A finite-time Gaussian pointer

D A finite-time Gaussian pointer

A simple effective measurement model illustrates exact propagation and controlled record overlap. Let a two-level label $Z$ have eigenvalues $\pm1$, and let a scalar pointer $y$ of mass $M$ have

with real smooth $F$ on a finite interval. This is a specified effective apparatus coupling; its unbounded linear profile is not being identified with the compact preparation controls in theorem 6.1.

Define

If $\chi_{\rm free}$ is the free evolution of $\chi_0$, direct substitution gives the two exact packets

Their position densities are Gaussians of variance $s(t)^2$, centered at $\pm d(t)$. This formula supplies a unitary finite-time propagator by translations, phases, and free evolution, preserving Schwartz states.

For the initial spinor $c_+|+\rangle+c_-|-\rangle$, put $w_\pm=|c_\pm|^2$, with $w_++w_-=1$. The joint norm density is $r=w_+|\chi_+|^2+w_-|\chi_-|^2>0$. The branch velocities are

The actual velocity is their local density-weighted mean. With $a=\dot s/s$,

For an explicit derivative bound, put $\lambda=w_+|\chi_+|^2/r$. Then

The formulas also hold at $w_+=0$ or $w_-=0$ by the constant-weight limits. Since $s\ge\sigma>0$, this derivative and the linear-growth coefficients are bounded on every fixed finite interval. The guidance equation therefore has a complete all-point flow on that interval. To avoid a collision with the force notation, write its cumulative density as $\mathcal F_t$; the continuity equation and zero flux at infinity give $\mathcal F_t(Y_t)=\mathcal F_0(Y_0)$.

At a readout time with $d(T)>0$, take $y>0$ to record $+$. If $\Phi_{\rm G}$ denotes the standard normal distribution function, set

Joint equilibrium gives

The packets need not have disjoint support. The error is explicitly controlled, while both the wavefunction evolution and the guidance calculation are exact.