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In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to begin with a system whose mathematical solution is known, and then add a weak perturbing Hamiltonian representing a small disturbance. If the disturbance is not too large, the physical quantities of the perturbed system, such as its energy levels and eigenstates, can be written as corrections to those of the simpler system. In this way, perturbation theory makes it possible to study complicated unsolved systems by expanding around simpler solvable models.

Introduction

Perturbation theory is an important tool in quantum physics because exact solutions of the Schrödinger equation are known only for a limited number of idealized systems, such as the hydrogen atom, the quantum harmonic oscillator, and the particle in a box. By starting from such exactly solvable models, one can construct approximate solutions for more realistic systems.

A standard example is the addition of an external electric field to the hydrogen atom. This produces small shifts in the spectral lines of hydrogen, known as the Stark effect. Although the resulting series expansions are generally not exact, they are often highly accurate when the expansion parameter is small. In quantum electrodynamics (QED), perturbative calculations of the electron's magnetic moment agree with experiment to extremely high precision.^[1]

Perturbative expansions are usually asymptotic rather than convergent, so after sufficiently high order the approximation may worsen rather than improve. Nevertheless, perturbation theory remains one of the central methods of modern theoretical physics.^[2]

Approximate Hamiltonians

In perturbation theory one writes the Hamiltonian as $H = H_{0} + λ V,$ where $H_{0}$ is the exactly solvable unperturbed Hamiltonian, $V$ is the perturbation, and $λ$ is a dimensionless parameter that measures the strength of the perturbation. The parameter is often introduced formally and set equal to 1 at the end of the calculation.

The goal is to determine how the eigenvalues and eigenstates of $H_{0}$ are modified by the perturbation. If the perturbation is weak, the energies and states can be expanded in powers of $λ$ .

Applying perturbation theory

Perturbation theory is applicable when the problem cannot be solved exactly but can be formulated as a small correction to an exactly solvable one. The method works best when the matrix elements of the perturbation are small compared with the relevant differences of unperturbed energy levels.

If the perturbation is too large, or if the system contains qualitatively new states not connected smoothly to the unperturbed states, perturbation theory may fail. This occurs, for example, in low-energy quantum chromodynamics, where the coupling constant becomes too large for a perturbative expansion to remain valid. It also fails for genuinely non-perturbative phenomena such as certain bound states, solitons, and collective effects like superconductivity, where other methods such as the variational method, WKB approximation, or numerical approaches such as density functional theory may be required.^[3]

Time-independent perturbation theory

Time-independent perturbation theory deals with perturbations that do not depend explicitly on time. It was presented by Erwin Schrödinger in 1926, building on earlier work of Lord Rayleigh, and is therefore often called Rayleigh–Schrödinger perturbation theory.^[4]^[5]^[6]

Suppose the unperturbed Hamiltonian satisfies $H_{0} | n^{(0)} ⟩ = E_{n}^{(0)} | n^{(0)} ⟩ .$

For a weak perturbation, $H = H_{0} + λ V .$

The perturbed energies and states are expanded as $\begin{aligned} E_{n} & = E_{n}^{(0)} + λ E_{n}^{(1)} + λ^{2} E_{n}^{(2)} + \dots, \\ | n ⟩ & = | n^{(0)} ⟩ + λ | n^{(1)} ⟩ + λ^{2} | n^{(2)} ⟩ + \dots . \end{aligned}$

First-order corrections

For a non-degenerate level, the first-order shift in the energy is $E_{n}^{(1)} = ⟨ n^{(0)} | V | n^{(0)} ⟩,$ which is the expectation value of the perturbation in the unperturbed state.

The first-order correction to the eigenstate is $| n^{(1)} ⟩ = \sum_{k \neq n} \frac{⟨ k^{(0)} | V | n^{(0)} ⟩}{E_{n}^{(0)} - E_{k}^{(0)}} | k^{(0)} ⟩ .$

This expression shows that nearby energy levels contribute most strongly to the mixing of states.

Second-order and higher-order corrections

To second order, the energy becomes $E_{n} (λ) = E_{n}^{(0)} + λ ⟨ n^{(0)} | V | n^{(0)} ⟩ + λ^{2} \sum_{k \neq n} \frac{{| ⟨ k^{(0)} | V | n^{(0)} ⟩ |}^{2}}{E_{n}^{(0)} - E_{k}^{(0)}} + O (λ^{3}) .$

The third-order energy correction can also be written explicitly as^[7] $E_{n}^{(3)} = \sum_{k \neq n} \sum_{m \neq n} \frac{⟨ n^{(0)} | V | m^{(0)} ⟩ ⟨ m^{(0)} | V | k^{(0)} ⟩ ⟨ k^{(0)} | V | n^{(0)} ⟩}{(E_{n}^{(0)} - E_{m}^{(0)}) (E_{n}^{(0)} - E_{k}^{(0)})} - ⟨ n^{(0)} | V | n^{(0)} ⟩ \sum_{m \neq n} \frac{| ⟨ n^{(0)} | V | m^{(0)} ⟩ |^{2}}{{(E_{n}^{(0)} - E_{m}^{(0)})}^{2}} .$

Higher-order corrections can be developed systematically, though the expressions rapidly become cumbersome.

