Anharmonic Oscillator

The anharmonic oscillator is the standard sandbox for approximation methods. It is close enough to the harmonic oscillator to use ladder operators, but rich enough to expose perturbative corrections, asymptotic series, variational estimates, WKB ideas, and numerical benchmarking.

The simplest stable version is the quartic oscillator:

$$H = \frac{P^2}{2m} + \frac12m\omega^2X^2 + \lambda X^4, \qquad \lambda\gt 0.$$

Here

$$H_0 = \frac{P^2}{2m} + \frac12m\omega^2X^2$$

is exactly solvable, and the perturbation is $V=X^4$ with coefficient $\lambda$.

Regimes

For small $\lambda$, perturbation theory around the harmonic oscillator is natural. For large quantum number or slowly varying classical motion, semiclassical estimates become useful. For the ground state, a Gaussian variational estimate gives a quick upper bound. For precision and benchmarking, numerical diagonalization in a harmonic oscillator basis is straightforward.

The value of the model is that these methods can be compared on the same Hamiltonian.

First-Order Perturbative Correction

Using

$$X = \sqrt{\frac{\hbar}{2m\omega}} \left(a+a^\dagger\right),$$

one finds

$$\langle n|X^4|n\rangle = \left( \frac{\hbar}{2m\omega} \right)^2 (6n^2+6n+3).$$

Therefore the first-order energy correction for the quartic perturbation is

$$\Delta E_n^{(1)} = \lambda \frac{3\hbar^2}{4m^2\omega^2} \left( 2n^2+2n+1 \right).$$

The general interpretation, symmetry tests, and degenerate warning for this formula are collected on First-Order Energy Corrections.

The corrected energy through first order is

$$E_n = \hbar\omega \left(n+\frac12\right) + \lambda \frac{3\hbar^2}{4m^2\omega^2} \left( 2n^2+2n+1 \right) + O(\lambda^2).$$

Symmetry Checks

The harmonic oscillator eigenstates have definite parity. Since $X^4$ is even under parity, it has nonzero diagonal matrix elements. A cubic perturbation $\lambda X^3$ would have zero first-order energy correction in every oscillator eigenstate because $X^3$ is odd:

$$\langle n|X^3|n\rangle=0.$$

This does not mean a cubic perturbation has no effect. It means the first nonzero energy shift comes at higher order, assuming the Hamiltonian remains physically well-defined in the regime considered.

Perturbative Validity

A natural dimensionless measure for the ground state is the ratio of quartic potential energy to oscillator energy scale:

$$g = \frac{\lambda x_0^4}{\hbar\omega}, \qquad x_0=\sqrt{\frac{\hbar}{m\omega}}.$$

Up to numerical factors,

$$g \sim \frac{\lambda\hbar}{m^2\omega^3}.$$

Small $g$ supports a low-order perturbative calculation. Large $n$ increases the size of the quartic correction, so a perturbative treatment valid for the ground state may fail for highly excited states.

Asymptotic Series Preview

The quartic oscillator perturbation series is a classic example of a useful but subtle expansion. For the stable $\lambda X^4$ oscillator, low orders can be highly informative when the coupling is small, but the full perturbative series is asymptotic rather than a simple convergent power series. Asymptotic Series and Nonperturbative Corrections derives the least-term stopping rule and explains how the instability at negative coupling controls factorial large-order growth.

This matters pedagogically: perturbation theory is not judged only by convergence of the infinite series. It is judged by whether a controlled truncation gives a reliable approximation in a stated regime.

Variational Estimate Preview

Use a Gaussian trial ground state: the ground state of a harmonic oscillator with adjustable frequency $\Omega$. Its energy expectation is

$$E(\Omega) = \frac{\hbar\Omega}{4} + \frac{\hbar\omega^2}{4\Omega} + \frac{3\lambda\hbar^2}{4m^2\Omega^2}.$$

Minimizing over $\Omega\gt 0$ gives a variational upper bound to the true ground-state energy. For small $\lambda$, the minimizing $\Omega$ is close to $\omega$. For positive $\lambda$, the optimum has $\Omega>\omega$ and the Gaussian narrows. Anharmonic Oscillator by Variational Methods owns the full optimization, weak- and strong-coupling limits, residual test, and numerical benchmark.

This estimate is not a replacement for perturbation theory. It answers a different question: how good a chosen trial family can be as a bound.

Variational Perturbation Theory turns the same adjustable frequency into an order-dependent reference scale. Its first order is this Gaussian upper bound, while subsequent orders reorganize the divergent weak-coupling series and are not generally variational bounds.

Numerical Benchmark

A practical numerical benchmark expands the Hamiltonian in the harmonic oscillator basis and truncates to a finite matrix. The matrix elements of $X^4$ connect states with the same parity and with oscillator quantum numbers differing by $0,2,$ or $4$.

Increasing the basis size tests convergence. This gives a way to compare perturbative, variational, and semiclassical estimates against a controlled numerical calculation.

Bridge to Field Theory

The quartic oscillator is the one-degree-of-freedom cousin of quartic field interactions. The analogy is useful for intuition about perturbative expansions and interaction terms, but it should not be overstated. Field theory adds infinitely many modes, locality, renormalization, and relativistic or many-body structure.

Common Mistakes

• Treating the first-order quartic shift as valid for all $\lambda$.
• Ignoring the growth of corrections with $n$.
• Concluding that an odd perturbation has no effect just because the first-order diagonal matrix element vanishes.
• Confusing an asymptotic perturbation series with a convergent one.
• Using a variational estimate without stating the trial family.
• Trusting a truncated numerical basis without checking convergence.

Cross-Links

• Time-Independent Perturbation Theory
• Nondegenerate Perturbation Theory
• Second-Order Energy Corrections
• Higher-Order Structure
• Asymptotic Series and Nonperturbative Corrections
• Perturbation Theory for the Harmonic Oscillator
• Anharmonic Oscillator by Perturbation Theory
• Anharmonic Oscillator by Variational Methods
• Small Parameters and Error Estimates
• Quantum Harmonic Oscillator
• Ladder-Operator Solution
• Matrix Diagonalization
• Harmonic Oscillator Reference
• Vibrations of Diatomics applies anharmonic reasoning to molecular term values, the Morse potential, isotope shifts, and infrared overtones.

References

• C. M. Bender and T. T. Wu, “Anharmonic oscillator,” Physical Review 184, 1231-1260, 1969.
• C. M. Bender and T. T. Wu, “Anharmonic oscillator. II. A study of perturbation theory in large order,” Physical Review D 7, 1620-1636, 1973.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., Oxford University Press, 2002.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.

Exercises

1. Use parity to show that $\langle n|X^3|n\rangle=0$ for harmonic oscillator eigenstates.

Solution

The state $\lvert n\rangle$ has definite parity, so its probability density is even. The operator $X^3$ is odd under parity. Therefore the integrand in position space is odd:

$$\psi_n^*(x)x^3\psi_n(x)$$

changes sign under $x\mapsto -x$. Its integral over the real line vanishes.

2. Compute the first-order quartic shift for the ground state.

Solution

Set $n=0$ in

$$\Delta E_n^{(1)} = \lambda \frac{3\hbar^2}{4m^2\omega^2} \left( 2n^2+2n+1 \right).$$

Then

$$\Delta E_0^{(1)} = \frac{3\lambda\hbar^2}{4m^2\omega^2}.$$