7: Orbital Angular Momentum
- Page ID
- 15770
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As is well known, angular momentum plays a vitally important role in the classical description of three-dimensional motion . Let us now investigate the role of angular momentum in the quantum mechanical description of such motion.
- 7.1: Angular Momenum Operators
- This page examines vector angular momentum in quantum mechanics, defining it as the cross product of position and momentum vectors. It describes the components of angular momentum as Hermitian operators and derives their commutation relations, establishing that \(L^2\) can be measured with one component, typically \(L_z\). The page also introduces the operators \(L_+\) and \(L_-\), detailing their properties and relationships in quantum contexts.
- 7.2: Representation of Angular Momentum
- This page explores angular momentum operators in quantum mechanics, representing them as spatial differential operators akin to linear momentum operators. Using spherical polar coordinates (r, θ, φ), it derives expressions for L_x, L_y, L_z, and the squared angular momentum operator L², highlighting that these operators depend on the angular coordinates θ and φ while excluding the radial coordinate r.
- 7.3: Eigenstates of Angular Momentum
- This page explores the simultaneous eigenstates of the angular momentum operators \(L_z\) and \(L^2\), denoting them as \(Y_{l,m}(\theta,\phi)\). It highlights their eigenvalues, \(m\hbar\) for \(L_z\) and \(l(l+1)\hbar^2\) for \(L^2\), and discusses how the raising and lowering operators \(L_+\) and \(L_-\) alter these states' eigenvalues. The page establishes expressions for these operations and emphasizes the structure and properties of angular momentum eigenstates in quantum mechanics.
- 7.4: Eigenvalues of Lz
- This page explains the separation of the eigenstate \(Y_{l,m}(\theta,\phi)\) into orthonormal components \( {\mit\Theta}_{l,m}(\theta) \) and \( {\mit\Phi}_{m}(\phi) \). It notes that the \(L_z\) operator's dependence on the azimuthal angle leads to \({\mit\Phi}_m(\phi) \sim {\rm e}^{\,{\rm i}\,m\,\phi}\), with the single-valued wavefunction requiring integer quantization of \(m\).
- 7.5: Eigenvalues of L²
- This page details the constraints on quantum numbers \(l\) and \(m\) based on the angular wavefunction. It establishes inequalities that must be met, specifically \(l(l+1) \geq m(m+1)\) and \(l(l+1) \geq m(m-1)\), resulting in the condition \(-l \leq m \leq l\). The page specifies the values \(m_- = -l\) and \(m_+ = l\), concluding that \(l\) must be a non-negative integer with possible values of \(0, 1, 2, \ldots\), and \(m\) ranging within those limits.
- 7.6: Spherical Harmonics
- This page covers the properties and functional forms of spherical harmonics \(Y_{l,m}(\theta, \phi)\), highlighting their role as simultaneous eigenstates of angular momentum operators \(L_z\) and \(L^2\). It derives expressions for various harmonics, including \(Y_{l,l}\) and \(Y_{l,-l}\), and shows how to obtain lower states using the lowering operator \(L_-\).
- 7.E: Orbital Angular Momentum (Exercises)
- This page covers the expectation values \(\langle L_x\rangle\) and \(\langle L_x^{\,2}\rangle\) for quantum states described by spherical harmonics, particularly focusing on the angular momentum operator \(L_x\) for \(l=1\). It discusses the measurements related to \(L_x\) and subsequent \(L_z\) outcomes and probabilities. Additionally, the page introduces the Hamiltonian for an axially symmetric rotator, examining its eigenvalues.