Qm Identical Particles And Exclusion — Examples

Adopt the state and composition postulate P1 (State and composition postulate P1: Hilbert spaces, preparations and composite systems) and the identical-particle and spin postulate P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations) in the readings of Declared species sectors and their preparations: the framework's reading of P6, and assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Declare a spin-half fermionic species with an arbitrary model-assigned mass parameter $M_{\mathrm{s}\mathrm{p}}>0\mathrm{k}\mathrm{g}$, unused in this calculation and not assigned an empirical value here, whose one-particle space is the completed product $K\widehat{\otimes }V$ of a nonzero finite-dimensional complex Hilbert space $K$ (the orbital factor) with $V={\mathbb{C}}^2$ (the internal factor of P6 clause (ii) for $s=\frac{1}{2}$), with $V$ carrying the orthonormal basis $(e_{+},e_{-})$ of Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet, and declare $n=2$ particles in the adopted antisymmetric sector ${\mathfrak{H}}_2^{-}=\mathrm{ran}{P}_{-}^{(K\otimes V)}\subseteq (K\otimes V)\widehat{\otimes }(K\otimes V)$ of the declared species, with the regrouping unitary $\mathrm{\Theta }$ of Regrouping two factors: the symmetry decomposition of a two-particle sector over an orbital and an internal factor. Write $u\wedge v:=\sqrt{2}{P}_{-}^{(K)}(u\otimes v)$ and $u\odot v:=\sqrt{2}{P}_{+}^{(K)}(u\otimes v)$ for $u,v\in K$, and ${\sigma }_{\pm }:=(e_{+}\otimes e_{-}\pm e_{-}\otimes e_{+})\mathrm{/}\sqrt{2}$ for the spin combinations. Then, as conditional statements of the declared model:

(a) sector decomposition: $\mathrm{\Theta }$ carries ${\mathfrak{H}}_2^{-}$ onto the orthogonal direct sum $(\mathrm{ran}{P}_{-}^{(K)}\otimes \mathrm{ran}{P}_{+}^{(V)})+(\mathrm{ran}{P}_{+}^{(K)}\otimes \mathrm{ran}{P}_{-}^{(V)})$, where $\mathrm{ran}{P}_{-}^{(V)}=\mathbb{C}\mathrm{\Psi }$ is the one-dimensional spin-singlet line spanned by the singlet $\mathrm{\Psi }$ of Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet clause (d) and $\mathrm{ran}{P}_{+}^{(V)}=W_1$ is the three-dimensional spin-triplet space of that theorem; consequently the antisymmetric two-particle sector is the sum of a spin-singlet part $\mathrm{ran}{P}_{+}^{(K)}\otimes \mathbb{C}\mathrm{\Psi }$, of dimension $\left(\genfrac{}{}{0px}{}{m+1}{2}\right)$ for $m=\dim K$, with symmetric orbital factor, and a spin-triplet part $\mathrm{ran}{P}_{-}^{(K)}\otimes W_1$, of dimension $3\left(\genfrac{}{}{0px}{}{m}{2}\right)$, with antisymmetric orbital factor;

(b) opposite orbital symmetry: for all $u,v\in K$ and $\chi ,{\chi }^{\mathrm{\prime }}\in V$, $\mathrm{\Theta }\left(P_{-}^{(K\otimes V)}\right((u\otimes \chi )\otimes (v\otimes {\chi }^{\mathrm{\prime }})\left)\right)=P_{+}^{(K)}(u\otimes v)\otimes P_{-}^{(V)}(\chi \otimes {\chi }^{\mathrm{\prime }})+P_{-}^{(K)}(u\otimes v)\otimes P_{+}^{(V)}(\chi \otimes {\chi }^{\mathrm{\prime }});$ the spin-singlet component therefore multiplies the symmetric orbital component and the spin-triplet component multiplies the antisymmetric orbital component, so within each component the orbital exchange symmetry is opposite to the spin exchange symmetry;

(c) parallel spins: if $u,v\in K$ are orthonormal and $\chi \in V$ is a unit vector, then $\mathrm{\Theta }$ carries the ray of $P_{-}^{(K\otimes V)}((u\otimes \chi )\otimes (v\otimes \chi ))$ to the ray of $P_{-}^{(K)}(u\otimes v)\otimes (\chi \otimes \chi )$, whose unit representative is $(u\wedge v)\otimes (\chi \otimes \chi )$; for $\chi =e_{+}$ this is the $S_z^{\mathrm{t}\mathrm{o}\mathrm{t}}=\mathrm{\hslash }$ triplet component with an antisymmetric orbital factor;

(d) antiparallel spins: if $u,v\in K$ are orthonormal, then $\mathrm{\Theta }$ carries the ray of $P_{-}^{(K\otimes V)}((u\otimes e_{+})\otimes (v\otimes e_{-}))$ to the ray of $\frac{1}{\sqrt{2}}((u\odot v)\otimes {\sigma }_{-}+(u\wedge v)\otimes {\sigma }_{+})$, an equal-amplitude superposition of the singlet component with symmetric orbital factor and the $S_z^{\mathrm{t}\mathrm{o}\mathrm{t}}=0$ triplet component with antisymmetric orbital factor; in particular such an antisymmetrised product of two orthonormal one-particle states has no definite orbital exchange symmetry.

