Time-Independent Perturbation Theory

Time-independent perturbation theory is a cornerstone of quantum mechanics, enabling you to approximate the energy levels and wavefunctions of systems where exact solutions are intractable. From understanding atomic spectra to designing quantum devices, this method provides essential insights by treating small deviations from solvable models. Mastering it allows you to tackle real-world problems where perfect symmetry is broken, such as atoms in electric or magnetic fields.

The Perturbative Approach: Breaking Down the Problem

In quantum mechanics, many systems have Hamiltonians that cannot be solved exactly. Time-independent perturbation theory addresses this by assuming the Hamiltonian can be written as $H=H_0+\lambda V$, where $H_0$ is an unperturbed Hamiltonian with known solutions, $V$ is a perturbation, and $\lambda$ is a small parameter. The eigenstates and eigenvalues of $H_0$ form the basis for approximating those of $H$. This is akin to slightly adjusting a familiar recipe—you know the original dish well, and small changes allow you to predict the new outcome without starting from scratch.

The unperturbed system has eigenvalues $E_n^{(0)}$ and eigenstates $|n^{(0)}⟩$ satisfying $H_0|n^{(0)}⟩=E_n^{(0)}|n^{(0)}⟩$. Your goal is to find corrections to $E_n$ and $|n⟩$ due to $V$. The theory proceeds by expanding these in powers of $\lambda$: $E_n=E_n^{(0)}+\lambda E_n^{(1)}+{\lambda }^2{E}_n^{(2)}+\cdots$ and $|n⟩=|n^{(0)}⟩+\lambda |n^{(1)}⟩+{\lambda }^2|n^{(2)}⟩+\cdots$. Substituting into the Schrödinger equation $H|n⟩=E_n|n⟩$ and matching orders of $\lambda$ yields systematic equations for the corrections. This expansion hinges on the assumption that $\lambda$ is small, ensuring convergence and physical relevance.

Non-Degenerate Perturbation Theory: First and Second Order

When the unperturbed energy levels are non-degenerate (each $E_n^{(0)}$ is unique), the corrections simplify. The first-order energy correction is given by the expectation value of the perturbation: $E_n^{(1)}=⟨n^{(0)}|V|n^{(0)}⟩$. This means you only need the unperturbed state to compute the leading energy shift. For example, if $V$ represents a small electric field, $E_n^{(1)}$ is the average potential energy in the unperturbed state.

The first-order correction to the wavefunction involves mixing with other unperturbed states: $$|n^{(1)}⟩=\sum _{m\ne n}\frac{⟨m^{(0)}|V|n^{(0)}⟩}{E_n^{(0)}-E_m^{(0)}}|m^{(0)}⟩.$$ This sum excludes $m=n$ to avoid divergence, and it shows how the perturbation introduces components from other states, weighted by matrix elements and energy differences. Wavefunction modifications thus reflect virtual transitions induced by $V$.

Second-order corrections become important when $E_n^{(1)}$ vanishes due to symmetry, such as in systems with parity invariance. The second-order energy correction is $$E_n^{(2)}=\sum _{m\ne n}\frac{|⟨m^{(0)}|V|n^{(0)}⟩{|}^2}{E_n^{(0)}-E_m^{(0)}}.$$ This term always lowers the ground state energy (since $E_n^{(0)}-E_m^{(0)}<0$ for $m$ higher than $n$), reflecting increased stability from virtual transitions. To estimate these corrections, you often truncate the sum to dominant terms based on small matrix elements or large energy denominators.

Degenerate Perturbation Theory

When unperturbed energy levels are degenerate, non-degenerate theory fails because energy denominators vanish. For a degenerate subspace with states $|n_i^{(0)}⟩$ having the same $E_n^{(0)}$, the first-order corrections are found by diagonalizing the matrix of $V$ within this subspace. The eigenvalues of this matrix give the first-order energy shifts $E_n^{(1)}$, and the eigenvectors are the correct zeroth-order states. Higher-order corrections follow similar but more complex procedures.

Applications to Physical Systems

• Stark Effect: In atoms, an external electric field $E$ induces a perturbation $V=-eEz$. For hydrogen, first-order correction vanishes for ground state due to parity, so second-order correction gives quadratic Stark effect, leading to energy shifts proportional to $E^2$.
• Spin-Orbit Coupling: In atoms, relativistic effects couple electron spin and orbital angular momentum, with perturbation $V\propto \mathbf{L}\cdot \mathbf{S}$. This splits degenerate levels, such as in fine structure of hydrogen, with energy corrections depending on total angular momentum $j$.
• Zeeman Effect: A magnetic field $B$ perturbs the Hamiltonian with $V=-\mathbf{\mu }\cdot \mathbf{B}$, where $\mathbf{\mu }$ is the magnetic moment. For weak fields, linear Zeeman effect shifts energies proportionally to $m_j{g}_j{\mu }_{B}B$, with $g_j$ the Landé g-factor.

Common Pitfalls

• Applying non-degenerate theory to degenerate systems without modification.
• Assuming perturbation series always converges; for large $\lambda$, higher-order terms may diverge.
• Neglecting off-diagonal matrix elements that can significantly affect corrections.
• Forgetting to use correct zeroth-order states after diagonalization in degenerate case.

Summary

• Time-independent perturbation theory approximates energies and wavefunctions for systems with small perturbations $H=H_0+\lambda V$.
• Non-degenerate theory gives first-order energy correction $E_n^{(1)}=⟨n^{(0)}|V|n^{(0)}⟩$ and second-order correction involving sums over other states.
• Degenerate theory requires diagonalizing the perturbation within degenerate subspaces to find first-order shifts.
• Applications include Stark effect (electric field), spin-orbit coupling (relativistic), and Zeeman effect (magnetic field), each with characteristic energy corrections.
• Care must be taken to ensure non-degeneracy or handle degeneracy appropriately, and to truncate series wisely.