Rayleigh–Ritz Method

The Rayleigh–Ritz method is the finite-dimensional form of the variational principle. Choose a trial subspace, represent the Hamiltonian on that subspace, and diagonalize the resulting matrix. The lowest eigenvalue is a variational upper bound to the true ground-state energy.

This is the basic structure behind basis-set quantum mechanics, configuration interaction, oscillator-basis truncations, finite-element discretizations, and many numerical diagonalization methods.

Required background. The Variational Principle supplies the bound; Matrix Diagonalization supplies the finite eigenproblem. Variational Parameters treats outer optimization when basis functions carry nonlinear widths or centers.

Trial Subspaces

Let $H$ be self-adjoint and bounded below. Choose an $N$-dimensional subspace $\mathcal V_N\subset Q(H)$ of its quadratic-form domain, spanned by linearly independent trial vectors

$$\mathcal V_N = \operatorname{span} \{ \lvert\phi_1\rangle,\ldots,\lvert\phi_N\rangle \}.$$

A general trial vector in this subspace has the form

$$\lvert\psi\rangle = \sum_{j=1}^N c_j\lvert\phi_j\rangle.$$

Rayleigh–Ritz minimizes the energy quotient over all coefficient vectors $c_j$ in the chosen subspace:

$$E_N = \min_{\psi\in\mathcal V_N,\ \psi\ne0} \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}.$$

By the variational principle,

$$E_N\ge E_0.$$

Orthonormal Basis

If the basis vectors are orthonormal, define

$$H_{ij}=q_H(\phi_i,\phi_j).$$

On the operator domain this is $\langle\phi_i|H|\phi_j\rangle$; for form-domain trial functions use the weak kinetic form below. Then

$$q_H[\psi] = \sum_{i,j}c_i^*H_{ij}c_j, \qquad \langle\psi|\psi\rangle = \sum_i |c_i|^2.$$

Minimizing the quotient gives the finite-dimensional eigenvalue problem

$$\sum_{j=1}^N H_{ij}c_j = E c_i, \qquad i=1,\ldots,N.$$

The lowest eigenvalue of this matrix is the best variational energy inside $\mathcal V_N$. The corresponding eigenvector gives a state of minimum energy in that subspace. It need not be the state with largest overlap with the exact ground state.

Nonorthogonal Basis

Many useful basis functions are not orthogonal. Define both the Hamiltonian matrix and overlap matrix:

$$H_{ij}=q_H(\phi_i,\phi_j), \qquad S_{ij} = \langle\phi_i|\phi_j\rangle.$$

The variational condition becomes the generalized eigenvalue problem

$$\sum_{j=1}^N H_{ij}c_j = E \sum_{j=1}^N S_{ij}c_j.$$

Equivalently,

$$Hc=ESc.$$

Linear independence makes $S$ positive definite on coefficient space. If two basis functions are nearly linearly dependent, $S$ becomes ill-conditioned and the numerical eigenvalues may become unstable.

Why Diagonalization Appears

The quotient

$$R(c) = \frac{c^\dagger Hc}{c^\dagger Sc}$$

is stationary when variations with respect to $c_i^*$ vanish. For the nonorthogonal case,

$$\delta \left( c^\dagger Hc - E c^\dagger Sc \right) = 0$$

gives

$$Hc=ESc.$$

Thus matrix diagonalization is not an extra approximation after choosing the subspace. It is the exact minimization problem inside that subspace.

Convergence with Basis Size

If the subspaces are nested,

$$\mathcal V_1\subset\mathcal V_2\subset\cdots,$$

then the corresponding lowest Ritz values satisfy

$$E_1\ge E_2\ge\cdots\ge E_0.$$

The sequence decreases because each larger subspace contains all previous trial states. If the union of the nested subspaces is dense in the form norm, their lowest Ritz values converge to the spectral bottom. Hilbert-space norm density alone does not control the kinetic energy.