The model assignment of the antisymmetric sector is that of P6; the statements above are conditional on it and assert no spin-statistics theorem, no interaction, no dynamics and no measurement.

• The singlet and triplet names. The names refer to the total spin of the pair in the sense of the internal-factor composition theorem: $W_0$ is the kernel of $S^{\mathrm{t}\mathrm{o}\mathrm{t}2}$ and $W_1$ is the triplet block. The orbital factor carries no spin and the theorem attributes no spin to the orbital degrees of freedom.
• A Slater determinant is a superposition. As clause (d) shows, the antisymmetrised product of two orthonormal one-particle states with opposite spin orientations contains both the singlet and the triplet $S_z=0$ components with equal amplitudes; a definite orbital exchange symmetry belongs to the components, not to every vector of the sector.

Adopt the state and composition postulate P1 and the identical-particle and spin postulate P6 in the readings of Declared species sectors and their preparations: the framework's reading of P6, 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). Declare the following two-orbital spin-half model:

(a) the species is a spin-half fermion with an arbitrary model-assigned mass parameter $M_{\mathrm{s}\mathrm{p}}>0\mathrm{k}\mathrm{g}$, unused in this computation and with no empirical value asserted; its orbital factor is $K:={\mathrm{span}}_{\mathbb{C}}\{u,v\}$ where $u,v\in W^{1,2}({\mathbb{R}}^3;\mathbb{C})\cap L^2({\mathbb{R}}^3;\mathbb{C})$ are real-valued, normalized and orthogonal in $L^2({\mathbb{R}}^3)$ (Integer-order Sobolev spaces and their norms, The space $L^p(\mu )$ as the quotient by null functions), with $uv\mathrm{\ne }0$ in $L^1$; the internal factor is $V={\mathbb{C}}^2$ with basis $(e_{+},e_{-})$; the one-particle space is $K\widehat{\otimes }V$ and $n=2$ particles are declared in the antisymmetric sector ${\mathfrak{H}}_2^{-}=\mathrm{ran}{P}_{-}^{(K\otimes V)}$ (The two spin-half fermion sector: the spin-singlet and spin-triplet parts and their orbital symmetry);

(b) write $E:=K\widehat{\otimes }K$ and let $J:E\to L^2({\mathbb{R}}^6)$ be the restriction of the Euclidean product unitary ${\mathrm{\Theta }}_6$ (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})), so $J(a\otimes b)(x,y)=a(x)b(y)$. Here $L^2({\mathbb{R}}^r)$ consists of almost-everywhere classes $[f]$ of complex Lebesgue-measurable functions with ${\int }_{{\mathbb{R}}^r}\mathrm{\mid }f{\mathrm{\mid }}^{2}dx<\mathrm{\infty }$ (The space $L^p(\mu )$ as the quotient by null functions). In SI units, let $x,y\in {\mathbb{R}}^3$ be spatial coordinates measured in metres and define $W:{\mathbb{R}}^3\to \mathbb{R}$ by $W(z)=\kappa \mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ for $z\mathrm{\ne }0$ and $W(0)=0$. The value at zero is an arbitrary representative on a measure-zero set. Here $\kappa >0$ is an adopted parameter of unit $\mathrm{J}\mathrm{m}$, with no relation to a charge assumed, so $W$ has unit $\mathrm{J}$. Define the multiplication operator $M_W:D(M_W)\subseteq L^2({\mathbb{R}}^6)\to L^2({\mathbb{R}}^6),D(M_W):=\{[g]\in L^2({\mathbb{R}}^6):[(x,y)\mapsto W(x-y)g(x,y)]\in L^2({\mathbb{R}}^6)\},M_W[g]:=[(x,y)\mapsto W(x-y)g(x,y)]$ where $g$ is any representative. This is independent of the representative because $W$ is finite and measurable everywhere. The declared pair interaction is the compression of this multiplier to $E$: for $z,z^{\mathrm{\prime }}\in E$, set $q(z,z^{\mathrm{\prime }}):={\int }_{{\mathbb{R}}^6}W(x-y)(Jz)(x,y)\overline{(J{z}^{\mathrm{\prime }})(x,y)}dxdy\mathrm{.}$ The normalized position-space orbitals have units ${\mathrm{m}}^{-3\mathrm{/}2}$, and $D$, $X$ and the pair-interaction expectations below have units $\mathrm{J}$. All $Jz$ lie in $D(M_W)$, as verified below. For the orthonormal basis $(b_1,b_2,b_3,b_4):=(u\otimes u,u\otimes v,v\otimes u,v\otimes v)$ of $E$, define $Az:={\sum }_{j=1}^{4}q(z,b_j)b_j\mathrm{.}$ Thus $⟨Az,z^{\mathrm{\prime }}⟩=q(z,z^{\mathrm{\prime }})$; $A$ is a bounded self-adjoint pair-energy operator on $E$, has units $\mathrm{J}$, and has real matrix entries in this basis. It commutes with spatial exchange. The model's sector interaction is the restriction to ${\mathfrak{H}}_2^{-}$ of ${\mathrm{\Theta }}^{-1}(A\otimes I_{V\otimes V})\mathrm{\Theta }$, with $\mathrm{\Theta }$ the finite-dimensional regrouping unitary of the symmetry-decomposition theorem. This restriction is well defined because the conjugated operator commutes with total exchange. On a factored preparation with unit $\chi$ its expectation is $q(\phi ,\phi )=⟨M_{W}J\phi ,J\phi ⟩$;

