The Variational Method

Author

Daniel Fischer

Introduction

The variational method is one of the most elegant and broadly useful ideas in quantum mechanics. It provides a way to approximate the energy and wavefunction of a quantum system without ever solving the Schrödinger equation exactly.

The idea originated in classical mechanics with Lord Rayleigh (1873), who used similar reasoning to approximate the frequencies of vibrating systems. In 1909, Walther Ritz adapted this principle to quantum mechanics—long before the Schrödinger equation was formulated. Decades later, it became one of the first practical methods to estimate the ground-state energy of the helium atom, achieving impressive accuracy at a time when no numerical computers existed.

At its heart, the variational principle expresses a simple but profound statement:

Any reasonable guess for a system’s wavefunction will predict an energy no lower than the true ground-state energy.

This makes it both a conceptual tool—offering insight into why certain wavefunctions “work”—and a computational strategy, forming the foundation of modern quantum chemistry and solid-state theory (including Hartree–Fock, configuration interaction, and density functional methods).

In this chapter, we will:
- derive the variational principle and understand why it works,
- practice applying it through simple examples, and
- connect it to the systematic Ritz method, which generalizes the idea to linear combinations of trial wavefunctions.

By the end, you’ll see how this principle transforms an impossible differential equation into a solvable problem of optimization—a theme that reappears throughout all of modern quantum physics.

1. The Variational Principle in Quantum Mechanics

In quantum mechanics, we often seek the ground-state energy (lowest possible energy) and its corresponding wavefunction. The variational principle provides a general and remarkably useful way to approximate both.

TipVariational principle

For any trial wavefunction \(\psi\), the expectation value of the Hamiltonian satisfies

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

where \(E_0\) is the true ground-state energy.

This means: no matter what function you try, its energy expectation value can never be below the true ground-state energy. The best possible trial function gives an upper bound to \(E_0\).

Note

Even if your trial function isn’t normalized, the principle still holds as long as you divide by \(\langle \psi | \psi \rangle\).

1.1 Why It Works

Let \(\{|n\rangle\}\) be the complete set of energy eigenstates of \(\hat{H}\):

\[ \hat{H}|n\rangle = E_n |n\rangle, \qquad E_0 \le E_1 \le E_2 \le \dots \]

Any normalized state can be written as a superposition:

\[ |\psi\rangle = \sum_n c_n |n\rangle, \qquad \sum_n |c_n|^2 = 1 \]

Then the energy expectation value is

\[ \langle \psi | \hat{H} | \psi \rangle = \sum_n |c_n|^2 E_n \ge E_0 \sum_n |c_n|^2 = E_0 \]

since every \(E_n \ge E_0\). Equality holds only when all coefficients except \(c_0\) vanish, i.e. \(\psi\) is the true ground state.

Thus, the variational principle is a direct consequence of the spectral decomposition of the Hamiltonian.

1.2 How to Use It

In practice, the method becomes a systematic algorithm:

  1. Choose a trial wavefunction \(\psi(\mathbf{r};\alpha_1,\alpha_2,\dots)\) containing one or more adjustable parameters (e.g. width, decay rate).

  2. Compute the expected energy \[ \langle E(\boldsymbol{\alpha}) \rangle = \frac{\int \psi^* \hat{H} \psi\, dV} {\int |\psi|^2\, dV} \]

  3. Minimize with respect to the parameters: \[ \frac{\partial E}{\partial \alpha_i} = 0 \]

  4. Interpret the result.
    The minimized \(\langle E\rangle\) gives your best approximation to \(E_0\) and serves as an upper bound. The optimized parameters yield an approximate ground-state wavefunction.

1.3 Example: The Hydrogen Ground State

To see the method in action, let’s approximate the ground-state energy of the hydrogen atom without solving Schrödinger’s equation directly.

The exact ground state is known to be an exponential, so we choose a trial function of the same form but with adjustable decay rate:

\[ \psi(r) = N e^{-\alpha r} \]

where \(\alpha\) is the variational parameter.

Step 1: Normalize

\[ N = \sqrt{\frac{\alpha^3}{\pi}} \]

Step 2: Compute the Energy Expectation

For the hydrogen Hamiltonian, \[ \hat{H} = -\frac{\hbar^2}{2m} \nabla^2 - \frac{e^2}{4\pi\varepsilon_0 r}, \] the expectation value is

\[ \langle E(\alpha)\rangle = \frac{\hbar^2 \alpha^2}{2m} - \frac{e^2\alpha}{4\pi\varepsilon_0}. \]

Step 3: Minimize

Setting \(dE/d\alpha=0\) gives

\[ \alpha_\text{opt} = \frac{m e^2}{4\pi\varepsilon_0 \hbar^2}. \]

Step 4: Evaluate

Substituting \(\alpha_\text{opt}\) back in:

\[ E_\text{min} = -\frac{m e^4}{2(4\pi\varepsilon_0)^2 \hbar^2} = -13.6\ \text{eV}. \]

This is exactly the true ground-state energy! Had we chosen a less perfect functional form, the result would still have been an upper bound but slightly above \(-13.6\) eV.

