---
title: The Variational Method
module: Approximation Methods for Bound States
moduleNumber: 10
lessonNumber: 4
order: 1004
summary: >
  The expectation of the Hamiltonian in any trial state is an upper bound on the
  true ground-state energy. Minimizing that expectation over a parametrized
  family of trial functions turns the ground-state problem into ordinary
  calculus and needs no small parameter. We prove the bound, apply it to the
  helium atom with a screened effective charge, use a two-center trial to
  predict binding in the hydrogen molecular ion, and extend the method to
  excited states through orthogonality.
topics: [Approximation Methods for Bound States]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 8; §8.1 The Variational Principle, §8.2 The Ground State of Helium, §8.3 The Hydrogen Molecule Ion"
  - book: Shankar
    ref: "Ch. 16; §16.1 The Variational Method"
  - book: Sakurai & Napolitano
    ref: "Ch. 5; §5.4 Variational Methods"
draft: false
---

[Perturbation theory](/quantum-mechanics/approximation-methods/time-independent-perturbation-theory)
needs a Hamiltonian split into a solvable piece and a small correction. Many
problems have no such split: the two electrons of helium repel each other with a
term as large as their attraction to the nucleus, and no piece is small. The
variational method sidesteps the requirement entirely. It rests on a single
inequality — the energy of any trial state overestimates the ground-state energy —
and converts finding the ground state into minimizing a function of a few
parameters. The method gives no error bar by itself, but it gives a rigorous
ceiling, and a well-chosen trial family brings that ceiling within a percent of
the truth.

## The variational theorem

Let $\hat H$ have a discrete spectrum bounded below, with ground-state energy
$E_{gs}$. For any normalizable state $\ket{\psi}$, the expectation of the energy
cannot fall below $E_{gs}$.

> **Theorem (Variational principle).** For any trial state $\ket{\psi}$,
> $$\braket{H} = \frac{\bra{\psi}\hat H\ket{\psi}}{\braket{\psi | \psi}} \ge E_{gs},$$
> with equality if and only if $\ket{\psi}$ is the ground state.

> **Proof.** Expand the trial state in the exact orthonormal eigenstates
> $\hat H\ket{n} = E_n\ket{n}$, so $\ket{\psi} = \sum_n c_n\ket{n}$. Assuming
> $\braket{\psi | \psi} = \sum_n\lvert c_n\rvert^2 = 1$,
> $$\braket{H} = \sum_n E_n\lvert c_n\rvert^2 \ge \sum_n E_{gs}\lvert c_n\rvert^2 = E_{gs},$$
> using $E_n \ge E_{gs}$ for every $n$. Equality requires $c_n = 0$ for every
> state above the ground level, i.e. $\ket{\psi} = \ket{gs}$.

The bound is one-sided and cheap: it needs only an integral, not a solution.
Its power comes from the weak dependence of $\braket{H}$ on errors in $\ket{\psi}$.
If the trial state differs from the true ground state by an amount of order
$\epsilon$, its energy differs by order $\epsilon^2$, because the ground state is
a stationary point of $\braket{H}$. A crude trial wavefunction, wrong by 10% in
shape, can give an energy wrong by only 1%.

$$
% caption: Every trial state's energy lies at or above the true ground level;
% the closer the trial to the true ground state the tighter the ceiling, and the
% minimum over a family is the best available upper bound.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% energy axis
\draw[black, ->] (0,0) -- (0,4.0) node[above, black!70] {energy};
% true levels
\draw[black, very thick] (0.3,0.6) -- (3.3,0.6);
\node[anchor=west, black!70] at (3.4,0.6) {$E_{gs}$ (true)};
\draw[black] (0.3,2.0) -- (3.3,2.0);
\node[anchor=west, black!70] at (3.4,2.0) {$E_1$};
% trial values above
\fill[acc] (1.2,1.1) circle (2.2pt);
\fill[acc] (2.0,1.7) circle (2.2pt);
\fill[acc] (2.6,2.6) circle (2.2pt);
\node[anchor=east, acc] at (1.05,1.1) {best trial};
\draw[acc, ->, thick] (2.6,2.5) .. controls (1.9,1.9) .. (1.25,1.25);
\node[anchor=west, acc] at (2.7,3.0) {minimize $E$};
\end{tikzpicture}
$$

## The recipe

The method is mechanical once a trial family is chosen.

