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.
Figure \(\PageIndex{1}\): Perturbed Energy Spectrum. (CC BY-SA 2.0, Frontier).
We begin with a Hamiltonian \(\hat{H}^0\) having known eigenkets and eigenenergies:
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:
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 \(\lambda\),
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 \(\lambda=1\)):
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’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
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
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 \(\ref{7.4.12}\):
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 \(\langle n^o | \):
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
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 value of \(\hat{H}^1\) in the unperturbed wavefunctions.
That is, the first order energies (Equation \ref{7.4.13}) are given by
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 \(m \neq n\),
Equation \(\ref{7.4.24}\) is essentially is an expansion of the unknown wavefunction correction as a linear combination of known unperturbed wavefunctions \(\ref{7.4.24.2}\):
This is justified since the set of original zero-order wavefunctions forms a complete basis set that can describe any function.
Figure \(\PageIndex{2}\): The first order perturbation of the ground-state wavefunction for a perturbed (left potential) can be expressed as a linear combination of all excited-state wavefunctions of the unperturbed potential (Equation \(\ref{7.4.24.2}\)), shown as a harmonic oscillator in this example (right potential). Note that the ground-state harmonic oscillator wavefuncion is not part of this expression and technically all wavefunctions need to be included in the expression, not just the first eight wavefunctions shown here. (CC BY; Delmar Larsen)
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 \(\PageIndex{1}\).
Figure \(\PageIndex{1}\): Simplified algorithmic flowchart of the Perturbation Theory approximation showing the first two perturbation orders. The process can be continued to third and higher orders. (CC BY; Delmar Larsen)
