Perturbative Expansion: Order and Convergence

Welcome back. In our last lesson, we drew a crucial line in the sand, distinguishing between regular and singular perturbation problems. We saw that singular problems, where the $ϵ \to 0$ limit fundamentally alters the nature of the equations, require their own special toolkit.

Today, we turn our attention to the mechanics of regular perturbations. We will construct the formal machinery for calculating corrections to physical quantities, order by order, as a power series in a small parameter. This process is the bread and butter of theoretical physics. However, generating the series is only half the story. The more profound and subtle part of a physicist's craft lies in assessing what that series actually means. Is it a convergent series that precisely represents the answer, or something else entirely? As you'll see, some of the most important expansions in physics, from quantum mechanics to your own field of QFT, are not convergent at all. This lesson will equip you to perform perturbative calculations and, crucially, to interpret the results with physical insight.

1. The Machinery of Perturbation Theory: A Quantum Mechanics Framework

The archetypal example of regular perturbation theory is the time-independent, non-degenerate case in quantum mechanics. You have likely encountered the final formulas in your quantum courses. Our goal here is to re-examine the methodology of its derivation, as it provides a template for any regular perturbation problem.

The setup is a Hamiltonian $H$ that is a small perturbation away from a solvable Hamiltonian $H^{(0)}$ :

H (λ) = H^{(0)} + λ δ H

We assume we know the complete set of eigenvalues $E_{n}^{(0)}$ and orthonormal eigenstates $∣ n^{(0)} ⟩$ for $H^{(0)}$ . We then posit that the eigenvalues $E_{n} (λ)$ and eigenstates $∣ n ⟩_{λ}$ of the full Hamiltonian $H (λ)$ can be expressed as a power series in the small, dimensionless parameter $λ$ :

E_{n} (λ) = E_{n}^{(0)} + λ E_{n}^{(1)} + λ^{2} E_{n}^{(2)} + \dots

∣ n ⟩_{λ} = ∣ n^{(0)} ⟩ + λ ∣ n^{(1)} ⟩ + λ^{2} ∣ n^{(2)} ⟩ + \dots

The core of the method is to substitute these series into the full Schrödinger equation $H (λ) ∣ n ⟩_{λ} = E_{n} (λ) ∣ n ⟩_{λ}$ and collect terms with the same power of $λ$ . Since the resulting equation must hold for any small $λ$ , the coefficient of each power of $λ$ must vanish independently. This generates a hierarchy of equations that can be solved order by order.

Chapter 1 - Perturbation theory

This chapter from the MIT OpenCourseWare notes on Quantum Physics III provides a clear and systematic derivation of the standard formulas. We will use it to ground our understanding of the general procedure.

Please read section 1.1, "Nondegenerate perturbation theory". Follow the derivation from the initial setup of the power series in Eq. (1.1.7), through the hierarchy of equations in Eq. (1.1.13), to the final summary formulas in Eq. (1.1.37). Focus on the logic of the derivation: How is the equation for a given order, say \lambda^k, constructed? How is the inner product with the unperturbed state \langle n^{(0)}| used to isolate the energy corrections? How is the completeness relation used to determine the state corrections?

The derivation you just studied yields the foundational results of non-degenerate perturbation theory:

First-Order Energy Correction: The expectation value of the perturbation in the unperturbed state.
$E_{n}^{(1)} = ⟨ n^{(0)} ∣ δ H ∣ n^{(0)} ⟩$
First-Order State Correction: A superposition of all other unperturbed states, weighted by the off-diagonal matrix elements of the perturbation and the energy difference.
$∣ n^{(1)} ⟩ = \sum_{k \neq = n} \frac{⟨ k ^{(0)} ∣ δ H ∣ n ^{(0)} ⟩}{E _{n}^{(0)} - E _{k}^{(0)}} ∣ k^{(0)} ⟩$
Second-Order Energy Correction: Determined by the "mixing" of the state $∣ n^{(0)} ⟩$ with all other states.
$E_{n}^{(2)} = \sum_{k \neq = n} \frac{∣ ⟨ k ^{(0)} ∣ δ H ∣ n ^{(0)} ⟩ ∣ ^{2}}{E _{n}^{(0)} - E _{k}^{(0)}}$

This systematic procedure—expand, substitute, collect powers, and solve the hierarchy—is the universal algorithm for regular perturbation problems.