- **Choose a trial family** $\psi(\vec r; a_1, \dots, a_k)$ with adjustable
  parameters, guided by the qualitative shape expected of the ground state
  (nodeless, decaying, symmetric).
- **Compute the energy** $E(a_1, \dots, a_k) = \bra{\psi}\hat H\ket{\psi} / \braket{\psi|\psi}$
  as a function of the parameters.
- **Minimize** by solving $\partial E/\partial a_i = 0$. The minimum value is the
  best upper bound the family can give, and the minimizing parameters give the
  best approximate ground state in that family.

When the family contains the true ground state, the method returns it exactly. A
Gaussian trial for the [harmonic oscillator](/quantum-mechanics/wave-mechanics-1d/operators-expectation-values-and-the-harmonic-oscillator),
$\psi(x) = e^{-b x^2}$, minimizes to $b = m\omega/2\hbar$ and
$E_{\min} = \tfrac12\hbar\omega$, the exact ground state, because the true ground
state is a Gaussian. The interesting cases are those where it is not.

> **Worked example.** Estimate the hydrogen ground state with a Gaussian trial
> $\psi(r) = e^{-\beta r^2}$, deliberately the wrong shape (the true state decays
> as $e^{-r/a_0}$). Computing the kinetic and Coulomb expectations gives
> $$E(\beta) = \frac{3\hbar^2\beta}{2m} - \frac{e^2}{4\pi\epsilon_0}\sqrt{\frac{2\beta}{\pi}},$$
> and $\d E/\d\beta = 0$ yields the minimum
> $$E_{\min} = -\frac{8}{3\pi}\,\text{Ry} \approx -11.5\ \text{eV}.$$
> This lies above the exact $-13.6\ \text{eV}$, as the theorem demands, and
> recovers 85% of the binding energy from a trial function with the wrong
> asymptotic form. The missing 15% is the price of the cusp at the origin,
> which the true state has and the Gaussian lacks.

## The ground state of helium

Helium is the first problem with no exact solution and no small parameter. The
Hamiltonian for two electrons about a charge-$2$ nucleus is

$$
\hat H = -\frac{\hbar^2}{2m}(\nabla_1^2 + \nabla_2^2)
- \frac{e^2}{4\pi\epsilon_0}\left(\frac{2}{r_1} + \frac{2}{r_2}\right)
+ \frac{e^2}{4\pi\epsilon_0}\frac{1}{\lvert\vec r_1 - \vec r_2\rvert},
$$

where the last term, the electron–electron repulsion, is comparable in size to
the attraction and cannot be treated as a perturbation. The physical picture
suggests the trial: each electron partially screens the nucleus from the other,
so each sees an effective charge $Z$ somewhat less than $2$. Take a product of
hydrogenic $1s$ orbitals with $Z$ as the variational parameter,

$$
\psi(\vec r_1, \vec r_2) = \frac{Z^3}{\pi a_0^3}\,
e^{-Z(r_1 + r_2)/a_0}.
$$

Evaluating each piece in Rydberg units gives the kinetic energy $2Z^2$, the
nuclear attraction $-8Z$ (the actual charge is $2$, not $Z$), and the
electron–electron repulsion $+\tfrac54 Z$, so

$$
E(Z) = \left(2Z^2 - \frac{27}{4}Z\right)\text{Ry}.
$$

Minimizing, $\d E/\d Z = 4Z - \tfrac{27}{4} = 0$ gives the effective charge and
energy

$$
Z = \frac{27}{16} = 1.69, \qquad
E_{\min} = -\frac{729}{128}\,\text{Ry} \approx -77.5\ \text{eV}.
$$

