Qm Angular Momentum And Spin — Examples

Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the spin postulate P6, read for the massive spin-$\frac{1}{2}$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m>0$, and the angular momentum readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations). Fix a unit vector $f\in L^2({\mathbb{R}}^3;\mathbb{C})$ and a unit internal spinor $v=(v_1,v_2)\in {\mathbb{C}}^2$, and put $⟨\mathit{\sigma }{⟩}_v:=(2\mathrm{Re}(v_1\overline{v_2}),-2\mathrm{Im}(v_1\overline{v_2}),\mathrm{\mid }{v}_1{\mathrm{\mid }}^2-\mathrm{\mid }{v}_2{\mathrm{\mid }}^2)\in {\mathbb{R}}^3\mathrm{.}$ Let $\psi :=f\otimes v\in H_{1\mathrm{/}2}$ and let $S_n={\sum }_i{n}_i{S}_i$ be the directional spin of Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities. Then:

(i) $\psi$ is a unit vector, $\mathrm{\mid }⟨\mathit{\sigma }{⟩}_v\mathrm{\mid }=1$, and for every unit $n\in {\mathbb{R}}^3$ ${\mu }_{\psi }^{S_n}\left(\right\{\frac{\mathrm{\hslash }}{2}\left\}\right)=\frac{1+n\cdot ⟨\mathit{\sigma }{⟩}_v}{2},{\mu }_{\psi }^{S_n}\left(\right\{-\frac{\mathrm{\hslash }}{2}\left\}\right)=\frac{1-n\cdot ⟨\mathit{\sigma }{⟩}_v}{2};$

(ii) for the parametrization $v=(e^{i\phi }\cos (\theta \mathrm{/}2),\sin (\theta \mathrm{/}2))$ with $\theta \in [0,\pi ]$ and $\phi \in \mathbb{R}$ one has $⟨\mathit{\sigma }{⟩}_v=(\sin \theta \cos \phi ,-\sin \theta \sin \phi ,\cos \theta )$, and for the coordinate directions ${\mu }_{\psi }^{S_3}\left(\right\{\pm \frac{\mathrm{\hslash }}{2}\left\}\right)={\cos }^2\frac{\theta }{2},{\sin }^2\frac{\theta }{2},{\mu }_{\psi }^{S_1}\left(\right\{\pm \frac{\mathrm{\hslash }}{2}\left\}\right)=\frac{1\pm \sin \theta \cos \phi }{2};$

(iii) for the basis state $v=e_{1\mathrm{/}2}$ (the case $\theta =0$) the component $S_3$ has the laws ${\mu }_{\psi }^{S_3}(\{\mathrm{\hslash }\mathrm{/}2\})=1$ and ${\mu }_{\psi }^{S_3}(\{-\mathrm{\hslash }\mathrm{/}2\})=0$, while ${\mu }_{\psi }^{S_1}(\{\pm \mathrm{\hslash }\mathrm{/}2\})=\frac{1}{2}$ for both signs; and for $v=(e_{1\mathrm{/}2}+e_{-1\mathrm{/}2})\mathrm{/}\sqrt{2}$ the component $S_1$ has ${\mu }_{\psi }^{S_1}(\{\mathrm{\hslash }\mathrm{/}2\})=1$ and ${\mu }_{\psi }^{S_1}(\{-\mathrm{\hslash }\mathrm{/}2\})=0$.

All quantities are model probabilities for the pure preparation $[\psi ]$; no measurement procedure, apparatus, calibration or empirical frequency is asserted, and the example makes no claim that a laboratory device realizes the observable $S_n$.

• One preparation, all directions. The Bloch vector determines the whole family of directional laws of this preparation. These marginal laws alone do not specify joint probabilities for measurements in two directions. Such probabilities require a declared joint or sequential measurement model, including any state change between measurements; none is supplied here. No impossibility of all joint probability laws is asserted.
• No apparatus. The probabilities are Born weights of the model; whether a given Stern-Gerlach or optical apparatus realizes $S_n$ is a tested identification under the preparation and apparatus postulate P7 and is not established here.

