Time-Independent Perturbation Theory
Time-independent (and non-degenerate) perturbation theory
Problem setup
We consider a system whose Hamiltonian is given by
\[ \hat{H} = \hat{H}_0 + \hat{V}, \]
where \(\hat{V}\) is a weak perturbation of the system.
The eigenstates \(\left\lvert n^{(0)} \right\rangle\) and eigenenergies \(E_n^{(0)}\) of the unperturbed system are assumed to be known, i.e.
\[ \hat{H}_0 \left\lvert n^{(0)} \right\rangle= E_n^{(0)} \left\lvert n^{(0)} \right\rangle. \]
Question: How does the perturbation \(\hat{V}\) modify the eigenenergies and eigenstates?
Ansatz
We introduce a real-valued “tuning” parameter \(\lambda\) (between 0 and 1) to interpolate between the unperturbed (\(\lambda=0\)) and the fully perturbed (\(\lambda=1\)) system:
\[ \hat{H} = \hat{H}_0 + \lambda \hat{V}. \]
We then expand the eigenenergies and eigenstates in powers of \(\lambda\):
\[ E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \lambda^3 E_n^{(3)} + \dots \]
\[ \left\lvert n \right\rangle= N(\lambda)\left( \left\lvert n^{(0)} \right\rangle+ \sum_{k \neq n} c_{nk}(\lambda) \left\lvert k^{(0)} \right\rangle\right), \]
with expansion coefficients
\[ c_{nk}(\lambda) = \lambda c_{nk}^{(1)} + \lambda^2 c_{nk}^{(2)} + \lambda^3 c_{nk}^{(3)} + \dots \]
Solution strategy
We plug this expansion into the Schrödinger equation:
\[ (\hat{H}_0 + \lambda \hat{V}) \left( \left\lvert n^{(0)} \right\rangle+ \sum_{\substack{k\neq n \\ i \geq 1}} \lambda^i c_{nk}^{(i)} \left\lvert k^{(0)} \right\rangle\right) = \left( \sum_{i\geq 0} \lambda^i E_n^{(i)} \right) \left( \left\lvert n^{(0)} \right\rangle+ \sum_{\substack{k\neq n \\ i \geq 1}} \lambda^i c_{nk}^{(i)} \left\lvert k^{(0)} \right\rangle\right). \]
By comparing powers of \(\lambda\) on both sides, we obtain recursive equations for the corrections \(E_n^{(i)}\) and \(c_{nk}^{(i)}\).
At the end, we set \(\lambda=1\).
Remark. This method works because the perturbation is assumed small. If \(\hat{V}\) is comparable to \(\hat{H}_0\) in magnitude, the perturbation expansion does not converge.
Zeroth order (\(\lambda^0\))
At order \(\lambda^0\) we recover the unperturbed Schrödinger equation:
\[ \hat{H}_0 \left\lvert n^{(0)} \right\rangle= E_n^{(0)} \left\lvert n^{(0)} \right\rangle, \]
which is already solved.
First order (\(\lambda^1\))
At order \(\lambda^1\) we find
\[ \sum_{k' \neq n} c_{nk'}^{(1)} \hat{H}_0 \left\lvert k'^{(0)} \right\rangle+ \hat{V}\left\lvert n^{(0)} \right\rangle= \sum_{k' \neq n} c_{nk'}^{(1)} E_n^{(0)} \left\lvert k'^{(0)} \right\rangle+ E_n^{(1)} \left\lvert n^{(0)} \right\rangle. \]
Taking the inner product with \(\left\langle k^{(0)} \right\rvert\) and using orthonormality \(\left\langle k^{(0)} \middle| n^{(0)} \right\rangle= \delta_{nk}\) gives
\[ c_{nk}^{(1)} E_k^{(0)} + \left\langle k^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle= c_{nk}^{(1)} E_n^{(0)} + E_n^{(1)} \delta_{nk}. \]
First-order energy correction
For \(n=k\), this reduces to:
\[ E_n^{(1)} = \left\langle n^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle. \]
So the energy to first order is:
\[ E_n \approx E_n^{(0)} + \left\langle n^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle. \]
First-order correction to the state
For \(k \neq n\), we obtain:
\[ c_{nk}^{(1)} = \frac{\left\langle k^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle}{E_n^{(0)} - E_k^{(0)}}. \]
Thus the perturbed state (to first order) becomes:
\[ \left\lvert n \right\rangle\approx N(1) \left( \left\lvert n^{(0)} \right\rangle+ \sum_{k \neq n} \left\lvert k^{(0)} \right\rangle\frac{\left\langle k^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle}{E_n^{(0)} - E_k^{(0)}} \right). \]
Remark. Notice the denominators \(E_n^{(0)} - E_k^{(0)}\).
If two states are nearly degenerate (energies close together), the correction becomes very large. This is the reason we need degenerate perturbation theory in those cases.
Second order (\(\lambda^2\))
At second order, one finds for the energy correction:
\[ E_n^{(2)} = \sum_{k \neq n} \frac{|\left\langle k^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle|^2}{E_n^{(0)} - E_k^{(0)}}. \]
This expression shows that states closer in energy contribute more strongly.
Additional notes
- The perturbation expansion breaks down for nearly degenerate levels because denominators approach zero. Degenerate perturbation theory must be used instead.
- The normalization factor \(N(1)\) must be chosen such that \[ \left\langle n \middle| n \right\rangle= 1. \]
Summary
First-order correction to the energy:
\[ E_n^{(1)} = \left\langle n^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle. \]First-order correction to the state:
\[ \left\lvert n \right\rangle\approx \left\lvert n^{(0)} \right\rangle+ \sum_{k \neq n} \left\lvert k^{(0)} \right\rangle\frac{\left\langle k^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle}{E_n^{(0)} - E_k^{(0)}}. \]Second-order correction to the energy:
\[ E_n^{(2)} = \sum_{k \neq n} \frac{|\left\langle k^{(0)} \right\rvert\hat{V} \left\lvert n^{(0)} \right\rangle|^2}{E_n^{(0)} - E_k^{(0)}}. \]