11: Time-Independent Perturbation Theory
- Page ID
- 15798
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consider the following very commonly occurring problem. The Hamiltonian of a quantum mechanical system is written
\[H = H_0 + H_1. \nonumber \]
Here, \(H_0\) is a simple Hamiltonian whose eigenvalues and eigenstates are known exactly. \(H_1\) introduces some interesting additional physics into the problem, but is sufficiently complicated that when we add it to \(H_0\) we can no longer find the exact energy eigenvalues and eigenstates. However, \(H_1\) can, in some sense (which we shall specify more precisely later on), be regarded as small compared to \(H_0\). Can we find approximate eigenvalues and eigenstates of the modified Hamiltonian, \(H_0+H_1\), by performing some sort of perturbation expansion about the eigenvalues and eigenstates of the original Hamiltonian, \(H_0\)? Let us investigate.
Incidentally, in this chapter, we shall only discuss so-called time-independent perturbation theory, in which the modification to the Hamiltonian, \(H_1\), has no explicit dependence on time. It is also assumed that the unperturbed Hamiltonian, \(H_0\), is time independent.
- 11.1: Improved Notation
- This page presents improved notation for quantum mechanics, defining eigenstates of the Hamiltonian and their eigenvalues. It covers the orthonormality of these states in various dimensions and for spinors, alongside the expansion of wavefunctions in terms of eigenstates. Additionally, it introduces the expression for the expectation value of an operator using matrix elements and emphasizes the completeness of eigenstates represented by the identity operator.
- 11.2: Two-State System
- This page explores a quantum mechanical system with two eigenstates, focusing on the effects of perturbation from an additional Hamiltonian \(H_1\) on the energy eigenvalue problem. It derives coupled equations in matrix form using eigenstates and eigenvalues of the unperturbed Hamiltonian \(H_0\).
- 11.3: Non-Degenerate Perturbation Theory
- This page explores the extension of perturbation analysis to systems with multiple energy eigenstates, defining eigenstates for the unperturbed Hamiltonian and detailing the treatment of the perturbed Hamiltonian. It provides a series of equations to derive adjusted eigenvalue and wavefunction expansions, ensuring orthonormality up to second-order perturbation terms. The discussion emphasizes how the energy eigenvalue is modified by perturbation-related terms.
- 11.4: Quadratic Stark Effect
- This page explores the Stark effect in hydrogen atoms exposed to an electric field, detailing the Hamiltonian's structure and energy shifts. It covers selection rules governing matrix elements, emphasizing conditions for non-zero values influenced by angular momentum quantum numbers. The discussion also includes the quadratic Stark effect and challenges with energy shift divergences in degenerate states.
- 11.5: Degenerate Perturbation Theory
- This page explores the Stark effect in excited hydrogen atom states using non-degenerate perturbation theory. It highlights the complexity of energy levels and eigenstates for quantum number \(n>1\) due to degeneracies. To address the resulting singularities, new linear combinations of eigenstates are proposed, enabling them to be simultaneous eigenstates of both Hamiltonians.
- 11.6: Linear Stark Effect
- This page explores the influence of an external electric field on the \(n=2\) energy states of a hydrogen atom, focusing on the \(2S\) and \(2P\) states. It utilizes a matrix eigenvalue equation to assess the coupling between these states, revealing that the energy levels shift linearly with the electric field, illustrating the linear Stark effect.
- 11.7: Fine Structure of Hydrogen
- This page covers the relativistic kinetic energy and its effects on hydrogen atom energy levels, incorporating kinetic energy formulas and perturbation theory to analyze energy shifts due to relativistic corrections. It examines spin-orbit coupling and its role in generating fine structure shifts for various quantum states, emphasizing the impact on degeneracy and the consistency of combined corrections, especially for states with \(l=0\).
- 11.8: Zeeman Effect
- This page explains the Zeeman effect, detailing the energy shifts in hydrogen atom states under a weak magnetic field. It describes modifications to the Hamiltonian considering the electron's magnetic moment and applies first-order perturbation theory to derive energy shifts.
- 11.9: Hyperfine Structure
- This page explores the proton's magnetic moment in hydrogen atoms, detailing its effect on energy levels through spin-spin coupling with electrons. It includes mathematical formulations for the magnetic moment and field, as well as the associated Hamiltonian. Perturbation theory is discussed in relation to energy shifts causing hyperfine structure, highlighting differences in energy levels between singlet and triplet states.
- 11.10: Exercises
- This page explores perturbation theory in a two-state quantum system, detailing the effects of various conditions on energy eigenvalues and eigenstates. It provides general formulas for perturbed energies and states, emphasizing the significance of perturbations on degenerate states.