Adopt the mass parameter $m>0$, a spin label $s$ and the angular momentum readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings for the massive spin-$s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings, and assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $Y_{lm}$ be the spherical harmonics of Associated Legendre function normalization and $h_{lm}$ their harmonic polynomial representatives of Harmonic polynomial dimension clause (c), and let $L_i^{\mathrm{s}\mathrm{p}\mathrm{h}}$ be the self-adjoint orbital generators on $L^2(S^2,\sigma )$ of Orbital self-adjoint generators, and let $D_i=(x\times \mathrm{\nabla }{)}_i$ act on polynomials on ${\mathbb{R}}^3$. The polynomial and sphere actions are related by $L_i^{\mathrm{s}\mathrm{p}\mathrm{h}}{Y}_{lm}={\mathrm{res}}_l(-i\mathrm{\hslash }{D}_i{h}_{lm})$ as specified by that theorem. Then:

(i) the harmonic polynomial representatives of the degrees $l=0$ and $l=1$ are $h_{00}=\frac{1}{\sqrt{4\pi }},h_{10}=\sqrt{\frac{3}{4\pi }}{x}_3,h_{1,1}=-\sqrt{\frac{3}{8\pi }}(x_1+i{x}_2),h_{1,-1}=\sqrt{\frac{3}{8\pi }}(x_1-i{x}_2),$ and the family $(Y_{00},Y_{1,-1},Y_{10},Y_{11})$ is orthonormal in $L^2(S^2,\sigma )$;

(ii) on the degree-one block, $L_3^{\mathrm{s}\mathrm{p}\mathrm{h}}{Y}_{1m}=\mathrm{\hslash }m{Y}_{1m},{\sum }_{i=1}^3(L_i^{\mathrm{s}\mathrm{p}\mathrm{h}}{)}^{2}F=2{\mathrm{\hslash }}^{2}F(F\in V_1),$ while the corresponding polynomial identity is $-i\mathrm{\hslash }{D}_3{h}_{1m}=\mathrm{\hslash }m{h}_{1m}$ on ${\mathbb{R}}^3$; on the degree-zero block $L_i^{\mathrm{s}\mathrm{p}\mathrm{h}}{Y}_{00}=0$ for every $i$, so ${\sum }_i(L_i^{\mathrm{s}\mathrm{p}\mathrm{h}}{)}^2{Y}_{00}=0$;

(iii) more generally, for every $l\ge 0$ and $m\in \{-l,\dots ,l\}$ one has $L_3^{\mathrm{s}\mathrm{p}\mathrm{h}}{Y}_{lm}=\mathrm{\hslash }m{Y}_{lm}$; for every nonzero $R\in C_c^{\mathrm{\infty }}((0,\mathrm{\infty }))$ and unit $w\in {\mathbb{C}}^r$, the normalized model vector $\phi =R(\mathrm{\mid }x\mathrm{\mid })h_{lm}(x)w\mathrm{/}\mathrm{\parallel }R(\mathrm{\mid }x\mathrm{\mid })h_{lm}(x)w\mathrm{\parallel }$ satisfies ${\sum }_i\mathrm{\parallel }{L}_i\phi {\mathrm{\parallel }}^2={\mathrm{\hslash }}^{2}l(l+1)$. Thus the low-degree computations of (i),(ii) are the first two cases of the sphere identities.

The low-degree sphere computations are independent of the radial profile; clause (iii) uses a nonzero radial profile to form a normalized model vector. The example involves no dynamics, no detector and no empirical input.

• The $l=0$ exception. The vanishing of all three components on $Y_{00}$ illustrates that a state with zero total orbital angular momentum has every component sharp at value zero; the noncommutation of two components on other states is not contradicted.
• Radial scope. Clause (iii) is verified locally from the defined rotation generators on the displayed compactly supported radial harmonics. The broader eigenstate and outcome-law theorem Orbital angular momentum eigenstates of the massive model: spherical harmonics, the l,m quantum numbers and the Casimir value also addresses their measurement laws under its additional Choice premise; those laws are not used here. No radial dynamics or radial energy equation is solved here.