2. Assessing the Expansion: Convergence vs. Asymptoticity

We have generated a power series for the energy. The immediate mathematical question is: does this series converge? A physicist's question is more pragmatic: is this series useful? You might assume the two are synonymous, but in theoretical physics, they are decidedly not. Many of the most celebrated and useful perturbation series are, in fact, divergent.

To understand this crucial concept, let's explore a simple, non-quantum example.

Asymptotics and perturbation methods - Lecture 1: Asymptotic expansions

In this classic lecture from his course on asymptotics, Steven Strogatz introduces the concept of an asymptotic expansion through an integral that cannot be solved in closed form. This example perfectly encapsulates the paradox of useful divergent series.

Please watch the segment from the opening example. The goal is to approximate the integral f(x) = \int_0^\infty \frac{e^{-xt}}{1+t} dt for large x. The lecturer proceeds by: Formally expanding \frac{1}{1+t} as a geometric series. Integrating term-by-term to generate a power series for f(x) in powers of 1/x. Applying the ratio test to this series and discovering that it diverges for all x. Plugging in a numerical value (x=10) and showing that the first few terms of the divergent series give a surprisingly accurate approximation to the true value of the integral.

The video presents a striking paradox. A mathematically divergent series, which you were taught in calculus to discard, provides an excellent numerical approximation. This is not an anomaly; it is a central feature of perturbation theory in physics. The resolution lies in the concept of an asymptotic series.

The Nature of Asymptotic Series

A convergent series gets more accurate as you add more terms. The error goes to zero as the number of terms $N \to \infty$ . An asymptotic series behaves differently. For a fixed, small value of the expansion parameter $ϵ$ , the terms in the series typically decrease at first, reach a minimum, and then start to grow and diverge.

Behavior of partial sums $s_n$ for an asymptotic series. For a fixed value of the expansion parameter, the partial sums approach the true value up to an optimal number of terms, $n \approx 5$ in this case. Adding more terms makes the approximation worse.

The utility of an asymptotic series stems from the fact that the error made by truncating the series at $N$ terms is on the order of the first neglected term.

Asymptotics and perturbation methods - Lecture 1: Asymptotic expansions

Let's return to the Strogatz lecture to see a more careful analysis of the error.

Please watch the segment from on estimating the remainder. Here, the lecturer avoids the illegal infinite expansion by using an exact finite geometric series with a remainder term. By integrating this exact expression, he proves that the remainder R_n(x) of the perturbation series is smaller in magnitude than the first neglected term, |a_{n+1}|.

This is the key property: $∣ f (x) - S_{N} (x) ∣ < ∣ a_{N + 1} (x) ∣$ , where $S_{N}$ is the partial sum up to term $N$ . This means we can get a good approximation by summing the terms until they start getting bigger, and then stopping. This procedure is known as optimal truncation. The error is minimized around the smallest term in the series.

Formal Definitions

With this physical intuition, we can now introduce the formal language used to describe these approximations.

Asymptotics and perturbation methods - Lecture 1: Asymptotic expansions

The final part of the Strogatz lecture provides the rigorous definitions that underpin this field.

Watch the concluding segment from on formal definitions. This will introduce you to: The "is asymptotic to" symbol: f(x) \sim g(x). Big-O notation: f(x) = O(g(x)). Little-o notation: f(x) = o(g(x)). The definition of an asymptotic sequence. Poincaré's formal definition of an asymptotic expansion.

Let's summarize the main ideas:

Asymptotic Equivalence: $f (x) \sim g (x)$ as $x \to x_{0}$ means $lim_{x \to x_{0}} f (x) / g (x) = 1$ . This captures the leading-order behavior.
Big-O Notation: $f (x) = O (g (x))$ means that $∣ f (x) / g (x) ∣$ is bounded as $x \to x_{0}$ . It gives an upper bound on the order of magnitude.
Little-o Notation: $f (x) = o (g (x))$ means $lim_{x \to x_{0}} f (x) / g (x) = 0$ . It signifies that $f$ is of a smaller order than $g$ .
Asymptotic Expansion: A series $\sum_{n = 0}^{\infty} a_{n} ϕ_{n} (x)$ is an asymptotic expansion of $f (x)$ if for every fixed $N$ , the remainder is of smaller order than the last included term: $f (x) - n = 0 \sum N a_{n} ϕ_{n} (x) = o (ϕ_{N} (x)) as x \to x_{0}$

This definition precisely captures the behavior we observed: the error is controlled relative to the terms in the series, even if the sum itself diverges.

