3.1: Time-Independent Degenerate Perturbation Theory
- Page ID
- 28753
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We have seen how we can find approximate solutions for a system whose Hamiltonian is of the form
\[\hat{H} = \hat{H}_0 + \hat{V} \nonumber\]
When we assumed that \(\hat{H}\) and \(\hat{H}_0\) possess discrete, non-degenerate eigenvalues only. This led to a mixing of states where
\[|\phi_0 \rangle = |n_0 \rangle + \sum_{k \neq 0} \frac{V_{k0}}{(E_0 − E_k)} |n_k \rangle \nonumber\]
Clearly, if \(E_0 = E_k\) this diverges. As do the higher order energy shifts (see 2.4). Thus for the degenerate case we cannot associate a particular perturbed state \(|\phi_0 \rangle\) with a particular unperturbed state \(|n_0 \rangle\): we need to take a different approach. In fact, the approximation we make is completely different: we assume that the small perturbation only mixes those states which are degenerate. We then solve the problem exactly for that subset of states.
Assume that \(\hat{H}_0\) possesses \(N\) degenerate eigenstates \(|m \rangle\) with eigenvalue \(E_{deg}\). It may also possesses non-degenerate eigenstates, which can be treated separately by non-degenerate perturbation theory. We write a perturbed eigenstate \(|\phi_j \rangle\) as an linear expansion in the unperturbed degenerate eigenstates only:
\[|\phi_j \rangle = \sum_i |m_i \rangle \langle m_i | \phi_j \rangle = \sum_i c_{ji} | m_i \rangle \nonumber\]
Where \(i\) here runs over degenerate states only. The TISE now becomes:
\[[\hat{H}_0 + \hat{V} ] |\phi_j \rangle = [\hat{H}_0 + \hat{V} ] \sum_i c_{ji} |m_i \rangle = E_j \sum_i c_{ji} |m_i \rangle \nonumber\]
but we know that for all degenerate eigenstates \(\hat{H}_0|m_i \rangle = E_{deg} |m_i \rangle\). So we obtain:
\[\sum_i c_{ji} \hat{V} | m_i \rangle = (E_j - E_{deg}) \sum_i c_{ji} | m_i \rangle \nonumber\]
premultiplying by some unperturbed state \(\langle m_k|\) gives
\[\sum_i c_{ji} \left[ \langle m_k | \hat{V} | m_i \rangle - \delta_{ik} (E_j - E_{deg}) \right] = 0 \nonumber\]
We can get a similar equation from each unperturbed state \(|m_k \rangle\). We thus have an eigenvalue problem: the eigenvector has elements \(c_{ji}\) and the eigenvalues are \(\Delta E_j = E_j − E_{deg}\). Writing the matrix elements between the \(i^{th}\) and \(k^{th}\) unperturbed degenerate states as \(V_{ik} \equiv \langle m_i |\hat{V} |m_k \rangle\) we recover the determinantal equation:
\[\begin{vmatrix} V_{11} − \Delta E_j & V_{12} & ... & V_{1N} \\ V_{21} & V_{22} − \Delta E_j & ... & V_{2N} \\ ... & ... & ... & ... \\ V_{N1} & V_{N2} & ... & V_{NN} − \Delta E_j \end{vmatrix} = 0 \nonumber\]
The \(N\) eigenvalues obtained by solving this equation give the shifts in energy due to the perturbation, and the eigenvectors give the perturbed states \(|\phi \rangle\) in the unperturbed, degenerate basis set \(|m \rangle\).