1.4 Summary

Concept Formula Comment
Variational energy \(\displaystyle \langle E \rangle = \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle}\) Always ≥ \(E_0\)
Minimization \(\displaystyle \frac{\partial E}{\partial \alpha_i}=0\) Optimize parameters
Physical meaning “Best guess” wavefunction yields lowest possible energy estimate Upper bound to \(E_0\)

2. Ritz’s Method

The Ritz variational method generalizes the basic variational principle into a systematic and computationally tractable form. Instead of relying on a single trial function with a few adjustable parameters, it allows the wavefunction to be expressed as a flexible linear combination of simpler building blocks.

2.1 The Linear Variational Method

Rather than guessing one function \(\psi(r)\), we represent the trial wavefunction as a linear combination of known basis functions:

\[ \psi(r) = \sum_{i=1}^{n} c_i\, \phi_i(r) \]

The coefficients \(c_i\) are unknown and will be determined by minimizing the total energy.

This approach is extremely general:
- The basis functions \(\{\phi_i\}\) could be orthonormal (like sine or spherical harmonics) or non-orthogonal (like Gaussian orbitals).
- The set can be finite or infinite. A larger or more complete basis provides a better approximation to the exact wavefunction.
- In the limit of a complete basis, the expansion becomes exact and the variational approximation converges to the true eigenvalues and eigenstates.

In practice, we work with a finite subset that captures the essential physics of the problem while keeping the algebra manageable.

2.2 Derivation of the Secular Equation

We start from the energy expectation value for the trial wave function

\[ E[\psi] = \frac{\langle \psi | \hat{H} | \psi \rangle} {\langle \psi | \psi \rangle} = \frac{\displaystyle\sum_{ij} c_i^* H_{ij} c_j} {\displaystyle\sum_{ij} c_i^* S_{ij} c_j}, \]

with the matrix elements

\[ H_{ij} = \langle \phi_i | \hat{H} | \phi_j \rangle, \qquad S_{ij} = \langle \phi_i | \phi_j \rangle. \]

The coefficients \(c_i\) are our variational parameters. We want to find the coefficients for which the energy expectation value becomes stationary.

Since the coefficients are generally complex, we need to consider both their real and imaginary parts. Writing

\[ c_i=x_i+iy_i, \]

the stationary conditions are

\[ \frac{\partial E}{\partial x_i}=0, \qquad \frac{\partial E}{\partial y_i}=0. \]

Equivalently, we can formally differentiate with respect to the complex coefficients \(c_i\) or their complex conjugates \(c_i^*\). For a real-valued function, the two real stationary conditions are equivalent to the single complex condition

\[ \boxed{\frac{\partial E}{\partial c_i^*}=0.} \]

This is convenient because the energy expectation value is quadratic in the coefficients and their complex conjugates.

Applying this condition to the energy expectation value leads to

\[ \boxed{ \sum_{j=1}^{n}(H_{ij}-ES_{ij})c_j=0, \qquad i=1,\ldots,n. } \]

These equations can be written in matrix form as

\[ \boxed{ (\mathbf H-E\mathbf S)\vec c=0. } \]

This is a generalized eigenvalue problem. The matrix \(\mathbf H\) contains the Hamiltonian matrix elements, while \(\mathbf S\) accounts for the fact that our basis functions need not be orthogonal.

A nontrivial solution \(\vec c\neq0\) exists only if

\[ \boxed{\det(\mathbf H-E\mathbf S)=0.} \]

This is the secular equation. Its solutions determine the approximate energy eigenvalues, and the corresponding eigenvectors give the coefficients of the approximate wave functions.

The important result is that the variational method transforms the original differential-equation problem into a matrix eigenvalue problem.

We begin with

\[ E=\frac{N}{D}, \]

where

\[ N=\sum_{ij}c_i^*H_{ij}c_j, \qquad D=\sum_{ij}c_i^*S_{ij}c_j. \]

1. Differentiation with respect to a complex coefficient

A complex coefficient \(c_k=x_k+iy_k\) represents two independent real variables. For a real-valued energy functional \(E\), stationarity therefore requires

\[ \frac{\partial E}{\partial x_k}=0, \qquad \frac{\partial E}{\partial y_k}=0. \]

Instead of imposing these two conditions separately, we can combine them into a single complex condition. Using \(c_k^*=x_k-iy_k\) and the chain rule, we define

\[ \frac{\partial}{\partial c_k^*} = \frac12\left( \frac{\partial}{\partial x_k} +i\frac{\partial}{\partial y_k} \right). \]

Since \(E\) is real-valued, its derivatives with respect to \(x_k\) and \(y_k\) are real. Thus,

\[ \boxed{ \frac{\partial E}{\partial c_k^*}=0 \quad\Longleftrightarrow\quad \frac{\partial E}{\partial x_k} = \frac{\partial E}{\partial y_k}=0. } \]

This approach is called Wirtinger differentiation. It allows us to formally treat \(c_k\) and \(c_k^*\) as independent when differentiating, even though they are complex conjugates. In particular,

\[ \frac{\partial c_i^*}{\partial c_k^*}=\delta_{ik}, \qquad \frac{\partial c_i}{\partial c_k^*}=0. \]

We could equally well differentiate with respect to \(c_k\), but choosing \(c_k^*\) leads directly to the desired matrix eigenvalue equation.

