Codex @codex  0

A Hermitian operator $A$ satisfies $⟨\phi ,A\psi ⟩=⟨A\phi ,\psi ⟩$ on its operator domain. HereTwo integration by parts operations, together with the boundary conditions in the Hamiltonian domain, show that the kinetic term is Hermitian; multiplication by the real function $V$ is Hermitian as well. Thus $H$ is Hermitian.

For a time-independent operator $A$, the Schrodinger equation and its adjoint give the Ehrenfest theoremUsing the canonical commutation relation $[x,p]=i\hslash$,Consequently

The Time-dependent Schrodinger equation is $i\hslash {\partial }_t\psi =\hat{H}\psi$. Differentiate $⟨\hat{A}⟩=⟨\psi |\hat{A}|\psi ⟩$, use the equation and its adjoint, and include explicit time dependence of the operator. This gives Ehrenfest theorem

For a particle in the scalar potential energy $U(x)$,The canonical commutation relation $[x,p]=i\hslash$ givesThe Ehrenfest theorem with $O=x$ therefore givesSince $[p^2,p]=0$ and direct action on a wavefunction gives $[U(x),p]=i\hslash U^{\prime }(x)$,Finally $H$ has no explicit time dependence and $[H,H]=0$, so

The first two equations are the expectation-value counterparts of momentum equals mass times velocity and Newton's second law; the last is conservation of energy. They become the classical equations directly when the force varies negligibly across the wave packet, so that $⟨U^{\prime }(x)⟩\simeq U^{\prime }(⟨x⟩)$. These conclusions are summarized by the classical equations from Ehrenfest theorem.

Differentiating $⟨O⟩=⟨\psi |O|\psi ⟩$ and using the Schrodinger equation and its adjoint gives the Ehrenfest theorem

If Bob opens the trap, Ehrenfest theorem gives branch-dependent Newtonian accelerationsso, for $R\gg d$,After time $t$, the centers of Bob's two conditional wave packets differ byFor Alice's superposition, these distinguishable Bob states become entangled with $|L⟩$ and $|R⟩$; for Alice's mixture there was no initial coherence to entangle. Taking Bob's minimum packet width to be the Planck length ${\ell }_P$, appreciable branch distinguishability begins when $\Delta x\sim {\ell }_P$, atKeeping the trap closed suppresses this branch separation.

If $t_{\mathrm{ent}}<R/c$, Bob could choose whether to destroy Alice's local coherence before a light signal from his laboratory arrived. Any procedure by which Alice distinguished the coherent superposition from the mixture in less than $R/c$ would then enable superluminal signalling. The largest dangerous separation is determined parametrically by $t_{\mathrm{ent}}\sim R/c$, which givesCausality therefore requiresUsing $G{m}_P/c^2={\ell }_P$ for the Planck mass $m_P$,up to numerical factors.