Non-Degenerate Perturbation Theory
Systematic corrections to exactly solvable quantum systems
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Time-independent perturbation theory is one of the most powerful and widely-used approximation methods in quantum mechanics. When we encounter a system whose Hamiltonian can be written as a sum of a solvable "unperturbed" part and a small "perturbation," we can systematically calculate corrections to the energies and wave functions order by order.
Video Lecture
Introduction to Time-Independent Perturbation Theory
Comprehensive introduction to the fundamental concepts of perturbation theory, including the setup, first-order corrections, and physical interpretation.
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Development of Perturbation Theory
Schrodinger's Perturbation Method
1926Erwin Schrodinger
Developed the first systematic perturbation theory for quantum mechanics in his series of papers on wave mechanics. Used it to calculate corrections to energy levels in atoms.
Rayleigh-Schrodinger Perturbation Theory
1930Various physicists
Formalized the complete mathematical framework combining classical Rayleigh perturbation methods with quantum mechanics, establishing the systematic power series expansion.
Perturbative QED
1949Feynman, Schwinger, Tomonaga
Extended perturbation theory to quantum field theory, calculating corrections to quantum electrodynamics with extraordinary precision (Nobel Prize 1965).
The Setup
The fundamental idea is to split the Hamiltonian into two parts:
where:
- $\hat{H}_0$ is the "unperturbed" Hamiltonian with known exact solutions
- $\hat{H}'$ is the "perturbation" (typically small)
- $\lambda$ is a dimensionless parameter measuring the strength of the perturbation (often set to 1 at the end)
The strategy is to expand the true energies and wave functions as power series in $\lambda$, then solve order by order. This works remarkably well when $\hat{H}'$ introduces only small changes compared to the level spacings of $\hat{H}_0$. The parameter $\lambda$ serves as a bookkeeping device: we collect terms at each power of $\lambda$ and solve them sequentially.
The Unperturbed Problem
We assume we know the complete solution to the unperturbed problem:
The superscript (0) denotes unperturbed quantities. The states $\{|n^{(0)}\rangle\}$ form a complete orthonormal basis satisfying $\langle m^{(0)}|n^{(0)}\rangle = \delta_{mn}$ and the completeness relation $\sum_n |n^{(0)}\rangle\langle n^{(0)}| = \hat{I}$.
The Perturbation Series Expansion
We expand both the energy eigenvalues and eigenstates as power series in $\lambda$:
The corrections $E_n^{(k)}$ and $|n^{(k)}\rangle$ are called the k-th order energy and state corrections. Substituting these expansions into the eigenvalue equation $\hat{H}|n\rangle = E_n|n\rangle$ and collecting terms at each power of $\lambda$, we obtain a hierarchy of equations.
Deriving the Corrections
Substituting the expansions into $(\hat{H}_0 + \lambda\hat{H}')|n\rangle = E_n|n\rangle$ and collecting by powers of $\lambda$:
The zeroth-order equation is just the unperturbed problem. The first- and second-order equations yield the correction formulas below.
Video Lecture
Derivation of First-Order Perturbation Theory
Step-by-step derivation of the first-order energy correction and wave function correction formulas.
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
First-Order Energy Correction
Taking the inner product of the first-order equation with $\langle n^{(0)}|$, we obtain the remarkably simple result:
The first-order energy correction is simply the expectation value of the perturbation in the unperturbed state. This is the diagonal matrix element of $\hat{H}'$. It requires no sum over states and is straightforward to compute once you know the unperturbed wave functions.
First-Order State Correction
Taking the inner product of the first-order equation with $\langle m^{(0)}|$ for $m \neq n$, and expanding $|n^{(1)}\rangle$ in the complete basis, we find:
This formula tells us that the perturbation "mixes in" other unperturbed states. The mixing amplitude is proportional to the off-diagonal matrix element $\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle$ and inversely proportional to the energy difference $(E_n^{(0)} - E_m^{(0)})$.
Key insight: States with energies close to $E_n^{(0)}$ contribute more strongly to the correction. If any $E_m^{(0)} = E_n^{(0)}$ (degeneracy), the denominator vanishes and non-degenerate perturbation theory fails entirely. This is the signal that degenerate perturbation theory is required.
Second-Order Energy Correction
From the second-order equation, taking the inner product with $\langle n^{(0)}|$ and substituting the first-order state correction:
This second-order correction involves a sum over all intermediate states $|m^{(0)}\rangle$. Several important properties follow immediately:
- For the ground state ($n = 0$), every denominator $E_0^{(0)} - E_m^{(0)}$ is negative, so $E_0^{(2)} \leq 0$ always. Perturbations always lower the ground state energy at second order.
- The correction scales as $|\hat{H}'|^2$, making it generically smaller than the first-order term.
- States far in energy contribute less, providing a natural truncation for practical calculations.
Self-Check Question
Question: In second-order perturbation theory for the ground state $|0\rangle$, why is the energy correction $E_0^{(2)}$ always negative?
Show Answer
Answer: For the ground state, $E_0^{(0)} < E_m^{(0)}$ for all $m \neq 0$. Therefore every denominator $(E_0^{(0)} - E_m^{(0)})$ is negative, while every numerator $|\langle m^{(0)}|\hat{H}'|0^{(0)}\rangle|^2$ is non-negative. Each term in the sum is therefore non-positive, making the total negative.
Physical interpretation: Any perturbation allows the ground state to "mix in" excited state character, effectively lowering its energy through quantum fluctuations. This is consistent with the variational principle: the true ground state energy is always lower than or equal to the expectation value in any trial state.
Validity Conditions
Perturbation theory is valid when the corrections are small compared to the unperturbed quantities:
Equivalently, the perturbation matrix elements must be small compared to the level spacings. More precisely, the series should show apparent convergence:
- First-order correction much smaller than unperturbed energy: $|E_n^{(1)}| \ll |E_n^{(0)}|$
- Second-order smaller than first: $|E_n^{(2)}| \ll |E_n^{(1)}|$
- No near-degeneracies in the spectrum (otherwise use degenerate perturbation theory)
When Perturbation Theory Fails
- Degeneracy: If $E_n^{(0)} = E_m^{(0)}$, the denominators blow up. Must use degenerate perturbation theory.
- Near-degeneracy: Even if levels aren't exactly degenerate, small energy differences lead to large corrections that violate convergence.
- Strong perturbations: If $\hat{H}'$ is comparable to or larger than $\hat{H}_0$, the series doesn't converge.
- Asymptotic series: Many perturbation series are asymptotic (diverge at high order) even when low orders are accurate. The famous example is the anharmonic oscillator with $\lambda x^4$ perturbation.
Looking Ahead
On the next page, we will apply these formulas to concrete physical examples: the anharmonic oscillator, the hydrogen atom in an electric field (Stark effect), and the weak-field Zeeman effect. We will also discuss convergence and the conditions under which the perturbation series breaks down.