5.1.1 Toy Model#

Prompts

Why is perturbation theory useful in quantum mechanics when exact solutions are rare, and what does it mean to treat a system as a small deformation of a solvable one?
For \(\hat{H}(\lambda)=\hat{H}_0+\lambda\hat{V}\), what information from \(\hat{H}_0\) and \(\hat{V}\) is needed to predict behavior at small \(\lambda\)?
In a two-level model, what does off-diagonal coupling do physically to level crossing/avoided crossing and to state mixing?
A perturbative expansion is a Taylor series in \(\lambda\). How should you judge, in practice, whether a low-order truncation is reliable for a chosen real \(\lambda\)?

Lecture Notes#

Overview#

Exact solutions in quantum mechanics are rare, but many physically important systems are only small deformations of solvable ones. Perturbation theory is the framework that turns this observation into a controlled method: start from a known reference problem, then track how energies and states deform as a small control parameter is turned on.

This lecture sets the conceptual foundation for the chapter. The key idea is that perturbation theory is, at heart, a Taylor expansion problem: treat the eigenenergy \(E_n(\lambda)\) and eigenstate \(\vert\psi_n(\lambda)\rangle\) as functions of \(\lambda\), then expand them around \(\lambda=0\).

So the central question is not “can we diagonalize \(\hat{H}(\lambda)\) exactly?” but “how much of the eigensystem can we recover from local data near \(\lambda=0\)?”

General Setup#

Start from the standard perturbative decomposition:

Problem Setup

The Hamiltonian is parameterized as

\[ \hat{H}(\lambda)=\hat{H}_0+\lambda\hat{V}. \]

\(\hat{H}_0\): unperturbed Hamiltonian with known eigensystem at \(\lambda=0\).
\(\lambda\hat{V}\): perturbation term, where \(\hat{V}\) sets the deformation direction and \(\lambda\) sets its strength.

At \(\lambda=0\), the spectrum of \(\hat{H}_0\) is known. The goal is to find the eigenenergy \(E_n(\lambda)\) and corresponding eigenstate \(\vert \psi_n(\lambda)\rangle\) of \(\hat{H}(\lambda)\), defined by the eigenvalue equation \(\hat{H}(\lambda)\vert \psi_n(\lambda)\rangle=E_n(\lambda)\vert \psi_n(\lambda)\rangle\).

Perturbation theory expands these quantities as

\[ E_n(\lambda)=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^2E_n^{(2)}+\cdots, \]

\[ \vert \psi_n(\lambda)\rangle=\vert \psi_n^{(0)}\rangle+\lambda\vert \psi_n^{(1)}\rangle+\lambda^2\vert \psi_n^{(2)}\rangle+\cdots. \]

Each coefficient has physical meaning: zeroth order is the unperturbed problem, higher orders are corrections induced by \(\hat{V}\). We now test this framework on an exactly solvable two-level benchmark.

Toy Model#

Consider \(\hat{H}_0=\hat{Z}\) and \(\hat{V}=\hat{X}\). Then

(177)#\[\begin{split} \hat{H}(\lambda)=\hat{Z}+\lambda\hat{X}=\begin{pmatrix}1&\lambda\\\lambda&-1\end{pmatrix} \end{split}\]

At \(\lambda=0\), the unperturbed eigenstates are \(\vert 0\rangle,\vert 1\rangle\) with energies \(+1,-1\).

Exact Results#

The exact energies are

(178)#\[ E_\pm(\lambda)=\pm\sqrt{1+\lambda^2}. \]

The exact eigenstates are

\[ \vert \psi_+(\lambda)\rangle=\frac{\left(1+\sqrt{1+\lambda^2}\right)\vert 0\rangle+\lambda\vert 1\rangle}{\sqrt{2\left(1+\lambda^2+\sqrt{1+\lambda^2}\right)}}, \]

\[ \vert \psi_-(\lambda)\rangle=\frac{\left(1+\sqrt{1+\lambda^2}\right)\vert 1\rangle-\lambda\vert 0\rangle}{\sqrt{2\left(1+\lambda^2+\sqrt{1+\lambda^2}\right)}}. \]

Taylor Expansions#

Before applying to this model, recall the Taylor expansion of a function around \(x=0\):

Taylor Series Expansion

\[ f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+\cdots \]

This expresses a function near \(x=0\) as zeroth-order value plus first-, second-, and higher-order corrections.

For perturbation theory, the same idea is applied to \(E_\pm(\lambda)\) and \(\vert\psi_\pm(\lambda)\rangle\) as functions of \(\lambda\).

Energy and State Expansions

(179)#\[ E_\pm(\lambda)=\pm\left(1+\frac{\lambda^2}{2}-\frac{\lambda^4}{8}+O(\lambda^6)\right) \]

(180)#\[\begin{split} \begin{split} \vert \psi_+(\lambda)\rangle &= \vert 0\rangle+\frac{\lambda}{2}\vert 1\rangle-\frac{\lambda^2}{8}\vert 0\rangle+O(\lambda^3),\\ \vert \psi_-(\lambda)\rangle &= \vert 1\rangle-\frac{\lambda}{2}\vert 0\rangle-\frac{\lambda^2}{8}\vert 1\rangle+O(\lambda^3). \end{split} \end{split}\]

This toy model is a benchmark: it shows what perturbation theory should reproduce order by order. The next subsection develops the general method to compute these coefficients directly from \(\hat{H}_0\) and \(\hat{V}\), without exact diagonalization.