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the spin postulate P6, read for the massive spin-$\frac{1}{2}$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m>0$, together with the effective magnetic-moment coupling of Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator with parameter $\gamma :={\gamma }_{1\mathrm{/}2}\in \mathbb{R}$, and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix $B_z\in \mathbb{R}$ and the uniform field $\mathbf{B}=(0,0,B_z)$, let $H_Z=H_0-\gamma B_z{S}_3$ on $D(H_Z)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^2)$ be the model's Zeeman operator with flow $U_Z(t)=e^{-it{H}_Z\mathrm{/}\mathrm{\hslash }}$ and put $\omega :=\gamma B_z$. Then:

(i) for every unit $f\in L^2({\mathbb{R}}^3;\mathbb{C})$ and each sign, the product flow satisfies $U_Z(t)(f\otimes e_{\pm })=e^{\pm i\gamma B_{z}t\mathrm{/}2}\left(e^{-it{H}_0\mathrm{/}\mathrm{\hslash }}f\right)\otimes e_{\pm }(t\in \mathbb{R}),$ and the transverse spin means of such a product state precess as in Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component. If in addition $f\in D(H_0)=W^{2,2}({\mathbb{R}}^3;\mathbb{C})$, then $H_Z(f\otimes e_{\pm })=\left(H_{0}f\right)\otimes e_{\pm }\mp \frac{\gamma \mathrm{\hslash }{B}_z}{2}f\otimes e_{\pm },⟨H_Z{⟩}_{f\otimes e_{\pm }}:=⟨H_Z(f\otimes e_{\pm }),f\otimes e_{\pm }⟩=⟨H_{0}f,f⟩\mp \frac{\gamma \mathrm{\hslash }{B}_z}{2};$ the displayed expectation is defined only for these domain vectors.

(ii) toy truncation. Fix a real energy parameter $E_0\in \mathbb{R}$ in joules. Suppose in addition that, on a declared two-dimensional internal subspace, the free part is replaced by the scalar $E_{0}I$ (a truncation assumption of the toy model, not a theorem of the full model): then the truncated operator is $H_Z^{(2)}=E_{0}I-\gamma B_z{S}_3=(E_0-\frac{\gamma \mathrm{\hslash }{B}_z}{2})e_{+}{e}_{+}^{\ast }+(E_0+\frac{\gamma \mathrm{\hslash }{B}_z}{2})e_{-}{e}_{-}^{\ast }$ in the basis $(e_{+},e_{-})$, with energy levels $E_0\mp \frac{\gamma \mathrm{\hslash }{B}_z}{2}$ and signed level difference $\mathrm{\Delta }E:=\gamma \mathrm{\hslash }{B}_z=\mathrm{\hslash }\omega$; its evolution is $e^{-it{H}_Z^{(2)}\mathrm{/}\mathrm{\hslash }}=e^{-it{E}_0\mathrm{/}\mathrm{\hslash }}\left(\begin{array}{cc}{e}^{i\omega t\mathrm{/}2}& 0\\ 0& e^{-i\omega t\mathrm{/}2}\end{array}\right),$ so the relative phase of the two amplitudes accumulates at the angular frequency $\omega$ (equivalently at the ordinary frequency $\mathrm{\mid }\omega \mathrm{\mid }\mathrm{/}(2\pi )$);

(iii) in the truncation the conjugation identities of the precession theorem reduce to explicit matrix identities, $V(t{)}^{\ast }{\sigma }_{1}V(t)=\cos (\omega t){\sigma }_1+\sin (\omega t){\sigma }_2,V(t{)}^{\ast }{\sigma }_{2}V(t)=\cos (\omega t){\sigma }_2-\sin (\omega t){\sigma }_1,$ with $V(t)=e^{-it{H}_Z^{(2)}\mathrm{/}\mathrm{\hslash }}{e}^{it{E}_0\mathrm{/}\mathrm{\hslash }}$;

(iv) the toy model contains no driving field: its diagonal evolution preserves the two sector weights, hence there is no spin flip between $e_{+}$ and $e_{-}$ under $H_Z^{(2)}$ alone. Statements about spin-flip transitions, resonance or spectroscopy require a separately adopted drive or coupling, belong to the driven-dynamics and light-matter pages, and are not derived here. No apparatus, calibration, measured frequency or measured value of $\gamma$ is asserted.

• What the truncation discards. Replacing $H_0$ by $E_{0}I$ discards the kinetic and any potential structure of the spatial factor; the truncated model is a toy description of the internal degree of freedom, and no spatial dynamics or position-space measurement model follows from it.
• Frequency conventions. $\omega =\gamma B_z$ is an angular frequency in ${\mathrm{s}}^{-1}$; the ordinary frequency is $\mathrm{\mid }\omega \mathrm{\mid }\mathrm{/}(2\pi )$ in hertz, with the radian dimensionless, in the conventions of Dimensions, SI unit conventions and quantity types of the framework.

