12: Time-Dependent Perturbation Theory
- Page ID
- 15805
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consider a system whose Hamiltonian can be written
\[H(t) = H_0 + H_1(t). \nonumber \]
Here, \(H_0\) is again a simple time-independent Hamiltonian whose eigenvalues and eigenstates are known exactly. However, \(H_1\) now represents a small time-dependent external perturbation. Let the eigenstates of \(H_0\) take the form
\[H_0\,\psi_m = E_m\,\psi_m. \nonumber \]
We know (see Section [sstat]) that if the system is in one of these eigenstates then, in the absence of an external perturbation, it remains in this state for ever. However, the presence of a small time-dependent perturbation can, in principle, give rise to a finite probability that if the system is initially in some eigenstate \(\psi_n\) of the unperturbed Hamiltonian then it is found in some other eigenstate at a subsequent time (because \(\psi_n\) is no longer an exact eigenstate of the total Hamiltonian). In other words, a time-dependent perturbation allows the system to make transitions between its unperturbed energy eigenstates. Let us investigate such transitions.
- 12.1: Preliminary Analysis
- This page discusses the time evolution of a quantum system in a superposition of energy eigenstates, focusing on constant probabilities in the absence of perturbations. When a time-dependent perturbation is introduced, the coefficients change over time according to the time-dependent Schrödinger equation, resulting in coupled differential equations. Although exact solutions are difficult to obtain, they are achievable for simple two-state systems.
- 12.2: Two-State System
- This page explores a two-state quantum system governed by a time-independent Hamiltonian and a time-dependent perturbation. It examines the interaction Hamiltonian's off-diagonal elements, which facilitate state transitions. The derivation of probability amplitudes leads to Rabi's formula, illustrating oscillations between states based on perturbation frequency.
- 12.3: Spin Magnetic Resonance
- This page covers a spin one-half particle in a magnetic field, including a rotating time-dependent field. It details the Hamiltonian's division into the unperturbed and perturbation parts, with eigenstates for spin configurations. The interaction with the rotating field facilitates spin flips at resonance, which improves measurement accuracy for gyromagnetic ratios by scanning the oscillating field's frequency.
- 12.4: Perturbation Expansion
- This page covers the behavior of a quantum system influenced by a small time-dependent perturbation to its Hamiltonian, using stationary eigenstates. It presents the wavefunction as a sum of these eigenstates, with time evolution described by a differential equation. By employing perturbation theory, it derives a first-order solution for the coefficients, determining the transition probability between energy eigenstates, valid only for small transition probabilities.
- 12.5: Harmonic Perturbation
- This page examines harmonic perturbations in quantum mechanics, focusing on Hermitian perturbations that oscillate over time. It presents key findings on transition amplitudes and probabilities related to energy absorption and emission through stimulated processes.
- 12.6: Electromagnetic Radiation
- This page examines the interaction of atomic electrons with electromagnetic radiation, detailing both classical and quantum mechanics frameworks. It covers transition probabilities for radiation-induced absorption via perturbation theory, addressing factors like electromagnetic field parameters and non-monochromatic radiation.
- 12.7: Electric Dipole Approximation
- This page explains the electric dipole approximation for electromagnetic radiation interactions with atoms, focusing on calculating transition rates for absorption and stimulated emission based on the effective electric dipole moment. It details the averaging of these rates over unpolarized isotropic radiation, leading to simplified expressions, while also addressing the significance of polarization and angular dependencies in the calculations.
- 12.8: Spontaneous Emission
- This page explores radiation-induced atomic transitions, focusing on absorption, stimulated emission, and spontaneous emission. It defines spontaneous emission as the transition of an atom from higher to lower energy without external influence, resulting in photon emission.
- 12.9: Radiation from Harmonic Oscillator
- This page analyzes electron behavior in a one-dimensional harmonic oscillator potential, detailing energy eigenvalues and spontaneous emission from excited to lower energy states under non-zero electric dipole moments. It explains allowed transitions by quantum numbers and establishes a fixed photon emission frequency linked to the classical oscillator's frequency.
- 12.10: Selection Rules (Hydrogen Atoms)
- This page covers spontaneous energy level transitions in hydrogen atoms using the electric dipole approximation. It details selection rules for these transitions, specifying that a change in the orbital quantum number \( l \) must be \( \pm 1 \), while the magnetic quantum number \( m \) can remain the same or change by \( \pm 1 \). Additionally, it notes that the spin quantum number \( m_s \) does not change since the perturbing Hamiltonian lacks spin operators.
- 12.11: 2P→1S Transitions in Hydrogen
- This page examines the spontaneous emission rate from the 2P excited state to the 1S ground state of a hydrogen atom, following quantum selection rules. It derives the dipole moment and calculates the energy and wavelength of the emitted photon. The transition rate is mathematically expressed, with a specific numerical value provided.
- 12.12: Intensity Rules
- This page explains how electron spin and spin-orbit coupling lead to the splitting of the six \(2P\) states in hydrogen into higher-energy \(2P_{3/2}\) states and lower-energy \(2P_{1/2}\) states. This energy difference affects spectral line splitting during the \(2P \rightarrow 1S\) transition, with the \(2P_{3/2}\) states exhibiting a brighter spectral line due to their greater population, even though transition rates remain independent of angular momentum quantum numbers.
- 12.13: Forbidden Transitions
- This page covers forbidden atomic transitions that violate electric dipole selection rules. While these transitions typically have a zero matrix element, they can occur at reduced rates if non-zero residual matrix elements are present. The rate of forbidden transitions is much lower than that of allowed transitions, scaling as \( \alpha^{\,5}\,\omega_{if} \).
- 12.E: Time-Dependent Perturbation Theory (Exercises)
- This page explores the dynamics of a two-state quantum system influenced by a Hamiltonian, detailing the probability evolution during state transitions. It covers spin states in magnetic fields for various spin values and their responses to resonant and non-resonant frequencies. Furthermore, it highlights that spontaneous transitions between atomic states with zero orbital angular momentum are not allowed, underscoring the constraints imposed by quantum mechanics.