Effects of degeneracy

When two or more unperturbed states have the same energy, ordinary non-degenerate perturbation theory fails because denominators such as $E_{n}^{(0)} - E_{k}^{(0)}$ vanish. In this case one must first diagonalize the perturbation within the degenerate subspace. This leads to degenerate perturbation theory, in which the perturbation lifts the degeneracy and defines the correct zeroth-order basis for further expansion.

Near-degenerate states must also be treated with care, since even a small perturbation can produce substantial mixing and level splitting.

Generalization to multi-parameter case

Time-independent perturbation theory can be generalized to Hamiltonians depending on several small parameters $x^{μ}$ . In this formulation the Hamiltonian may be written $H (x^{μ}) = H (0) + x^{μ} F_{μ},$ where the $F_{μ}$ are generalized force operators. The derivatives of the energies and states with respect to these parameters can be computed systematically using the Hellmann–Feynman theorems, $\partial_{μ} E_{n} = ⟨ n | \partial_{μ} H | n ⟩,$ $⟨ m | \partial_{μ} n ⟩ = \frac{⟨ m | \partial_{μ} H | n ⟩}{E_{n} - E_{m}} .$

This differential-geometric viewpoint is especially useful in modern treatments of parameter-dependent quantum systems and effective Hamiltonians.^[8]^[9]^[10]

Time-dependent perturbation theory

Time-dependent perturbation theory, developed by Paul Dirac, studies a time-dependent perturbation $V (t)$ added to a time-independent Hamiltonian $H_{0}$ .^[11]^[12]

The Hamiltonian is $H = H_{0} + V (t) .$

If the unperturbed system has eigenstates $| n ⟩$ , the general state may be written as $| ψ (t) ⟩ = \sum_{n} c_{n} (t) e^{- i E_{n} t / ℏ} | n ⟩ .$

The coefficients $c_{n} (t)$ satisfy coupled differential equations, $\frac{d c_{n}}{d t} = \frac{- i}{ℏ} \sum_{k} ⟨ n | V (t) | k ⟩ c_{k} (t) e^{- i (E_{k} - E_{n}) t / ℏ} .$

This framework is used to calculate transition amplitudes and transition probabilities between states, and leads to important results such as Fermi's golden rule and the Dyson series. It is particularly useful in areas such as laser physics, atomic transitions, particle decay, and line broadening.

Strong perturbation theory

A complementary expansion exists for very large perturbations. In this case one may exchange the roles of the unperturbed Hamiltonian and the perturbation, leading to a dual Dyson series. This strong-coupling expansion is related to the adiabatic approximation and, in suitable limits, to the Wigner–Kirkwood series and semiclassical methods.^[13]^[14]^[15]

Examples

Quartic oscillator

For a quantum harmonic oscillator with quartic perturbation, $H = - \frac{ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + \frac{m ω^{2} x^{2}}{2} + λ x^{4},$ the first-order correction to the ground-state energy is $E_{0}^{(1)} = \frac{3}{4} \frac{ℏ^{2} λ}{m^{2} ω^{2}} .$

Quantum pendulum

For the quantum pendulum with perturbation $V = - \cos ϕ,$ the first-order correction vanishes, $E_{n}^{(1)} = 0,$ while the second-order correction is $E_{n}^{(2)} = \frac{m a^{2}}{ℏ^{2}} \frac{1}{4 n^{2} - 1} .$

Potential energy as a perturbation

For weak spatial potentials acting on free-particle states, perturbation theory yields approximate scattered wave functions in one, two, and three dimensions. These results are widely used in scattering theory.^[16]

Applications

Perturbation theory has many applications in quantum physics, including:

Table of contents (137 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Perturbation theory

Introduction

Approximate Hamiltonians

Applying perturbation theory

Time-independent perturbation theory

First-order corrections

Second-order and higher-order corrections

Effects of degeneracy

Generalization to multi-parameter case

Time-dependent perturbation theory

Strong perturbation theory

Examples

Quartic oscillator

Quantum pendulum

Potential energy as a perturbation

Applications

See also

Table of contents (137 articles)

Index

Full contents

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