10: Angular Momentum and Group Symmetries of Electronic Wavefunctions
- Page ID
- 60567
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Electronic wavefunctions must also possess proper symmetry. These include angular momentum and point group symmetries
- 10.1: Angular Momentum Symmetry and Strategies for Angular Momentum Coupling
- This page outlines the constraints on trial wavefunctions for many-electron atoms and molecules, highlighting the necessity for these wavefunctions to be eigenfunctions of certain symmetry operators like \(S^2\) and \(S_z\) that commute with the Hamiltonian \(H\).
- 10.2: Electron Spin Angular Momentum
- This page explores the intrinsic spin of electrons, represented by quantum numbers \(s\) and \(m_s\), specifically \(s = 1/2\) with states \(\alpha\) and \(\beta\). It details the formation of total spin eigenstates in multi-electron systems, particularly for three-electron configurations like 1s2s3s. The page discusses creating quartet and doublet states, adhering to the Pauli principle, and the challenges of dealing with doubly occupied orbitals and closed-shell configurations.
- 10.3: Coupling of Angular Momenta
- This page covers vector and non-vector coupling of angular momenta in quantum mechanics, particularly for electron configurations like \(p^1d^1\). It explains total angular momentum values from combining two angular momenta, highlights differences in treatment of indistinguishable versus distinguishable particles, and addresses the Pauli exclusion principle's effect on term symbols.
- 10.4: Atomic Term Symbols and Wavefunctions
- This page explains the coupling of non-equivalent and equivalent angular momenta in atomic physics. It details how different angular momenta lead to various energy levels due to electron-electron repulsions and spin-orbit interactions, introducing term symbols like ^3F and ^1D. For equivalent angular momenta, a "box" method helps account for unique product states in line with the Pauli principle, yielding term symbols like ^3P, ^1D, and ^1S for configurations like p^2 and p^2d^1.
- 10.5: Atomic Configuration Wavefunctions
- This page explains the derivation of atomic wavefunctions using Slater determinants, focusing on constructing orthogonal states with specific angular momentum combinations. It details the process of generating states via raising and lowering operators, and exemplifies this with the \(p^2\) case.
- 10.6: Inversion Symmetry
- This page explains the role of inversion symmetry in atomic systems through an additional quantum number. It details how potential energy remains unchanged under inversion and the determination of the inversion operator's effect on atomic wavefunctions, assigning +1 or -1 factors based on orbital types. The resulting sign classifies wavefunctions as "even" or "odd," illustrated with term symbols for various electron configurations.
- 10.7: Review of Atomic Cases
- This page provides a quantum mechanical overview of atomic orbitals, focusing on quantum numbers \(l\) and \(m_l\) and their degeneracy within the many-electron Hamiltonian framework. It outlines angular momentum operator commutation relationships, wave function construction from determinants, and symmetry adaptations.