9: Spin Angular Momentum
- Page ID
- 15784
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Broadly speaking, a classical extended object (e.g., the Earth) can possess two different types of angular momentum. The first type is due to the rotation of the object’s center of mass about some fixed external point (e.g., the Sun)—this is generally known as orbital angular momentum. The second type is due to the object’s internal motion—this is generally known as spin angular momentum (because, for a rigid object, the internal motion consists of spinning about an axis passing through the center of mass). By analogy, quantum particles can possess both orbital angular momentum due to their motion through space (see Chapter [sorb]), and spin angular momentum due to their internal motion. Actually, the analogy with classical extended objects is not entirely accurate, because electrons, for instance, are structureless point particles. In fact, in quantum mechanics, it is best to think of spin angular momentum as a kind of intrinsic angular momentum possessed by particles. It turns out that each type of elementary particle has a characteristic spin angular momentum, just as each type has a characteristic charge and mass.
- 9.1: Spin Operators
- This page covers the properties and operators of spin angular momentum, paralleling them with orbital angular momentum. It defines the spin operators \(S_x\), \(S_y\), and \(S_z\), details their commutation relations, and introduces the magnitude squared of the spin vector \(S^2\). It notes that \(S^2\) can be measured with one component, often \(S_z\), and discusses the raising and lowering operators \(S_\pm\) and their connections to the spin operators.
- 9.2: Spin Space
- This page explores spin wavefunctions in abstract vector spaces, detailing the impact of spin operators and the principle of superposition. It describes the definition of vectors through dual spaces and emphasizes the necessity of Hermitian spin operators for real eigenvalues. Mutual orthogonality and normalization of eigenstates are highlighted, along with the expression of general spin states as superpositions of these eigenstates.
- 9.3: Eigenstates of Sz and S²
- This page covers spin angular momentum operators \(S_z\) and \(S^2\), highlighting their commutation properties and shared eigenstates. It introduces raising and lowering operators \(S_+\) and \(S_-\), detailing their impact on the eigenstates represented by \(m_s\), which range from \(-s\) to \(s\).
- 9.4: Pauli Representation
- This page introduces spin eigenstates for an electron, specifically the states \(\chi_+\) and \(\chi_-\) for spin "up" and "down". It explores the orthonormal properties of these states and how any spin state can be represented as a spinor, a linear combination of these two states. The Pauli representation is used to illustrate spin space, employing \(2 \times 2\) matrices (Pauli matrices) for spin operators. The page highlights the probabilistic interpretation of the coefficients in the spinor.
- 9.5: Spin Precession
- This page explores the magnetic moment of a small current loop, linking it to current and area through the equation \(\mu = I\,A\). It connects magnetic moment to angular momentum for electrons, resulting in a gyromagnetic ratio \(g\) close to 2, which differs from classical predictions. The Hamiltonian for electron spin in a magnetic field is introduced, with solutions that illustrate time-dependent spin state probabilities and precession around the z-axis.
- 9.E: Spin Angular Momentum (Exercises)
- This page covers the Pauli representation of spin operators for spin-1 and spin-1/2 particles, detailing eigenstates of \(S_x\) and \(S_y\), along with measurement probabilities for \(S_z\) in a specific orientation. It includes calculations for normalizing spin states, expected outcomes, and the Hamiltonian for an electron in an oscillating magnetic field. The page also discusses time evolution of spin states and conditions for complete spin flips.