(c) the declared pair preparations are the images under the inverse regrouping map of the symmetry-decomposition theorem (Regrouping two factors: the symmetry decomposition of a two-particle sector over an orbital and an internal factor) of the factored vectors ${\mathrm{\Psi }}_T:={\mathrm{\Theta }}^{-1}({\phi }_{-}\otimes (e_{+}\otimes e_{+})),{\mathrm{\Psi }}_S:={\mathrm{\Theta }}^{-1}({\phi }_{+}\otimes {\sigma }_{-}),{\phi }_{\pm }:=\frac{u\otimes v\pm v\otimes u}{\sqrt{2}},{\sigma }_{-}:=\frac{e_{+}\otimes e_{-}-e_{-}\otimes e_{+}}{\sqrt{2}},$ which are unit vectors of ${\mathfrak{H}}_2^{-}$ by the cited thought experiment. ${\mathrm{\Psi }}_T$ is the parallel-spin (spin-triplet) preparation and ${\mathrm{\Psi }}_S$ the spin-singlet preparation of the declared two-orbital model.

The example computes the declared pair-interaction expectations $q({\phi }_{\pm },{\phi }_{\pm })$ of the two preparations and shows that the exchange integral $X$ separates them strictly.

• What "exchange" names. In this model the name refers to the difference $2X$ between the two pair expectations, a property of the declared orbitals and interaction; the calculation does not attribute a mechanical force to it and does not by itself identify the energy of any atom.
• Nonzero orbital overlap. Orthogonality of the real orbitals gives $\int uv=0$; a nonzero product consequently changes sign on sets of positive measure. The Gaussian-square argument proves $X>0$ for every nonzero product, without a sign assumption. If $uv=0$ a.e., then $X=0$ and the pair expectations coincide.
• Finite orbital model. The interaction is the compression of the Coulomb form to $K\widehat{\otimes }K$. The full multiplier need not preserve this finite spatial subspace. The separate maps $\mathrm{\Theta }$ and ${\mathrm{\Theta }}_6$ respectively regroup orbital and spin factors and identify spatial tensors with Euclidean wavefunctions.

What the computations establish. Both items are conditional statements of declared models. The two spin-half fermion sector: the spin-singlet and spin-triplet parts and their orbital symmetry decomposes the antisymmetric two-particle sector of a declared spin-half species with a finite orbital factor into its spin-singlet and spin-triplet parts, exhibits the opposite orbital exchange symmetry of the two parts, and computes the components of an antisymmetrised product of two declared one-particle states. Exchange versus direct Coulomb terms: the pair-energy split of the two-orbital spin-half model specialises a two-orbital model with a declared Coulomb-form interaction and shows, for real normalized orthogonal $W^{1,2}$ orbitals with a nonzero product, that the declared parallel-spin (triplet) preparation has strictly smaller pair-interaction expectation than the singlet preparation, the difference being twice the exchange integral.

What they rest on. Both rest on the adopted antisymmetric sector of P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations) in the reading of Declared species sectors and their preparations: the framework's reading of P6: the assignment of the species to the antisymmetric sector and the choice of orbital factor are model declarations, and in the second item the two orbitals and the interaction form are declared idealizations. The spin-statistics association is not derived, no interaction is derived from the sector and no dynamics is introduced.

What is not claimed. The page records no measured energy, no count and no apparatus response; the terms "singlet", "triplet" and "exchange" name components and differences of the declared models. Whether a laboratory pair of particles is described by one of the declared preparations is a tested identification with uncertainty under the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions, and agreement of these conditional numbers with data would test the combined model and apparatus assumptions rather than prove the model.