5.1 Time-Independent Perturbation Theory

5.1 Time-Independent Perturbation Theory#

Overview#

Perturbation theory is the art of extending exact solutions to nearly-solvable problems. Given \(\hat{H}_0\) with known spectrum and a small perturbation \(V\), we find the spectrum of \(\hat{H}(\lambda) = \hat{H}_0 + \lambda V\) as a power series in \(\lambda\). The organizing principle is the Hellmann-Feynman theorem: the exact derivative of an eigenvalue with respect to a parameter equals the expectation value of the Hamiltonian’s derivative in that eigenstate.

Topics#

Lesson

Title

Core Question

5.1.1

Toy Model

How does the exactly solvable 2-level model reveal the structure and breakdown of perturbation theory?

5.1.2

Non-Degenerate Perturbation Theory

How does the Hellmann-Feynman theorem organize first- and second-order energy corrections?

5.1.3

Degenerate Perturbation Theory

How do we handle degenerate energy levels, and what determines the correct zeroth-order basis?

Key Concepts#

  • Hellmann-Feynman theorem: \(\frac{\mathrm{d}E_n}{\mathrm{d}\lambda} = \langle n(\lambda)\vert \partial H/\partial\lambda \vert n(\lambda)\rangle\) — the energy derivative equals the expectation value of the Hamiltonian’s derivative.

  • Level repulsion: Perturbation pushes nearby levels apart; energy levels of the same symmetry never cross.

  • State hybridization: Perturbed eigenstates mix unperturbed states; mixing is proportional to the matrix element divided by the energy gap.

  • Convergence: The series in \(\lambda\) converges when \(\lambda\|V\| \ll \min(\Delta E)\) (coupling much smaller than the gap).

  • Effective Hamiltonian: \(\hat{H}_\text{eff}\) diagonalize within the degenerate subspace to repair the breakdown of non-degenerate perturbation theory.

Learning Objectives#

  • Apply the Hellmann-Feynman theorem to derive first- and second-order energy corrections without expanding eigenstates explicitly.

  • Compute energy shifts and wavefunction corrections using perturbation theory formulas and verify against exactly solvable models.

  • Derive the effective Hamiltonian for degenerate perturbation theory and apply it to resolve level splitting.

  • Identify when perturbation theory breaks down (level degeneracy, large coupling) and choose the appropriate method.

Project#

Project: Quantum Chemistry: Perturbative Corrections to the Helium Atom#

The helium atom is a prototypical test of perturbation theory: it has two electrons interacting via Coulomb repulsion, and no closed-form solution. This project explores how perturbation theory captures electron correlation.

Objective: Implement variational and perturbation-theoretic approaches to the helium ground state energy, and compare against exact (full configuration interaction) calculations.

Suggested Approach:

  1. Variational baseline: Use the variational principle with a trial wavefunction $\(\psi(r_1, r_2) = \psi_{1s}(r_1; Z_{\text{eff}}) \psi_{1s}(r_2; Z_{\text{eff}})\)$

    where \(Z_{\text{eff}}\) is screened by electron-electron repulsion. Optimize over \(Z_{\text{eff}}\) to find the best upper bound on the ground state energy.

  2. Perturbation theory: Treat the electron-electron repulsion \(V = e^2/|r_1 - r_2|\) as a perturbation around independent-particle hydrogen-like atoms. Compute first and second-order corrections to the ground state energy.

  3. Exact reference: Use numerical full configuration interaction or benchmark against published CI results (e.g., FCI energy for helium).

  4. Extension: Compute the hyperfine structure of hydrogen using relativistic perturbation theory (relativistic kinetic energy + Darwin term + spin-orbit coupling). Compare perturbative and Dirac equation results.

Expected Deliverable:

  • Numerical code (Python/Mathematica) implementing variational and perturbation methods

  • Plot: ground state energy as a function of perturbation order (variational, 1st order, 2nd order, exact)

  • Convergence analysis: at what order does the perturbative series saturate?

  • Brief report (2–3 pages) discussing accuracy, physical insights (electron correlation, screening), and relevance to quantum chemistry (e.g., density functional theory as an alternative)

Frontier relevance: Perturbative corrections to many-electron atoms underpin quantum chemistry calculations used in drug discovery, materials science, and catalysis. Understanding where perturbation theory succeeds/fails is critical for developing better approximation methods.