Adopt the postulates P1, P2, P4, P5 and P6 and the effective magnetic-moment coupling as in The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field, for the massive spin-$\frac{1}{2}$ model with mass parameter $m>0$ and parameter $\gamma :={\gamma }_{1\mathrm{/}2}\in \mathbb{R}$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix a real $B_z\in C^1(\mathbb{R})$ with $B_z$, $B_z^{\mathrm{\prime }}$ bounded, reals $0<a<\mathrm{\infty }$ and $G$, and the field profile $\mathbf{B}(x)=(0,0,B_z(x_1))$, and suppose that $B_z^{\mathrm{\prime }}(x_1)=G$ for every $x_1\in (-a,a)$. Let $H_Z$ be the Zeeman operator of the model with flow $u(t)=e^{-it{H}_Z\mathrm{/}\mathrm{\hslash }}{u}_0$ for a unit vector $u_0\in D(H_Z^2)$, let $u_{\pm }(t):=P_{\pm }u(t)$ be the two spin-sector components with $P_{\pm }=\frac{1}{2}(I\pm 2S_3\mathrm{/}\mathrm{\hslash })$, and write $w_{\pm }:=\mathrm{\parallel }{u}_{\pm }(0){\mathrm{\parallel }}^2$ and, for each sign, let ${\eta }_{\pm }(t,x_1):={\int }_{{\mathbb{R}}^2}\mid u_{\pm }^{\pm }(t,x_1,x_2,x_3){\mid }^{2}d{x}_{2}d{x}_3$ be the first-coordinate marginal density of the $\pm$ component of $u_{\pm }(t)$, where $u_{\pm }^{\pm }$ denotes that nonzero component. Fix $\tau >0$ and adopt the following hypothetical containment premise: for each sign and every $t\in [0,\tau ]$, ${\int }_{\mathbb{R}\setminus (-a,a)}{\eta }_{\pm }(t,x_1)d{x}_1=0.$ Then:

(i) the sector weights are conserved: $\mathrm{\parallel }{u}_{\pm }(t){\mathrm{\parallel }}^2=w_{\pm }$ for every $t$, and $⟨S_3{⟩}_{u(t)}=\frac{\mathrm{\hslash }}{2}(w_{+}-w_{-})$ is independent of $t$;

(ii) under the containment premise, for each sign with $w_{\pm }>0$ the normalized sector transverse mean $M_{\pm }(t):=w_{\pm }^{-1}⟨P_x{u}_{\pm }(t),u_{\pm }(t)⟩$ is affine in $t$ on $[0,\tau ]$: $\frac{d}{dt}{M}_{\pm }(t)=\pm \frac{\gamma \mathrm{\hslash }G}{2},M_{\pm }(\tau )=M_{\pm }(0)\pm \frac{\gamma \mathrm{\hslash }G\tau }{2},$ so, when $w_{+}{w}_{-}>0$, their difference changes by $\gamma \mathrm{\hslash }G\tau$: $M_{+}(\tau )-M_{-}(\tau )=M_{+}(0)-M_{-}(0)+\gamma \mathrm{\hslash }G\tau ,$ independently of the initial means; a sector with $w_{\pm }=0$ is trivial;

(iii) the total transverse mean evolves by the weighted rate $\frac{d}{dt}⟨P_{x}u(t),u(t)⟩=\frac{\gamma \mathrm{\hslash }G}{2}(w_{+}-w_{-})=\gamma G⟨S_3{⟩}_{u_0},t\in [0,\tau ],$ so for the equal-weight preparation $w_{+}=w_{-}=\frac{1}{2}$ the total transverse mean is conserved while the sector mean changes are equal and opposite; if $\gamma G=0$, both changes vanish;

(iv) Status and scope. This is an idealized hypothetical calculation: the containment premise is an explicitly assumed possible property of the evolved state, not an observed fact and not a consequence of the model; no position-space splitting, detector trace, count, calibration or measured value is asserted, no apparatus is modeled, and the historical Stern-Gerlach experiment is not described or explained by this item. The identification of the model with any real beam experiment remains a tested identification with uncertainty under Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions.