a special case of a stronger fact: $\langle H\rangle$ is stationary at every eigenstate, not only the ground state. Mix any two eigenstates, $|\psi(t)\rangle = \cos t\,|0\rangle + \sin t\,|1\rangle$, and the energy is $\langle H\rangle(t) = E_0\cos^2 t + E_1\sin^2 t$. Its derivative $\dot{\langle H\rangle} = (E_1-E_0)\sin 2t$ vanishes at $t=0$ (pure $|0\rangle$) and $t=\pi/2$ (pure $|1\rangle$), and only there. Each eigenstate is a critical point of the energy functional; the ground state is the minimum, the top state the maximum, the rest saddle points. That opens a door. If you restrict the trial family to states orthogonal to the ground state, the same averaging argument bounds the first excited energy from above: $\langle H\rangle \ge E_1$ whenever $\langle\psi|\text{gs}\rangle = 0$. In practice you rarely need to know the ground state to enforce this — a different symmetry does it for free. An odd-parity trial function is automatically orthogonal to the even-parity ground state, using the parity classification of Parity Symmetry of One-Dimensional Eigenstates, so the lowest odd trial energy bounds the lowest odd level. Extending this to a whole basis of trial functions is the linear (Rayleigh–Ritz) machinery owned by Ritz Variational Method and Excited States; this article supplies its foundation, the stationarity principle.

Nothing sharpens the method like watching it succeed and fail on the same atom. Take hydrogen, whose exact ground state is $\psi_{100}\propto e^{-r/a_0}$ with $E_1 = -13.6$ eV, and hand the variational method two different families. The exponential family $\psi(r) = e^{-r/a}$ carries one knob, the decay length $a$. In atomic units the kinetic and potential expectation values give a clean energy function,

where: the first term is the confinement (kinetic) cost, rising as the cloud is squeezed, and the second is the Coulomb attraction, deepening as it contracts — the two competing scalings whose balance sets the atom's size. Minimizing, $a = a_0$ and $\langle H\rangle = -\tfrac12$ Hartree $= -13.6$ eV, the exact answer. The reason is not luck: the trial family literally contains the true ground state at $a = a_0$, so the minimum of the bound coincides with the floor. The Gaussian family $\psi(r) = e^{-br^2}$ is the workhorse of quantum chemistry (Gaussians integrate easily), but it has the wrong shape at the origin: the true hydrogen state has a cusp (a kink, $d\psi/dr \ne 0$ at $r=0$) that no smooth Gaussian can reproduce. Optimizing the single width $b$ gives

where: $\text{Ry} = 13.6$ eV is the Rydberg. This is about $85\%$ of the true binding — a respectable number from a family that cannot even get the shape near the nucleus right, and a textbook demonstration that the energy forgives what the wavefunction cannot.

Example 1 — Hydrogen from an exponential trial, end to end. Take $\psi(r) = e^{-r/a}$ and find the best energy the family offers.

Step 1. Write the energy function in atomic units, kinetic plus Coulomb: $\langle H\rangle(a) = \dfrac{1}{2a^2} - \dfrac{1}{a}$ Hartree.

Step 2. Minimize: $\dfrac{d\langle H\rangle}{da} = -\dfrac{1}{a^3} + \dfrac{1}{a^2} = 0 \Rightarrow a = 1$ (in units of $a_0$), i.e. $a = a_0$.

Step 3. Evaluate at the minimum: $\langle H\rangle(1) = \tfrac12 - 1 = -\tfrac12$ Hartree.

Step 4. Convert: $-\tfrac12$ Hartree $\times\, 27.211$ eV/Hartree $= \mathbf{-13.6\ eV}$, the exact ground-state energy. The second derivative $+1/a^3 > 0$ confirms a minimum. The bound is saturated because the exponential family contains the true state at $a = a_0$ — the one case where "variational" and "exact" coincide.

Example 2 — A Gaussian on hydrogen: how good is a bad guess. Now the family $\psi(r) = e^{-br^2}$, which has the wrong cusp behavior.

Step 1. Compute the normalized expectations (both are standard Gaussian radial integrals): $\langle T\rangle = \tfrac{3}{2}b$ and $\langle V\rangle = -2\sqrt{2b/\pi}$ in atomic units.

Step 2. Minimize $\langle H\rangle(b) = \tfrac{3}{2}b - 2\sqrt{2b/\pi}$ over $b$.

Step 3. Setting $d\langle H\rangle/db = 0$ and substituting back gives $\langle H\rangle_{\min} = -\dfrac{4}{3\pi}$ Hartree $= -\dfrac{8}{3\pi}$ Ry.

Step 4. In eV: $-\dfrac{4}{3\pi}\times 27.211 \approx \mathbf{-11.5\ eV}$, which is $11.5/13.6 \approx \mathbf{85\%}$ of the true binding. The missing $15\%$ is the price of a family that cannot reproduce the cusp — the energy is far better than the wavefunction, exactly as the quadratic-accuracy theorem promised.

The famous payoff of the method — helium's ground state estimated at $-77.5$ eV against the experimental $-79.0$ eV using an effective-charge trial with $Z_{\text{eff}} = 27/16 = 1.6875$ — is stated here as the destination and computed in full by Variational Method with Trial Wavefunctions. That article owns the $\tfrac58 Z'$ electron–electron repulsion integral, the minimization, and the three-way comparison against perturbation theory; the correlated improvement toward the exact number belongs to Variational Treatment of Helium. This article supplies only what they all consume: the bound and the stationarity that license the search.

• Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 8 — the theorem, the hydrogen exhibits, and helium.
• Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 16 — the variational method and its relation to perturbation theory.
• Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §5.4 — the variational method in the operator formalism.
• Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Complement $E_{XI}$ — the variational method and the Ritz theorem, worked carefully.

Provenance. Statements and inequalities follow Griffiths (3rd ed., Ch. 8), Sakurai & Napolitano (2nd ed., Sec. 5.4), Shankar (2nd ed., Ch. 16) and Cohen-Tannoudji (Complement E_XI). Verifiable numbers: hydrogen $a_0=0.529$ Å and $E_1=-13.6$ eV; harmonic-oscillator ground state $\tfrac12\hbar\omega$; helium variational $Z_{\mathrm{eff}}=27/16=1.6875$ with energy $\approx-77.5$ eV versus experimental $-79.0$ eV (Griffiths Ch. 8). A verifier should reproduce the H-atom radial integrals leading to $\langle H\rangle(a)$, confirm the helium screening result, and check the quadratic-stationarity expansion. No non-standard numbers introduced.