# 7.4: Perturbation Theory Expresses the Solutions in Terms of Solved Problems

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It is easier to compute the changes in the energy levels and wavefunctions with a scheme of successive corrections to the zero-field values. This method, termed *perturbation theory*, is the single most important method of solving problems in quantum mechanics and is widely used in atomic physics, condensed matter and particle physics. Perturbation theory is another approach to finding approximate solutions to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbation" parts. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem.

We begin with a Hamiltonian \(\hat{H}^0\) having known eigenkets and eigenenergies:

\[ \hat{H}^o | n^o \rangle = E_n^o | n^o \rangle \label{7.4.1}\]

The task is to find how these eigenstates and eigenenergies change if a small term \(H^1\) (an external field, for example) is added to the Hamiltonian, so:

\[ ( \hat{H}^0 + \hat{H}^1 ) | n \rangle = E_n | n \rangle \label{7.4.2}\]

That is to say, on switching on \(\hat{H}^1\) changes the wavefunctions:

\[ \underbrace{ | n^o \rangle }_{\text{unperturbed}} \Rightarrow \underbrace{|n \rangle }_{\text{Perturbed}}\label{7.4.3}\]

and energies (Figure \(\PageIndex{1}\)):

\[ \underbrace{ E_n^o }_{\text{unperturbed}} \Rightarrow \underbrace{E_n }_{\text{Perturbed}} \label{7.4.4}\]

The basic assumption in perturbation theory is that \(H^1\) is sufficiently small that the leading corrections are the same order of magnitude as \(H^1\) itself, and the true energies can be better and better approximated by a successive series of corrections, each of order \(H^1/H^o\) compared with the previous one.

The strategy is to **expand** the true wavefunction and corresponding eigenenergy as series in \(\hat{H}^1/\hat{H}^o\). These series are then fed into Equation \(\ref{7.4.2}\), and terms of the same order of magnitude in \(\hat{H}^1/\hat{H}^o\) on the two sides are set equal. The equations thus generated are solved one by one to give progressively more accurate results.

To make it easier to identify terms of the same order in \(\hat{H}^1/\hat{H}^o\) on the two sides of the equation, it is convenient to introduce a dimensionless parameter \(\lambda\) which always goes with \(\hat{H}^1\), and then expand both eigenstates and eigenenergies as power series in \(\lambda\),

\[ \begin{align} | n \rangle &= \sum _ i^m \lambda ^i| n^i \rangle \label{7.4.5} \\[4pt] E_n &= \sum_{i=0}^m \lambda ^i E_n^i \label{7.4.6} \end{align}\]

where \(m\) is how many terms in the expansion we are considering. The ket \(|n^i \rangle\) is multiplied by \(\lambda^i\) and is therefore of order \((H^1/H^o)^i\).

\(\lambda\) is purely a bookkeeping device: we will set it equal to 1 when we are through! It’s just there to keep track of the orders of magnitudes of the various terms.

For example, in first order perturbation theory, Equations \(\ref{7.4.5}\) are truncated at \(m=1\) (and setting \(\lambda=1\)):

\[ \begin{align} | n \rangle &\approx | n^o \rangle + | n^1 \rangle \label{7.4.7} \\[4pt] E_n &\approx E_n^o + E_n^1 \label{7.4.8} \end{align}\]

However, let's consider the general case for now. Adding the full expansions for the eigenstate (Equation \(\ref{7.4.5}\)) and energies (Equation \(\ref{7.4.6}\)) into the Schrödinger equation for the perturbation Equation \(\ref{7.4.2}\) in

\[ ( \hat{H}^o + \lambda \hat{H}^1) | n \rangle = E_n| n \rangle \label{7.4.9}\]

we have

\[ (\hat{H}^o + \lambda \hat{H}^1) \left( \sum _ {i=0}^m \lambda ^i| n^i \rangle \right) = \left( \sum_{i=0}^m \lambda^i E_n^i \right) \left( \sum _ {i=0}^m \lambda ^i| n^i \rangle \right) \label{7.4.10}\]

We’re now ready to match the two sides term by term in powers of \(\lambda\). Note that the zeroth-order term, of course, just gives back the unperturbed Schrödinger Equation (Equation \(\ref{7.4.1}\)).

Let's look at Equation \(\ref{7.4.10}\) with the first few terms of the expansion:

\[ \begin{align} (\hat{H}^o + \lambda \hat{H}^1) \left( | n ^o \rangle + \lambda | n^1 \rangle \right) &= \left( E _n^0 + \lambda E_n^1 \right) \left( | n ^o \rangle + \lambda | n^1 \rangle \right) \label{7.4.11} \\[4pt] \hat{H}^o | n ^o \rangle + \lambda \hat{H}^1 | n ^o \rangle + \lambda H^o | n^1 \rangle + \lambda^2 \hat{H}^1| n^1 \rangle &= E _n^0 | n ^o \rangle + \lambda E_n^1 | n ^o \rangle + \lambda E _n^0 | n ^1 \rangle + \lambda^2 E_n^1 | n^1 \rangle \label{7.4.11A} \end{align}\]

Collecting terms in order of \(\lambda\) and coloring to indicate different orders

\[ \underset{\text{zero order}}{\hat{H}^o | n ^o \rangle} + \color{red} \underset{\text{1st order}}{\lambda ( \hat{H}^1 | n ^o \rangle + \hat{H}^o | n^1 \rangle )} + \color{blue} \underset{\text{2nd order}} {\lambda^2 \hat{H}^1| n^1 \rangle} =\color{black}\underset{\text{zero order}}{E _n^0 | n ^o \rangle} + \color{red} \underset{\text{1st order}}{ \lambda (E_n^1 | n ^o \rangle + E _n^0 | n ^1 \rangle )} +\color{blue}\underset{\text{2nd order}}{\lambda^2 E_n^1 | n^1 \rangle} \label{7.4.12}\]

If we expanded Equation \(\ref{7.4.10}\) further we could express the energies and wavefunctions in higher order components.

## Zero-Order Terms (\(\lambda=0\))

Collecting the zero order terms in the expansion (black terms in Equation \(\ref{7.4.10}\)) results in just the Schrödinger Equation for the unperturbed system

\[ \hat{H}^o | n^o \rangle = E_n^o | n^o \rangle \label{Zero}\]

## First-Order Expression of Energy (\(\lambda=1\))

The summations in Equations \(\ref{7.4.5}\), \(\ref{7.4.6}\), and \(\ref{7.4.10}\) can be truncated at any order of \(\lambda\). For example, the first order perturbation theory has the truncation at \(\lambda=1\). Matching the terms that linear in \(\lambda\) (red terms in Equation \(\ref{7.4.12}\)) and setting \(\lambda=1\) on both sides of Equation \(\ref{7.4.12}\):

\[ \hat{H}^o | n^1 \rangle + \hat{H}^1 | n^o \rangle = E_n^o | n^1 \rangle + E_n^1 | n^o \rangle \label{7.4.13}\]

Equation \(\ref{7.4.13}\) is the key to finding the first-order change in energy \(E_n^1\). Taking the **inner product** of both sides with \(\langle n^o | \):

\[ \langle n^o | \hat{H}^o | n^1 \rangle + \langle n^o | \hat{H}^1 | n^o \rangle = \langle n^o | E_n^o| n^1 \rangle + \langle n^o | E_n^1 | n^o \rangle \label{7.4.14}\]

since operating the zero-order Hamiltonian on the bra wavefunction (this is just the Schrödinger equation; Equation \(\ref{Zero}\)) is

\[ \langle n^o | \hat{H}^o = \langle n^o | E_n^o \label{7.4.15}\]

and via the orthonormality of the unperturbed \(| n^o \rangle\) wavefunctions both

\[ \langle n^o | n^o \rangle = 1 \label{7.4.16}\]

and Equation \(\ref{7.4.8}\) can be simplified

\[ \bcancel{E_n^o \langle n^o | n^1 \rangle} + \langle n^o | H^1 | n^o \rangle = \bcancel{ E_n^o \langle n^o | n^1 \rangle} + E_n^1 \cancelto{1}{\langle n^o | n^o} \rangle \label{7.4.14new}\]

since the unperturbed set of eigenstates are orthogonal (Equation \ref{7.4.16}) and we can cancel the other term on each side of the equation, we find that

\[ E_n^1 = \langle n^o | \hat{H}^1 | n^o \rangle \label{7.4.17}\]

The first-order change in the energy of a state resulting from adding a perturbing term \(\hat{H}^1\) to the Hamiltonian is just the

expectation valueof \(\hat{H}^1\) in theunperturbedwavefunctions.