> **Worked example.** The variational effective charge $Z = 1.69$ is measurably
> less than the bare $2$: each electron screens about a third of a proton's worth
> of charge from the other. The energy $-77.5\ \text{eV}$ sits above the
> experimental $-79.0\ \text{eV}$ (from the sum of the two ionization energies,
> $24.6 + 54.4\ \text{eV}$), a 2% ceiling. Fixing $Z = 2$ instead — no screening,
> no variation — gives only $-74.8\ \text{eV}$, so optimizing the one parameter
> recovers more than half of the remaining error.

$$
% caption: Each helium electron sees a nucleus partially screened by the other,
% so the best hydrogenic trial uses an effective charge below the bare value; the
% variational minimum lands within two percent of the measured energy.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% nucleus
\fill[black] (0,0) circle (3.5pt);
\node[anchor=north, black!75] at (0,-0.15) {$Z = 2$};
% screening cloud
\draw[acc, dashed] (0,0) circle (1.2);
% two electrons
\fill[acc] (1.7,0.6) circle (2.6pt);
\fill[acc] (-1.6,-0.7) circle (2.6pt);
\node[anchor=west, black!70] at (1.9,0.6) {reduced charge};
\node[anchor=east, black!70] at (-1.8,-0.7) {other electron screens};
\end{tikzpicture}
$$

## The hydrogen molecular ion

The variational method predicts chemical bonding. The ion $\mathrm{H}_2^+$ is one
electron shared by two protons a distance $R$ apart. Fix the protons (the
Born–Oppenheimer approximation: nuclei move slowly) and use a **linear
combination of atomic orbitals**, a sum of ground-state hydrogen orbitals
centered on each proton,

$$
\psi_\pm(\vec r) = A_\pm\left[\psi_0(\vec r - \vec r_1) \pm \psi_0(\vec r - \vec r_2)\right],
$$

with no free scale parameter but the internuclear distance $R$ as the variable to
minimize over. The symmetric (bonding) combination $\psi_+$ piles electron density
between the protons, where it is attracted to both; the antisymmetric
(antibonding) combination $\psi_-$ has a node between them and depletes that
density.

Computing $\braket{H}(R)$ and adding the proton–proton repulsion, the bonding
curve develops a minimum below the energy of a separated hydrogen atom plus a
proton, while the antibonding curve is repulsive at every $R$.

> **Theorem (Binding of the molecular ion).** The bonding trial state has an
> energy minimum below the dissociation limit, so the variational method predicts
> a bound molecule. The minimum occurs at $R \approx 2.4\,a_0 \approx 1.3\ \text{Å}$
> with a binding energy of about $1.8\ \text{eV}$; the antibonding state has no
> minimum and does not bind.

The prediction is qualitatively decisive (a bound state exists) and
quantitatively fair: the measured bond length is $R = 2.0\,a_0 = 1.06\ \text{Å}$
and the binding energy $2.8\ \text{eV}$. The gap is the price of a rigid trial
built from undistorted atomic orbitals; allowing the orbitals to contract toward
the bond tightens both numbers.

$$
% caption: The bonding combination's energy dips below the separated-atom limit
% and produces a minimum at finite internuclear distance, a bound molecule; the
% antibonding combination rises monotonically and does not bind.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (0,0) -- (6.2,0) node[right, black!70] {$R$};
\draw[black, ->] (0,-1.6) -- (0,2.4) node[above, black!70] {energy};
% dissociation limit
\draw[black, dashed] (0,0) -- (6.0,0);
\node[anchor=west, black!70] at (5.0,0.28) {separated atoms};
% bonding curve with minimum
\draw[acc, very thick] plot[domain=0.8:6.0, samples=100]
  (\x, {2.6/(\x*\x) - 2.6/\x*0.9});
\node[anchor=north, acc] at (4.2,-1.0) {bonding};
% antibonding curve monotonic
\draw[black, very thick, dashed] plot[domain=0.8:6.0, samples=100]
  (\x, {3.4/(\x*\x)});
\node[anchor=west, black!70] at (2.2,1.4) {antibonding};
\end{tikzpicture}
$$

## Excited states through orthogonality

The theorem bounds the ground state, but a refinement bounds excited states. If
the trial state is orthogonal to the exact ground state, its energy bounds the
first excited level.

