2.1: Small changes to the Hamiltonian
- Page ID
- 28739
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)There are very few problems in quantum mechanics which can be solved exactly. However, we are often interested in the effect of a small change to a system, and in such cases we can proceed by assuming that this causes only a small change in the eigenstates. Perturbation theory provides a method for finding approximate energy eigenvalues and eigenfunctions for a system whose Hamiltonian is of the form
\[\hat{H} = \hat{H}_0 + \hat{V} \nonumber\]
where \(\hat{H}_0\) is the ‘main bit’ of the Hamiltonian of an exactly solvable system, for which we know the eigenvalues, \(E_n\), and eigenfunctions, \(|n\rangle\), and \(\hat{V}\) is a small, time-independent perturbation. \(\hat{H}\), \(\hat{H}_0\) and \(\hat{V}\) are Hermitean operators. Using perturbation theory, we can get approximate solutions for \(\hat{H}\) using as basis functions eigenstates of the similar, exactly solvable system \(\hat{H}_0\).
Assuming that \(\hat{H}\) and \(\hat{H}_0\) possess discrete, non-degenerate eigenvalues only, we write
\[\hat{H}_0 |n_i \rangle = E_i |n_i \rangle \nonumber\]
in Dirac notation. The states \(|n_i\rangle\) are orthonormal. WLOG, consider a state \(i = 0\): the effect of the perturbation will be to modify the state and its corresponding energy slightly; The eigenstate \(|n_0 \rangle\) will become \(|\phi_0 \rangle\) and \(E_0\) will shift to \(E_0 + \Delta E_0\), where
\[\hat{H} |\phi_0 \rangle = E_0 + \Delta E_0 | \phi_0 \rangle \nonumber\]
WLOG, expanding \(|\phi_0 \rangle\) in the basis set \(|n_i\rangle\) with coefficients \(c_{i0}\) and premultiplying by \(\langle n_0|\)
\[ \langle n_k|(\hat{H}_0 + \hat{V} ) \sum_{i=0,\infty} c_{i0}|n_i \rangle = (E_0 + \Delta E_0) \langle n_0| \sum_{i=0, \infty} c_{i0} |n_i \rangle \nonumber\]
Which after a little algebra and cancellation yields the exact result:
\[\Delta E_0 = \langle n_0| \hat{V} |n_0 \rangle + \sum_{i=1,\infty} (c_{i0}/c_{00}) \langle n_0|\hat{V}|n_i \rangle \]
Similarly, expanding \(|\phi_0 \rangle\) in the basis set \(|n_i \rangle\) and premultiplying by another state \(\langle n_k|\)
\[ \langle n_k|(\hat{H}_0 + \hat{V} ) \sum_{i=0,\infty} c_{i0}|n_i \rangle = (E_0 + \Delta E_0) \langle n_k| \sum_{i=0, \infty} c_{i0} |n_i \rangle \nonumber\]
leading to \(|\phi_0 \rangle\) having a component of \(|n_k \rangle\)
\[c_{k0} (E_0 + \Delta E_0 − E_k) = \sum_{i=0, \infty} c_{i0} \langle n_k | \hat{V} |n_i \rangle \]
Note that although we have denoted the unperturbed state as \(|n_0 \rangle\), it is not necessarily the ground state.