That is, the first order energies (Equation \ref{7.4.13}) are given by

\[ \begin{align} E_n &\approx E_n^o + E_n^1 \\[4pt] &\approx \underbrace{ E_n^o + \langle n^o | H^1 | n^o \rangle}_{\text{First Order Perturbation}} \label{7.4.17.2} \end{align}\]

## First-Order Expression of Wavefunction (\(\lambda=1\))

The general expression for the first-order change in the *wavefunction* is found by taking the inner product of the first-order expansion (Equation \(\ref{7.4.13}\)) with the bra \( \langle m^o |\) with \(m \neq n\),

\[ \langle m^o | H^o | n^1 \rangle + \langle m^o |H^1 | n^o \rangle = \langle m^o | E_n^o | n^1 \rangle + \langle m^o |E_n^1 | n^o \rangle \label{7.4.18}\]

**Last term on right side of Equation \(\ref{7.4.18}\)**

The last integral on the right hand side of Equation \(\ref{7.4.18}\) is zero, since \(m \neq n\) so

\[ \langle m^o |E_n^1 | n^o \rangle = E_n^1 \langle m^o | n^o \rangle \label{7.4.19}\]

and

\[\langle m^o | n^0 \rangle = 0 \label{7.4.20}\]

**First term on right side of Equation \(\ref{7.4.18}\)**

The first integral is more complicated and can be expanded back into the \(H^o\)

\[ E_m^o \langle m^o | n^1 \rangle = \langle m^o|E_m^o | n^1 \rangle = \langle m^o | H^o | n^1 \rangle \label{7.4.21}\]

since

\[ \langle m^o | H^o = \langle m^o | E_m^o \label{7.4.22}\]

so

\[ \langle m^o | n^1 \rangle = \dfrac{\langle m^o | H^1 | n^o \rangle}{ E_n^o - E_m^o} \label{7.4.23}\]

and therefore the wavefunction corrected to first order is:

\[ \begin{align} | n \rangle &\approx | n^o \rangle + | n^1 \rangle \\[4pt] &\approx \underbrace{| n^o \rangle + \sum _{m \neq n} \dfrac{|m^o \rangle \langle m^o | H^1| n^o \rangle }{E_n^o - E_m^o}}_{\text{First Order Perturbation Theory}} \label{7.4.24} \end{align}\]

Equation \(\ref{7.4.24}\) is essentially is an expansion of the unknown wavefunction correction as a **linear combination** of known unperturbed wavefunctions \(\ref{7.4.24.2}\):

\[ \begin{align} | n \rangle &\approx | n^o \rangle + | n^1 \rangle \\[4pt] &\approx | n^o \rangle + \sum _{m \neq n} c_{m,n} |m^o \rangle \label{7.4.24.2} \end{align}\]

with the expansion coefficients determined by

\[ c_{m,n} = \dfrac{\langle m^o | H^1| n^o \rangle }{E_n^o - E_m^o} \label{7.4.24.3}\]

This is justified since the set of original zero-order wavefunctions forms a **complete basis set** that can describe any function.

Calculating the first order perturbation to the wavefunctions is in general an infinite sum of off diagonal matrix elements of \(H^1\) (Figure \(\PageIndex{2}\)).

- However, the denominator argues that terms in this sum will be weighted by states that are of
**comparable energy**. That means in principle, these sum can be truncated easily based off of some criterion. - Another point to consider is that many of these matrix elements will equal zero depending on the symmetry of the \(\{| n^o \rangle \}\) basis and \(H^1\) (e.g., some \(\langle m^o | H^1| n^o \rangle\) integrals in Equation \(\ref{7.4.24}\) could be zero due to the integrand having an odd symmetry; see Example \(\PageIndex{3}\)).

The denominators in Equation \(\ref{7.4.24}\) argues that terms in this sum will be preferentially dictated by states that are of

comparable energy. That is, eigenstates that have energies significantly greater or lower than the unperturbed eigenstate will weakly contribute to the perturbed wavefunction.

## Second-Order Terms (\(\lambda=2\))

There are higher energy terms in the expansion of Equation \(\ref{7.4.5}\) (e.g., the blue terms in Equation \(\ref{7.4.12}\)), but are not discussed further here other than noting the whole perturbation process is an infinite series of corrections that ideally converge to the correct answer. It is truncating this series as a finite number of steps that is the approximation. The general approach to perturbation theory applications is giving in the flowchart in Figure \(\PageIndex{1}\).

## Contributors

Michael Fowler (Beams Professor, Department of Physics, University of Virginia)