Radius of Convergence

A perturbative expansion is a Taylor series in \(\lambda\). Such series need not converge for all \(\lambda\); in general there is a finite radius of convergence \(R\) around \(\lambda=0\).

Within \(|\lambda|<R\), higher-order truncations approach the exact result asymptotically order by order. Outside that range, adding more terms does not reliably improve the approximation.

For practical use at real \(\lambda\), always check behavior numerically by comparing successive truncations (first, second, fourth order, …) and monitoring relative error as \(\lambda\) increases.

Poll: Practical convergence check

For \(E_+(\lambda)\approx 1+\lambda^2/2\), which workflow best checks whether second-order perturbation theory is trustworthy at a chosen real \(\lambda\)?

(A) Assume it is always valid whenever the unperturbed gap is nonzero.

(B) Compare the truncated result with exact (or higher-order) values and monitor relative error as \(\lambda\) increases.

(D) Check whether first-order correction vanishes.

Summary#

Perturbation theory starts from a solvable Hamiltonian and computes corrections order by order in \(\lambda\).
The general targets are \(E_n(\lambda)\) and \(\vert \psi_n(\lambda)\rangle\), expanded around \(\lambda=0\).
The qubit toy model provides exact energies and states, so perturbative coefficients can be benchmarked directly.
Practical validity is controlled by small perturbation scale and explicit error checks for real \(\lambda\).

Homework#

1. Accuracy window. Use the approximation \(E_+^{\text{(2nd)}}(\lambda)=1+\lambda^2/2\) as a model prediction for the upper level. Find the largest real \(\lambda>0\) such that the relative error

\[ \frac{\left|E_+^{\text{exact}}-E_+^{\text{(2nd)}}\right|}{\left|E_+^{\text{exact}}\right|} \]

is below \(5\%\). Interpret the result as a practical criterion for when second-order perturbation theory is trustworthy.

2. Inverse design. Suppose spectroscopy reports an upper-level energy \(E_+=1.18\) for the toy model. Estimate \(\lambda\) using second-order perturbation theory, then compute the exact \(\lambda\) from \(E_+=\sqrt{1+\lambda^2}\). Compare the two inferred couplings and discuss why the inverse problem can amplify approximation error.

3. Basis rotation: energies vs states. Consider

\[ \hat{H}(\lambda,\phi)=\hat{Z}+\lambda\big(\cos\phi\,\hat{X}+\sin\phi\,\hat{Y}\big). \]

(a) Find a unitary \(\hat U(\phi)\) that rotates about \(\hat Z\) and satisfies \(\hat U(\phi)^\dagger\,\hat H(\lambda,\phi)\,\hat U(\phi)=\hat H(\lambda,0)\).

(b) Use this unitary equivalence to argue that the eigenvalues \(E_\pm(\lambda,\phi)\) are independent of \(\phi\) — the spectrum depends on \(\lambda\) alone.

(c) Show that the eigenstates do depend on \(\phi\): \(\vert\psi_\pm(\lambda,\phi)\rangle=\hat U(\phi)\,\vert\psi_\pm(\lambda,0)\rangle\).

(d) Implication for a perturbative series in \(\phi\). Expand \(E_+(\lambda,\phi)\) as a Taylor series in \(\phi\) about \(\phi=0\). What must every coefficient be? Now expand \(\vert\psi_+(\lambda,\phi)\rangle\) in \(\phi\) — is the first-order \(\phi\)-correction zero? If not, compute it explicitly.

(e) State the general principle illustrated here: when a parameter enters \(\hat H\) only through a unitary similarity transformation, energies are invariant but eigenstates rotate.

4. Controlled asymmetry. Add a diagonal perturbation and study

\[ \hat{H}(\lambda,\mu)=\hat{Z}+\lambda\hat{X}+\mu\hat{Z}. \]

Treat \(\lambda\) and \(\mu\) as small independent parameters and expand \(E_+(\lambda,\mu)\) to second order in both. Which terms are linear in \(\mu\)? Which terms are linear in \(\lambda\)? Use symmetry arguments to justify the pattern.

5. Avoided crossing width. For

\[\begin{split} \hat{H}=\begin{pmatrix} \delta & v \\ v & -\delta \end{pmatrix}, \end{split}\]

show that the minimum gap over all \(\delta\) is \(2|v|\). Then solve the inverse question: if an experiment measures a minimum gap of \(0.12\,\mathrm{eV}\), what is \(|v|\)? Explain why this is a direct operational meaning of level repulsion.

6. Misconception test. One might claim: “If the unperturbed gap is nonzero, non-degenerate perturbation theory should remain valid even when \(\lambda\) is large.” Test this claim quantitatively by comparing exact and second-order energies at \(\lambda=0.2,0.8,1.5\). Identify where the claim fails and explain the failure using error growth as \(\lambda\) increases.

7. Near-degeneracy diagnosis. Consider

\[\begin{split} \hat{H}(\lambda,\epsilon)=\begin{pmatrix} \epsilon/2 & \lambda \\ \lambda & -\epsilon/2 \end{pmatrix}, \qquad \epsilon>0. \end{split}\]

(a) Compute the exact eigenvalues and expand for small \(\lambda\) at fixed \(\epsilon\).

(b) Determine the condition on \(\lambda, \epsilon\) under which the second-order term is a controlled correction.

(c) Use your condition to explain why the limits \(\epsilon\to 0\) and \(\lambda\to 0\) do not commute in practice for perturbation theory.