3. The Anharmonic Oscillator: A Canonical Divergent Series

The quantum anharmonic oscillator, with Hamiltonian $H = H_{S H O} + λ x^^{4}$ , is a cornerstone problem in theoretical physics. It serves as a toy model for interacting quantum field theories. Applying the perturbation theory machinery from Section 1 yields a power series in $λ$ for the energy levels.

L2.2 Anharmonic Oscillator via a quartic perturbation

In this clip from an MIT OCW course, the lecturer discusses the nature of the series obtained for the anharmonic oscillator's ground state energy.

Watch the segment from on the nature of the series. Pay close attention to the reveal that the series is, in fact, divergent, and the mention of the pioneering work by Bender and Wu.

As the video and numerous studies have shown, the coefficients $E_{n}^{(k)}$ in the perturbation series for the anharmonic oscillator grow factorially, like $k!$ . This causes the series to have a radius of convergence of zero. It diverges for any non-zero $λ$ .

Successive approximations to a function E(x) using its asymptotic series. The red line E(x) is the exact function. The blue, green, and black curves represent the partial sums of the series truncated at different orders. Initially, adding more terms improves the approximation (green is better than blue for small x), but eventually, higher-order terms cause the approximation to diverge wildly (black curve).

Yet, this is the very series used to make some of the most precise predictions in quantum electrodynamics (QED), which you know is a theory of anharmonic oscillators (the modes of the electromagnetic field). The reason is that the QED perturbation series is also an asymptotic series. The spectacular agreement between QED predictions and experiment is a testament to the physical utility of optimally truncated, divergent asymptotic series.

4. When Do Perturbation Series Converge?

Given the prevalence of asymptotic series, it is natural to ask under what conditions a perturbation series actually converges. The key lies not in the absolute size of the perturbation, but its size relative to the internal energy scales of the unperturbed system.

Chapter 1 - Perturbation theory

Let's return to the MIT OCW notes, which analyze a simple, exactly solvable two-level system.

Read section 1.1.1, "Validity of the perturbation expansion". The analysis of the 2x2 matrix model shows how the Taylor expansion of the exact solution only converges under a specific condition.

The analysis of the two-level system is illuminating. For the Hamiltonian
$H (λ) = (E_{1}^{(0)} λ V^{*} λV E_{2}^{(0)})$
the perturbation series for the eigenvalues converges only if
$∣ λV ∣ < \frac{1}{2} ∣ E_{1}^{(0)} - E_{2}^{(0)} ∣$
This is a profound result. Convergence requires that the perturbation, as measured by its ability to couple states, must be smaller than the energy separation between those states. It is this condition, not just $λ ≪ 1$ , that defines a "small" perturbation.

Conclusion

In this lesson, we have built and critically examined the tool of regular perturbation theory. You now understand not only how to generate a perturbative expansion but also how to assess its meaning—a far more subtle and important skill.

Key Takeaways:

Performing Expansions: Regular perturbation theory provides a systematic algorithm for finding corrections to eigenvalues and eigenstates by expanding in a small parameter $λ$ and solving a hierarchy of linear equations.
Assessing Convergence: The crucial step is to determine the nature of the resulting series. Many important series in physics (e.g., for the anharmonic oscillator, QED) are divergent asymptotic series.
Utility of Divergence: An asymptotic series is useful because the error from truncating it is of the order of the first neglected term. This allows for highly accurate approximations via optimal truncation—summing terms until they begin to grow, then stopping.
Condition for Convergence: A perturbation series for a quantum system generally converges only when the matrix elements of the perturbation are small compared to the energy differences between the states they couple.

This final point sets the stage for our next lesson. What happens when this condition is violated? What if the perturbation couples states that are degenerate or nearly degenerate? This leads to the breakdown of regular perturbation theory and the appearance of new phenomena, such as secular terms and level crossings, which require more sophisticated methods to handle.

Recommended resources

1. The Machinery of Perturbation Theory: A Quantum Mechanics Framework

2. Assessing the Expansion: Convergence vs. Asymptoticity

The Nature of Asymptotic Series

Formal Definitions

3. The Anharmonic Oscillator: A Canonical Divergent Series

4. When Do Perturbation Series Converge?

Conclusion