Continuity equation and local balance for the massive-model position density

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the kinematics and massive-particle model postulate P5 with its internal-factor postulate P6, read through The massive spin-s particle model of the framework: state space, operators, units and frame readings and Assigned energy observable, unitary flow and the two pictures of autonomous evolution (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). Assume Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $V_0\in L^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{R})$ be the real bounded measurable P5 potential and choose a bounded real measurable representative $V:{\mathbb{R}}^3\to \mathbb{R}$; on $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ use its restriction. Fix the P5 constants $\mathrm{\hslash }>0$ and $m>0$, a spin label $s$ with $r=2s+1$, a nonempty open spatial region $\mathrm{\Omega }$, and a nonempty open time interval $I\subseteq \mathbb{R}$. Let $\psi :\mathrm{\Omega }\times I\to {\mathbb{C}}^r$ have components ${\psi }_l$, ${\mathrm{\partial }}_t{\psi }_l$, ${\mathrm{\partial }}_j{\psi }_l$ and ${\mathrm{\partial }}_k{\mathrm{\partial }}_j{\psi }_l$ continuous on $\mathrm{\Omega }\times I$ for all $j,k\in \{1,2,3\}$, and suppose the model equation holds for Lebesgue almost every $(x,t)\in \mathrm{\Omega }\times I$: $i\mathrm{\hslash }{\mathrm{\partial }}_t{\psi }_l(x,t)=-\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\Delta }{\psi }_l(x,t)+V(x){\psi }_l(x,t)(l=1,\dots ,r)\mathrm{.}$ The displayed equation is the local differential expression assigned by P5/P4; it is a hypothesis here, and $\psi$ is not required to be a global P4 domain vector. P5 assigns $H_{\mathrm{o}\mathrm{p}}=-{\mathrm{\hslash }}^2\mathrm{\Delta }\mathrm{/}(2m)+V(Q)$ on the componentwise $W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r)$ domain, and P4 asserts the norm-differentiable equation for vectors in that domain. Existence of a smooth representative for such a global solution is a separate regularity hypothesis (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators clause (iv), Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution clause (iii)).

Define the squared-amplitude density and its associated current by $n(x,t):={\sum }_{l=1}^r\mathrm{\mid }{\psi }_l(x,t){\mathrm{\mid }}^2=\mathrm{\parallel }\psi (x,t){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2,j(x,t):=\frac{\mathrm{\hslash }}{m}{\sum }_{l=1}^r\mathrm{Im}(\overline{{\psi }_l(x,t)}\mathrm{\nabla }{\psi }_l(x,t))=-\frac{\mathrm{\hslash }}{m}\mathrm{Im}⟨\psi (x,t),\mathrm{\nabla }\psi (x,t){⟩}_{{\mathbb{C}}^r}\mathrm{.}$ For $\mathrm{\Omega }={\mathbb{R}}^3$, when $\psi (\cdot ,t)$ is a unit vector of $H_s$, $n(\cdot ,t)$ is the model's position probability density. Under the local hypotheses stated here, no global normalization or $H_s$ membership is assumed; $n$ is the squared-amplitude density and the same balance equations apply. Accordingly, $j$ is the probability current for a normalized state and the corresponding squared-amplitude current otherwise. The minus sign in the pairing expression follows from the convention that the $L^2$ pairing is linear in its first argument. For each component put $j_l=(\mathrm{\hslash }\mathrm{/}m)\mathrm{Im}(\overline{{\psi }_l}\mathrm{\nabla }{\psi }_l)$. Then:

(a) Pointwise continuity. For every $(x,t)\in \mathrm{\Omega }\times I$, ${\mathrm{\partial }}_{t}n(x,t)+\mathrm{div}j(x,t)=0.$

(b) Integrated balance. If $D\subseteq \mathrm{\Omega }$ is a bounded open set with $C^1$ boundary and $\bar{D}\subseteq \mathrm{\Omega }$ (Divergence on a bounded C1 Euclidean domain), then $t\mapsto {\int }_{D}n(x,t)dx$ is differentiable on $I$ with $\frac{d}{dt}{\int }_{D}n(x,t)dx=-{\int }_{\mathrm{\partial }D}j(x,t)\cdot \nu (x)dS,$ $\nu$ the outward unit normal.

(c) Global conservation under domination. If $\mathrm{\Omega }={\mathbb{R}}^3$, if for each $l$ there is a nonnegative measurable $g_l\in L^1({\mathbb{R}}^3)$ with $\mathrm{\mid }{\psi }_l(x,t){\mathrm{\mid }}^2\le g_l(x)$ for almost every $x$ and every $t\in I$, and if ${\lim }_{R\to \mathrm{\infty }}{\sup }_{t\in I}{\int }_{\{\mathrm{\mid }x\mathrm{\mid }=R\}}\mathrm{\mid }{j}_l(x,t)\mathrm{\mid }dS=0$ for each $l$, then $t\mapsto {\int }_{{\mathbb{R}}^3}n(x,t)dx$ is finite and constant on $I$. In particular, if this integral is $1$ at one time, it is $1$ throughout $I$ and $n(\cdot ,t)$ is a position probability density at every time.

(d) Flow conservation. For the model's assigned unitary flow $U(t)=\exp (-it{H}_{\mathrm{o}\mathrm{p}}\mathrm{/}\mathrm{\hslash })$ on the model's state space and every ${\psi }_0\in H_s$, one has $\mathrm{\parallel }U(t){\psi }_0\mathrm{\parallel }=\mathrm{\parallel }{\psi }_0\mathrm{\parallel }$ for every $t\in \mathbb{R}$; consequently the model's position probability of a unit preparation integrates to one at every time, without the regularity hypotheses of (a)-(c), because P4 assigns a unitary group (Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b)).

(e) Units, reading and scope. In the SI conventions of Dimensions, SI unit conventions and quantity types of the framework the position wave function has unit ${\mathrm{m}}^{-3\mathrm{/}2}$, the density $n$ has unit ${\mathrm{m}}^{-3}$ and the current $j$ has unit ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$; the display of (a) is the local balance of the model's squared-amplitude density in its smooth regime, and (b) reads the rate of change of its integral over a region as minus the outward current flux across its boundary. When $\psi$ is a normalized full-space state, these are the position probability density and probability flux, respectively. The statement does not assert that the flow of an arbitrary $L^2$ initial vector has a representative obeying the regularity hypotheses of (a)-(c), does not identify a recorded screen rate or any detection law with the surface flux (the detection model of this page is the declared effect and bin model), and records no empirical result.

• The smooth regime is an added hypothesis. The theorem concerns a componentwise $C^1$-in-time, $C^2$-in-space solution of the equation almost everywhere for the bounded measurable P5 potential. The model's assigned flow preserves every state norm by P4, but existence of a representative with the stated regularity is not derived here.
• Normalization controls the probability reading. The local equation and its global conservation conclusion concern the squared-amplitude density. Under the hypotheses of (c), normalization at one time makes it a probability density throughout the interval; (d) gives this conclusion directly for the unitary flow of a unit preparation.
• The current is not a detection rate. The current $j$ describes the local balance of the model's squared-amplitude density (or probability density when normalized); it is not by itself the recorded rate of a detector. The pair's recorded-outcome model assigns probabilities through the declared effects of the recorded detector.