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 $\lambda$.

Required background. Use the harmonic oscillator as the reference spectrum and nondegenerate perturbation theory for weak coupling.

The quartic oscillator across coupling 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. Parity permits diagonal matrix elements of $X^4$; their strict positivity follows from $\langle X^4\rangle=\|X^2\psi\|^2>0$ for a nonzero normalizable oscillator state. 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. The formal shift starts at second order. A pure real cubic potential is unbounded below on one side, however, so its local expansion describes metastable behavior rather than a stable oscillator spectrum.

Perturbative Validity

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

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

With the stated oscillator length this is exactly

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

Setting $X=x_0q$ gives $H/(\hbar\omega)=-(1/2)\partial_q^2+q^2/2+gq^4$. This dimensionless form makes comparisons at different masses and frequencies unambiguous.

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 derives the optimization and compares its weak- and strong-coupling limits with numerical energies.

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 $4$.

Use the projected matrix elements of $X^4$, not the fourth power of an already truncated $X$ matrix. Increasing nested basis spaces then gives Rayleigh–Ritz upper bounds for the corresponding ordered discrete levels and tests convergence.

At strong coupling, set $q=g^{-1/6}y$. The dimensionless Hamiltonian becomes

$$\frac{H}{\hbar\omega}=g^{1/3} \left[-\frac12\partial_y^2+y^4+\frac12g^{-2/3}y^2\right].$$

Thus each fixed low-lying energy scales as $g^{1/3}$ for $g\to\infty$, while the physical width scales as $x_0g^{-1/6}$. A weak-coupling polynomial cannot capture this limit without reorganization. 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.

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

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

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.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
• 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.