[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"lesson:en:72-qm-angular-momentum":3},"\u003Cp>Quantum angular momentum explains atomic structure, particle spin, and much of spectroscopy. Unlike a classical vector that can point in any direction, its components cannot all have definite values at once. The geometry of rotations becomes an algebra of operators.\u003C\u002Fp>\n\n\u003Cdiv class=\"key-concept\">\n  \u003Ch4>Orbital angular momentum and spin\u003C\u002Fh4>\n  \u003Cp>Orbital angular momentum $\\mathbf{L}=\\mathbf{r}\\times\\mathbf{p}$ comes from spatial motion. Spin $\\mathbf{S}$ is intrinsic and does not describe a tiny rotating sphere. Both obey the same rotation algebra.\u003C\u002Fp>\n\u003C\u002Fdiv>\n\n\u003Ch2>The Rotation Algebra\u003C\u002Fh2>\n\u003Cp>For any angular momentum $\\mathbf{J}$, the components satisfy:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$[J_i,J_j]=i\\hbar\\,\\varepsilon_{ijk}J_k$$\u003C\u002Fdiv>\n\u003Cp>Because $[J_x,J_y]\\neq0$, exact values cannot be assigned to all three components simultaneously. However, $J^2=J_x^2+J_y^2+J_z^2$ commutes with every component. We therefore use common eigenstates of $J^2$ and $J_z$:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$J^2|j,m\\rangle=\\hbar^2j(j+1)|j,m\\rangle,\\qquad J_z|j,m\\rangle=\\hbar m|j,m\\rangle$$\u003C\u002Fdiv>\n\u003Cp>The number $j$ is $0,\\frac12,1,\\frac32,\\ldots$, and $m$ takes the $2j+1$ values from $-j$ to $j$.\u003C\u002Fp>\n\n\u003Ch2>Ladder Operators\u003C\u002Fh2>\n\u003Cp>The operators $J_\\pm=J_x\\pm iJ_y$ change $m$ without changing $j$:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$J_\\pm|j,m\\rangle=\\hbar\\sqrt{j(j+1)-m(m\\pm1)}\\,|j,m\\pm1\\rangle$$\u003C\u002Fdiv>\n\u003Cp>The ladder must end at $m=\\pm j$, or the norm would become negative. This algebraic requirement produces angular-momentum quantization.\u003C\u002Fp>\n\n\u003Ch2>Spin One Half\u003C\u002Fh2>\n\u003Cp>For an electron, $s=1\u002F2$. In the $S_z$ basis, the operators are represented by Pauli matrices:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$S_i=\\frac{\\hbar}{2}\\sigma_i,\\quad\n\\sigma_x=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\quad\n\\sigma_y=\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix},\\quad\n\\sigma_z=\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}$$\u003C\u002Fdiv>\n\u003Cp>A pure spin-one-half state corresponds to a direction on the Bloch sphere:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$|\\psi\\rangle=\\cos\\frac{\\theta}{2}|\\uparrow\\rangle+e^{i\\phi}\\sin\\frac{\\theta}{2}|\\downarrow\\rangle$$\u003C\u002Fdiv>\n\n\u003Cdiv class=\"key-concept\">\n  \u003Ch4>Why the angle is halved\u003C\u002Fh4>\n  \u003Cp>Spin-one-half states form a representation of $SU(2)$, the double cover of the rotation group $SO(3)$. A $2\\pi$ rotation multiplies the ket by $-1$, and a $4\\pi$ rotation restores it exactly. The global sign is invisible for an isolated state but becomes observable through interference.\u003C\u002Fp>\n\u003C\u002Fdiv>\n\n\u003Ch2>Adding Angular Momenta\u003C\u002Fh2>\n\u003Cp>For two systems, $\\mathbf{J}=\\mathbf{J}_1+\\mathbf{J}_2$. The allowed total values are:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$j=|j_1-j_2|,\\ |j_1-j_2|+1,\\ldots,j_1+j_2$$\u003C\u002Fdiv>\n\u003Cp>Two spin-one-half systems give a triplet with $j=1$ and a singlet with $j=0$:\u003C\u002Fp>\n\u003Cdiv class=\"math-block\">$$|0,0\\rangle=\\frac{|\\uparrow\\downarrow\\rangle-|\\downarrow\\uparrow\\rangle}{\\sqrt2}$$\u003C\u002Fdiv>\n\u003Cp>The singlet is invariant under common rotations and is the entangled state used in Bell tests.\u003C\u002Fp>\n\n\u003Ch2>Orbital Angular Momentum\u003C\u002Fh2>\n\u003Cp>For a spatial wavefunction, the common eigenfunctions of $L^2$ and $L_z$ are the spherical harmonics $Y_\\ell^m(\\theta,\\phi)$. Here $\\ell$ is an integer because the orbital wavefunction must return to the same value after a full rotation. A spherical harmonic has parity $(-1)^\\ell$.\u003C\u002Fp>\n\n\u003Ch2>Exercises\u003C\u002Fh2>\n\u003Col>\n  \u003Cli>List all $m$ values for $j=2$.\u003C\u002Fli>\n  \u003Cli>Show that the singlet has eigenvalue zero under $J_z=S_{1z}+S_{2z}$.\u003C\u002Fli>\n  \u003Cli>Which total angular momenta result from combining $j_1=1$ and $j_2=1\u002F2$?\u003C\u002Fli>\n\u003C\u002Fol>\n\n\u003Ch2>Key Takeaways\u003C\u002Fh2>\n\u003Cul>\n  \u003Cli>Angular-momentum commutators encode the quantum geometry of rotations.\u003C\u002Fli>\n  \u003Cli>The states $|j,m\\rangle$ diagonalize $J^2$ and one component, usually $J_z$.\u003C\u002Fli>\n  \u003Cli>Spin is intrinsic and is not classical mechanical rotation.\u003C\u002Fli>\n  \u003Cli>Adding angular momenta creates superpositions described by Clebsch-Gordan coefficients.\u003C\u002Fli>\n  \u003Cli>The two-spin singlet connects rotational symmetry and entanglement.\u003C\u002Fli>\n\u003C\u002Ful>\n",1786528186704]