Raleigh-Ritz Method in Quantum Mechanics

• Raleigh-Ritz method is a variational approach for approximating eigenvalues and eigenstates of quantum Hamiltonians using finite-dimensional projections.
• It employs physically motivated basis functions and matrix diagonalization to ensure systematic convergence and high numerical accuracy.
• The method is especially reliable for complex potentials, such as Mexican hat and double-well profiles, where perturbative techniques fail.

The Raleigh Ritz method is a variational computational procedure for approximating the eigenstates and eigenvalues of quantum mechanical Hamiltonians, particularly prominent in applications where the analytic spectrum is inaccessible. It leverages the variational principle to generate lower bounds for the ground state energy by projecting the full spectral problem onto a finite-dimensional subspace, typically constructed from physically motivated basis functions. In quantum mechanical models with bounded below pure point spectra, especially those with nonlinear or double-well ("Mexican hat") potentials, the method is rigorously testable against exactly solvable cases and thus provides a benchmark for numerical reliability (Zarate et al., 25 Oct 2025).

1. Mathematical Formulation and Variational Principle

Given a self-adjoint Hamiltonian $H$ acting on a Hilbert space $\mathcal{H}$, the goal is to compute the ground state $\Omega$ and lowest eigenvalue $E_0$: $$H \Omega = E_0 \Omega, \qquad E_0 = \min \operatorname{Spec}(H)$$ The variational principle guarantees that for all normalized trial states $\psi$ ($\|\psi\|=1$),

$\langle \psi, H \psi \rangle \geq E_0$

with equality if and only if $\psi = \Omega$. The method constructs an optimal approximation by restricting $\psi$ to a variational family, typically parameterized by a truncation of a complete orthonormal basis $\{b_n\}$.

The Raleigh-Ritz procedure is as follows:

• Select the first $N$ basis functions $\{b_n\}_{n=0}^{N-1}$.
• Compute the finite-dimensional Hamiltonian matrix:

$$H^N_{mn} = \langle b_m, H b_n \rangle, \qquad m,n=0,\dots,N-1$$

• Diagonalize $H^N$ to obtain its eigenvalues $$\lambda_0^N \leq \lambda_1^N \leq \dots \leq \lambda_{N-1}^N$$ and eigenvectors $e_0^N, e_1^N, \dots$.
• The lowest eigenvalue $\lambda_0^N$ is the variational ground state estimate, and $e_0^N$ approximates $\Omega$.

2. Application to Mexican Hat and Double-Well Potentials

The method is especially suited for potentials of the form

$$V(q) = -\frac{m\omega^2}{2}q^2 + \frac{g}{2n}q^{2n}$$

with $n \geq 2$, generating Mexican hat profiles and double-well structures. In (Zarate et al., 25 Oct 2025), the analysis is extended via supersymmetric quantum mechanics by reverse-engineering analytic ground states: $$\Omega_b(x) = \exp\left[\sigma\frac{x^2}{2l^2} - \frac{x^{2k}}{2k L^{2k}}\right]$$ The associated Hamiltonian takes the form: $$H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V_b(x)\qquad V_b(x) = [S_b'(x)]^2 + S_b''(x)$$ where $S_b(x)$ is chosen so that $\Omega_b(x)$ is normalizable and $H\Omega_b=0$.

This construction allows a rigorous test: the exact ground state is available for comparison, and all expansion coefficients in a reference basis $\{b_n\}$ are calculable analytically.

3. Basis Selection and Matrix Construction

Basis choice in $L_2(\mathbb{R})$ typically favors the harmonic oscillator eigenfunctions (Hermite functions): $$b_n(x) = \frac{\pi^{-1/4}}{\sqrt{2^n n!}} e^{-x^2/2} H_n(x)$$ For each $N$, one computes $H^N$ analytically, using known orthogonality properties and integrals for polynomial potentials.

The truncated Hamiltonian matrix $H^N_{mn}$ is then diagonalized, yielding the variational ground state approximant $e_0^N$ and spectrum $\{\lambda_I^N\}$.

4. Convergence, Norm Errors, and Quantitative Benchmarks

The method is evaluated using two quantitative criteria:

• Ground state energies: $\lambda_0^N$ versus exact $E_0$ (which is zero by construction in the considered models).
• Expansion coefficients: Compute $c_n^N = \langle b_n, e_0^N \rangle$ and compare to exact $I_n = \langle b_n, \Omega_b \rangle / \|\Omega_b\|$.

As $N$ increases:

• $\lambda_0^N \to E_0$.
• $c_n^N \to I_n$ for all $n$.
• The truncated sum $\sum_{n=0}^{N-1} |c_n^N|^2$ approaches unity, ensuring norm preservation.

Empirical convergence is rapid for moderate nonlinearity (lower $k$, smaller $\epsilon$), achieving $\sim10^{-15}$ accuracy in $N \sim 200-300$ states. Greater nonlinearity (high $k$, strong coupling) slows convergence, requiring larger bases.

5. Limits of Perturbation Theory and Non-Perturbative Accuracy

Perturbation theory is ineffective at strong coupling or for highly nonlinear potentials. The Raleigh-Ritz method, by contrast, is non-perturbative, systematically improvable by increasing $N$, and is demonstrably robust in settings where perturbative approaches fail.

Reliability is established via rigorous numerical comparison to exact analytic ground states. The method’s variational nature ensures that $\lambda_0^N$ always overestimates $E_0$, and convergence behavior can be verified quantitatively.

6. Practical Implementation, Limitations, and Utility

Implementation requires computation of matrix elements $H^N_{mn}$, diagonalization of $H^N$, and extraction of the ground state vector. For models with known analytic solutions, the approach provides a benchmark for:

• Rate of convergence with $N$ (basis size)
• Precision in coefficient recovery and energy estimates
• Diagnostic for completeness and accuracy of the chosen basis

The main practical limitation is computational cost: rapid increase in required $N$ for strong coupling or high-degree potentials demands more memory and compute time.

7. Summary and Benchmark Status

The Raleigh-Ritz method constitutes a controlled, robust algorithm for approximating the ground state and spectrum of quantum Hamiltonians. When applied to Mexican hat-type potentials with known analytic solutions (Zarate et al., 25 Oct 2025), it achieves machine precision accuracy, far exceeds perturbation theory in challenging regimes, and provides a rigorous testbed for numerical spectral methods in quantum mechanics.

Key formulas:

Step	Expression	Notes
Hamiltonian	$H = \frac{p^2}{2m} + V(q)$	General quantum system
Truncation	$H^N_{mn} = \langle b_m | H | b_n \rangle$	$b_n$: orthonormal basis
Diagonalization	Find $\lambda_I^N$ from $H^N$	Approximates true energies
Ground state approx	$e_0^N$: eigenvector for lowest eigenvalue	Approximates $\Omega$
Norm convergence	$$\left| \sum_{n=0}^{N-1} |c_n^N|^2 - 1 \right| \to 0$$	Ensures wavefunction normalization

The method is thus validated for a fundamental class of quantum models, including those relevant for symmetry-breaking phenomena and field-theoretic analogs.