Perturbation Theory Benchmarks

This notebook benchmark compares perturbative predictions for the quartic anharmonic oscillator with direct diagonalization in a truncated harmonic-oscillator basis. Its purpose is to show where low-order perturbation theory is accurate, where it starts to drift, and how numerical truncation can be separated from approximation error.

Physical Problem

Use dimensionless oscillator units

$$\hbar=m=\omega=1$$

and study

$$H(g) = \frac12P^2+\frac12X^2+gX^4.$$

The unperturbed basis is the harmonic oscillator basis $\{\lvert n\rangle\}$ with

$$H_0\lvert n\rangle = \left(n+\frac12\right)\lvert n\rangle.$$

The coupling $g$ is the dimensionless perturbation parameter.

Analytic Targets

For the ground state, perturbation theory gives

$$E_0(g) = \frac12 + \frac34g - \frac{21}{8}g^2 + O(g^3).$$

For general $n$, the first-order shift is

$$E_n^{(1)} = \frac34g \left( 2n^2+2n+1 \right).$$

The notebook should compare:

• unperturbed energies,
• first-order perturbation theory,
• second-order ground-state perturbation theory,
• exact diagonalization in a truncated basis.

Numerical Setup

Construct the ladder operators in an $N$-dimensional basis:

$$a_{n-1,n}=\sqrt n, \qquad a^\dagger_{n+1,n}=\sqrt{n+1}.$$

Then

$$X=\frac{a+a^\dagger}{\sqrt2},$$

Build the unperturbed Hamiltonian as the exact diagonal matrix

$$H_{0,N} = \operatorname{diag} \left( \frac12,\frac32,\frac52,\ldots,N-\frac12 \right),$$

then add the truncated interaction matrix:

$$H_N(g) = H_{0,N}+gX^4$$

as an $N\times N$ Hermitian matrix and diagonalize it.

Constructing $H_0$ directly as a diagonal matrix avoids boundary artifacts that can appear if $P^2/2+X^2/2$ is formed after truncating the ladder operators.

Because $X^4$ preserves parity, even and odd oscillator states do not mix. The notebook may diagonalize the full matrix or separate parity blocks as a convergence check.

Parameter Sweep

Use a grid such as

$$g\in \{0,0.001,0.003,0.01,0.03,0.1,0.3\}.$$

For each $g$, compute low-lying eigenvalues for several truncations:

$$N=20,40,80,120.$$

The benchmark should report only eigenvalues that are stable under increasing $N$ to the chosen tolerance.

Expected Figures

The notebook should generate:

• ground-state energy versus $g$, with numerical diagonalization and first- and second-order perturbation curves;
• relative error versus $g$ on a log scale;
• convergence of $E_0$ with basis size $N$ for representative couplings;
• optionally, low-lying excited energies compared with first-order perturbation theory.

Axes should state the dimensionless units. Error plots should specify the numerical reference truncation.

Validation Checks

At $g=0$, the numerical spectrum must reproduce

$$E_n=n+\frac12$$

to machine precision for the represented states.

Hermiticity should be checked:

$$\|H_N-H_N^\dagger\|\approx0.$$

Eigenpair residuals should be small:

$$\|H_Nv_j-E_jv_j\|\ll1.$$

Parity should be conserved. Numerically, matrix elements coupling even and odd states should vanish up to roundoff.

Error Separation

There are two different errors:

• perturbative error: difference between the perturbative formula and the converged numerical eigenvalue;
• truncation error: difference between numerical eigenvalues at finite $N$ and the large-$N$ reference.

Do not interpret a perturbative comparison until the truncation error is smaller than the effect being studied.

For the ground state, a useful diagnostic is

$$\Delta_{\mathrm{pert}}(g) = E_0^{\mathrm{num}}(g) - \left( \frac12+\frac34g-\frac{21}{8}g^2 \right).$$

For small $g$, this should scale approximately like $g^3$ until numerical error or higher-order asymptotic behavior becomes visible.

Known Failure Modes

• Too small a basis at larger $g$, where the wavefunction samples larger $|x|$.
• Comparing high excited states that are close to the truncation boundary.
• Treating agreement at one $g$ as proof of convergence across the sweep.
• Forgetting that the perturbation series is asymptotic at high order.
• Losing parity structure through indexing mistakes in the ladder operators.

Reproducibility Metadata

Record:

• programming language and version;
• linear algebra library and version;
• matrix dimension $N$;
• coupling grid;
• sorting convention for eigenvalues;
• numerical precision;
• residual tolerance;
• hardware or backend if relevant.

The calculation is deterministic and should not require random seeds.

Cross-Links

• Computational Notebooks
• Variational Optimization Notebook
• Anharmonic Oscillator by Perturbation Theory
• Anharmonic Oscillator
• Higher-Order Structure
• Matrix Diagonalization
• Rayleigh-Ritz Method
• Small Parameters and Error Estimates

References

• C. M. Bender and T. T. Wu, “Anharmonic oscillator,” Physical Review 184, 1231-1260, 1969.
• G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, 2013.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Why should $E_0^{\mathrm{num}}(g)-E_0^{(2)}(g)$ scale approximately as $g^3$ for sufficiently small $g$?

Solution

The second-order perturbative expression includes terms through $g^2$. If the expansion is valid and the numerical result is converged, the first omitted perturbative term is order $g^3$. Thus the difference should scale like $g^3$ until numerical error or asymptotic effects dominate.

2. Why is parity a useful diagnostic in this benchmark?

Solution

The Hamiltonian contains only even powers of $X$, so it commutes with parity. Even and odd oscillator basis states should not mix. Nonzero even-odd matrix elements indicate an indexing, basis, or operator-construction error.