Chapter 17 Key Takeaways

The Big Picture

Almost no quantum system can be solved exactly. Perturbation theory provides a systematic way to compute approximate energies and states for problems that are close to exactly solvable ones. By decomposing the Hamiltonian as $\hat{H} = \hat{H}_0 + \lambda\hat{H}'$ and expanding in powers of $\lambda$, we obtain corrections order by order. The first-order energy correction is just an expectation value; the first-order wavefunction correction reveals how the perturbation mixes unperturbed states; and the second-order energy correction captures the effect of this mixing on the energy. Non-degenerate perturbation theory is the foundation for understanding atoms in external fields, anharmonic molecular vibrations, and countless other realistic quantum systems.

Key Equations

The Perturbation Expansion

$$\hat{H} = \hat{H}_0 + \lambda\hat{H}', \qquad E_n = \sum_{k=0}^{\infty}\lambda^k E_n^{(k)}, \qquad |n\rangle = \sum_{k=0}^{\infty}\lambda^k|n^{(k)}\rangle$$

First-Order Energy Correction

$$E_n^{(1)} = \langle n^{(0)}|\hat{H}'|n^{(0)}\rangle$$

• Expectation value of the perturbation in the unperturbed state
• Vanishes when $\hat{H}'$ has opposite parity to $|\langle n^{(0)}|^2$

First-Order Wavefunction Correction

$$|n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle}{E_n^{(0)} - E_m^{(0)}} |m^{(0)}\rangle$$

• Nearby states (small energy gap) mix more strongly
• States not connected by $\hat{H}'$ (vanishing matrix element) do not mix
• Requires non-degeneracy: $E_n^{(0)} \neq E_m^{(0)}$ for $m \neq n$

Second-Order Energy Correction

$$E_n^{(2)} = \sum_{m \neq n} \frac{|\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}}$$

• Always $\le 0$ for the ground state
• Produces level repulsion between coupled states
• Requires knowledge of all unperturbed states (including continuum)

Complete Energy Through Second Order

$$E_n \approx E_n^{(0)} + \langle n^{(0)}|\hat{H}'|n^{(0)}\rangle + \sum_{m \neq n} \frac{|\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}}$$

Key Applications

Anharmonic Oscillator ($\hat{H}' = \hat{x}^4$)

$$E_n^{(1)} = \lambda\left(\frac{\hbar}{2m\omega}\right)^2(6n^2 + 6n + 3)$$

• Raises all energy levels
• Breaks the equal spacing of the QHO
• Correction grows as $n^2$ for large $n$

Stark Effect (Hydrogen Ground State)

• First order: $E_1^{(1)} = 0$ (parity symmetry)
• Second order: $E_1^{(2)} = -\frac{1}{2}\alpha\mathcal{E}^2$ where $\alpha/(4\pi\epsilon_0) = 4.5 a_0^3$
• Energy shift is quadratic in the electric field (induced dipole, not permanent)

Comparison Table: First vs. Second Order

Validity Conditions

Perturbation theory works when:

Perturbation theory fails when:

• Unperturbed levels are degenerate $\to$ use degenerate PT (Chapter 18)
• Perturbation is not small $\to$ use variational method (Chapter 19) or numerical diagonalization
• Non-perturbative effects dominate $\to$ tunneling, instantons, lattice methods

Concepts to Remember

1. Perturbation theory is the primary computational tool of professional quantum mechanics, not a sign of failure.
2. Parity selection rules can force first-order corrections to vanish, making second-order the leading effect.
3. Level repulsion is a universal consequence of perturbative coupling between quantum states.
4. The perturbation series is typically asymptotic, not convergent. Optimal truncation gives exponentially accurate results for small coupling.
5. Non-perturbative effects ($\sim e^{-c/\lambda}$) are invisible to perturbation theory at any order.

Looking Ahead