> **Theorem (Bound on the first excited state).** If $\braket{gs | \psi} = 0$,
> then $\braket{H} \ge E_1$, the first excited energy. The proof repeats the
> variational argument with the ground-state component removed:
> $\braket{H} = \sum_{n\ge 1}E_n\lvert c_n\rvert^2 \ge E_1$.

> **Worked example.** For the harmonic oscillator an odd trial
> $\psi(x) = x\,e^{-b x^2}$ is orthogonal to the even ground state and bounds the
> first excited level. Computing the energy expectation gives
> $$\langle H\rangle(b) = \frac{3\hbar^2 b}{2m} + \frac{3 m\omega^2}{8 b},$$
> minimized at $b = m\omega/2\hbar$ to $\langle H\rangle_{\min} = \tfrac32\hbar\omega$,
> the exact first excited energy $E_1 = (1 + \tfrac12)\hbar\omega$. The bound is
> saturated because the trial family contains the true eigenfunction:
> $x\,e^{-m\omega x^2/2\hbar}$ is precisely $\psi_1$. Where the family misses the
> exact state, the same construction still returns a rigorous upper bound on the
> lowest state of that symmetry.

The obstacle is that the exact ground state is usually unknown, so exact
orthogonality is hard to impose. Symmetry rescues the common cases. When the
ground state is even under a symmetry (parity, for a symmetric potential), any
odd trial state is automatically orthogonal to it, so an odd trial family bounds
the lowest odd state without knowing the ground state at all. More generally, a
trial state built in a symmetry sector distinct from the ground state's is
orthogonal to it by construction, and the variational minimum in that sector
bounds the lowest state carrying those quantum numbers.

$$
% caption: A trial state chosen odd under parity is orthogonal to an even ground
% state for free, so its variational minimum bounds the lowest odd state from
% above, extending the method up the spectrum sector by sector.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% ground state even
\draw[black, ->] (0,0) -- (3.2,0);
\draw[acc, very thick] plot[domain=0.2:3.0, samples=60] (\x, {0.9*exp(-2*(\x-1.6)^2)});
\node[anchor=south, acc] at (1.6,0.95) {even ground state};
% odd trial
\begin{scope}[yshift=-2.4cm]
\draw[black, ->] (0,0) -- (3.2,0);
\draw[black, very thick, dashed] plot[domain=0.2:3.0, samples=80]
  (\x, {1.1*(\x-1.6)*exp(-2*(\x-1.6)^2)});
\node[anchor=north, black!70] at (2.4,-0.5) {odd trial: orthogonal};
\end{scope}
\end{tikzpicture}
$$

The variational method and perturbation theory are complementary. Perturbation
theory gives a systematic series when a small parameter exists; the variational
method gives a rigorous bound when none does, at the cost of a guess and a
one-sided estimate. The final method of the module, the
[WKB approximation](/quantum-mechanics/approximation-methods/the-wkb-approximation),
takes a third route: it exploits a slowly varying potential to build the
wavefunction semiclassically, recovering both bound-state quantization and
tunneling rates.

[^griffiths-var]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_ 3rd ed., §8.1 — the variational principle and its proof; §8.2 — the helium ground state with a screened effective charge $Z = 27/16$ giving $-77.5\ \text{eV}$; §8.3 — the hydrogen molecular ion and the LCAO bonding/antibonding trial. Cambridge, 2018.
[^shankar-var]: **Shankar**, _Principles of Quantum Mechanics_ 2nd ed., §16.1 — the variational bound, the second-order insensitivity to trial-function error, and excited-state bounds through orthogonality. Springer, 1994.
[^sakurai-var]: **Sakurai & Napolitano**, _Modern Quantum Mechanics_ 3rd ed., §5.4 — variational methods and the helium estimate. Cambridge, 2021.