2. Apply the quotient rule

Differentiating \(E=N/D\) gives

\[ \frac{\partial E}{\partial c_k^*} = \frac{ D\frac{\partial N}{\partial c_k^*} - N\frac{\partial D}{\partial c_k^*} }{D^2}. \]

At a stationary point, the derivative vanishes:

\[ D\frac{\partial N}{\partial c_k^*} - N\frac{\partial D}{\partial c_k^*}=0. \]

Since \(N=ED\), we obtain

\[ \boxed{ \frac{\partial N}{\partial c_k^*} - E\frac{\partial D}{\partial c_k^*}=0. } \]

Here, \(E\) denotes the value of the energy functional at the stationary point. We do not need to differentiate \(E\) again in this expression.

3. Evaluate the derivatives

For the numerator,

\[ \begin{aligned} \frac{\partial N}{\partial c_k^*} &= \frac{\partial}{\partial c_k^*} \sum_{ij}c_i^*H_{ij}c_j\\ &=\sum_{ij}\delta_{ik}H_{ij}c_j\\ &=\sum_jH_{kj}c_j. \end{aligned} \]

Similarly, for the denominator,

\[ \frac{\partial D}{\partial c_k^*} =\sum_jS_{kj}c_j. \]

Substituting into the stationary condition gives

\[ \sum_jH_{kj}c_j -E\sum_jS_{kj}c_j=0. \]

Therefore,

\[ \boxed{ \sum_j(H_{kj}-ES_{kj})c_j=0. } \]

Since this condition must hold for every \(k\), we obtain

\[ \boxed{\mathbf H\vec c=E\mathbf S\vec c.} \]

4. The secular determinant

Rearranging gives

\[ (\mathbf H-E\mathbf S)\vec c=0. \]

This homogeneous system has a nonzero solution only when the coefficient matrix is singular:

\[ \boxed{\det(\mathbf H-E\mathbf S)=0.} \]

Solving this equation determines the allowed variational energies.

2.3 Interpretation and Computational Aspects

The secular equation represents a generalized eigenvalue problem:

\[ \mathbf{H} \vec{c} = E\, \mathbf{S} \vec{c}. \]

If the basis \(\{\phi_i\}\) is orthonormal, then \(S = I\) and this reduces to the standard matrix eigenvalue equation:

\[ \mathbf{H} \vec{c} = E \vec{c}. \]

In practice:
- \(\mathbf{H}\) and \(\mathbf{S}\) are finite-dimensional matrices.
- Solving the secular equation corresponds to diagonalizing the Hamiltonian matrix in the chosen basis.
- The lowest eigenvalue gives the best variational estimate of the ground-state energy, and the higher ones approximate excited states.

2.4 Advantages of the Ritz Approach

  1. Systematic improvement
    Increasing the number of basis functions (\(n\)) always lowers the ground-state energy estimate, converging toward the exact value.

  2. Access to excited states
    Higher eigenvalues correspond to approximate excited states, often with surprisingly good accuracy.

  3. Flexibility
    Any convenient basis can be used: plane waves for periodic systems, Gaussians for molecules, hydrogenic orbitals for atoms, and so on.

  4. Guaranteed upper bounds
    Each approximate eigenvalue \(E_k^{(\text{approx})}\) is an upper bound to the corresponding exact eigenvalue \(E_k^{(\text{exact})}\).

2.5 Applications

The Ritz variational method underlies many modern computational frameworks:

  • Hartree–Fock theory: Represents many-electron wavefunctions as Slater determinants optimized variationally.
  • Configuration interaction (CI): Expands the total wavefunction in a linear combination of electronic configurations.
  • Density functional theory (DFT): Variational in the electron density rather than the wavefunction itself.
  • Tight-binding and LCAO models: Use localized atomic orbitals as basis functions to describe solids and molecules.

Together, these methods form the foundation of nearly all modern quantum-chemical and condensed-matter calculations.

2.6 Summary and Takeaways

Concept Formula Comment
Variational energy \(\displaystyle \langle E \rangle = \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle}\) Always ≥ \(E_0\)
Minimization condition \(\displaystyle \frac{dE}{d\alpha}=0\) Optimize parameters
Ritz secular equation \((\mathbf{H} - E\mathbf{S})\vec{c} = 0\) Generalized eigenproblem

3. Conclusion

The variational method is one of the most powerful and conceptually elegant tools in quantum mechanics. Even a rough trial function yields a rigorous upper bound to the true energy. Combined with the Ritz method, it becomes a general computational strategy — the foundation of modern quantum chemistry and condensed matter physics.