For practical calculations, convergence should be checked by increasing the basis size, changing basis families, or comparing with independent approximations such as perturbation theory or WKB when those are available.

Excited-State Estimates

The Rayleigh–Ritz spectrum also gives approximations to excited states. For a self-adjoint Hamiltonian with discrete low-lying spectrum, the min–max principle implies that the $k$th Ritz value in any admissible trial subspace of dimension at least $k$ is an upper bound to the $k$th exact eigenvalue below essential spectrum, with eigenvalues counted with multiplicity. The precise theorem, its proof, and its limits at essential spectrum are given in Upper Bounds and the Min–Max Principle.

Excited Ritz values are ordered eigenvalues of the whole projected problem. Merely imposing orthogonality to an approximate ground-state vector on a separately optimized family does not establish an excited-state upper bound. Preserve the intended symmetry sector and monitor convergence of eigenvectors as well as eigenvalues. A computed state with the wrong nodal structure or wrong symmetry may approximate the wrong physical level.

Choosing a Basis

A good basis should make the important physics cheap to represent. Common choices include:

• exact eigenstates of a nearby solvable Hamiltonian,
• harmonic oscillator functions for confined or approximately quadratic problems,
• plane waves for translation-invariant or periodic systems,
• localized finite elements or grid functions for spatially inhomogeneous problems,
• symmetry-adapted states for angular momentum, parity, or internal symmetries.

The best choice depends on the Hamiltonian, boundary conditions, target energy range, and required observables. The anharmonic oscillator is a standard test case because a harmonic oscillator basis gives a transparent comparison between perturbative and variational calculations.

Matrix Elements

The method is only as reliable as the matrix elements. For a coordinate-space Hamiltonian

$$H = - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x),$$

one computes

$$H_{ij} = \int dx\, \phi_i^*(x) \left[ - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right] \phi_j(x).$$

If the basis functions obey the boundary conditions, integration by parts often gives a symmetric kinetic-energy matrix:

$$T_{ij} = \frac{\hbar^2}{2m} \int dx\, \frac{d\phi_i^*}{dx} \frac{d\phi_j}{dx},$$

with boundary terms vanishing. This form is often numerically better because it makes positivity of the kinetic energy explicit.

Exercises

1. In an orthonormal two-state basis, let

$$H = \begin{pmatrix} A & B \\ B^* & C \end{pmatrix}.$$

Find the two Ritz values.

Solution

The characteristic equation is

$$(A-E)(C-E)-|B|^2=0.$$

Therefore

$$E_\pm = \frac{A+C}{2} \pm \sqrt{ \left(\frac{A-C}{2}\right)^2 + |B|^2 }.$$

The lower value $E_-$ is the best variational energy inside the two-dimensional subspace.

2. For a normalized nonorthogonal pair with $\langle\phi_1|\phi_2\rangle=s\in(0,1)$, suppose $H_{ij}=E_*S_{ij}$. Compare the correct generalized eigenvalues with diagonalizing $H$ alone.

Solution

The generalized problem gives two eigenvalues $E_*$. Diagonalizing $H=E_*\begin{pmatrix}1&s\\s&1\end{pmatrix}$ gives $E_*(1\pm s)$ instead: these reflect a coordinate metric error, not physical splitting. Whitening gives the Hermitian problem $S^{-1/2}HS^{-1/2}u=Eu$, with $c=S^{-1/2}u$; a Cholesky reduction avoids forming inverse matrices explicitly.

3. Explain why a nonorthogonal basis with nearly dependent vectors can cause numerical trouble even if the variational principle remains mathematically true.

Solution

Nearly dependent vectors make the overlap matrix $S$ nearly singular. Then small errors in matrix elements, arithmetic, or basis construction can produce large changes in the generalized eigenvectors and sometimes spurious eigenvalues. The variational statement assumes exact matrix elements in the exact trial subspace; numerical computation also needs a well-conditioned representation of that subspace.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013.
• B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, 1998.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.