Quantum wall dynamics: boundary choice, reflected packets and impulse
Result, 2026-10-27. For the specified incident Gaussian and free Dirichlet half-line dynamics below, the nonnegative boundary-force measures converge weakly on the whole time axis: \[ \frac{\hbar^2}{2m}\,|\partial_q\psi_{\hbar,D}(0,t)|^2\,dt \;\Longrightarrow\;2|p_0|\,\delta_{t_*}, \qquad t_*=-\frac{mq_0}{p_0}>0.\tag{1} \] This includes the impact layer excluded from a uniform off-impact momentum-current estimate. In fact, on the \(\sqrt\hbar\) time scale the force has an explicit Gaussian profile and the momentum current tends to an error-function transition.
For the corresponding Neumann half-line packet, the derivative in (1) vanishes identically. The complete boundary stress has another term; it has the same limiting impulse and the same leading impact-layer profile. The particle’s classical bounce therefore does not determine whether the quantum wave satisfies Dirichlet or Neumann boundary conditions. General Robin conditions are discussed through their scattering phases; no wavepacket limit is claimed here for arbitrary Robin parameters.
The boundary contribution to Ehrenfest’s theorem is established literature: see De Vincenzo (2013), equations (38)–(39) for a wall on the opposite half-line (passage read), and Albrecht et al. (2023) (abstract) on boundary momentum and bouncing packets. A particularly close precedent is Belloni–Doncheski–Robinett (2004), equations (8)–(11), (16)–(19): they normalize the same odd-image Gaussian on the opposite half-line and analyze its near-collision position and force (relevant passages read). The computation below gives a particular weak impulse measure and compares the Neumann stress; it does not claim priority for the reflected Gaussian or wall forces.
Classical impact and quantum boundary data
Take the half-line \(q\ge0\) and a free particle starting at \(q_0>0\) with \(p_0<0\). For an ideal elastic wall, the prescribed impact law is \(p(t_*+)=-p(t_* -)\) at \(t_*=-mq_0/p_0\). The motion is \[q(t)=|q_0+p_0t/m|,\qquad p(t)=p_0\operatorname{sgn}(q_0+p_0t/m)\quad(t\ne t_*).\] Position is continuous; momentum has a jump. A smooth-force equation with a finite smooth force does not describe that jump. Nonpenetration alone does not select the elastic law: a restitution coefficient or another contact rule would be additional data.
For a smooth wall in several dimensions, the elastic specular rule reverses the normal component and retains the tangential component: \(v_+=v_--2(v_-\cdot n)n\). This requires a specified unit normal and a well-defined collision. Corners, grazing contacts and an accumulation of impacts require separate analysis; one should not infer their uniqueness merely from the local smooth-force theorem.
A mathematically defined quantum impenetrable wall can be represented by the Dirichlet Laplacian \[H_{\hbar,D}=-\frac{\hbar^2}{2m}\frac{d^2}{dq^2}, \qquad D(H_{\hbar,D})=H^2(0,\infty)\cap H^1_0(0,\infty).\] It is self-adjoint and gives a unitary group. Its Euclidean kernel is \[ K_D(T;x,y)=k_{\hbar,T}(x-y)-k_{\hbar,T}(x+y), \quad k_{\hbar,T}(r)=\sqrt{\frac m{2\pi\hbar T}} e^{-mr^2/(2\hbar T)},\quad x,y>0. \] This is the method of images: the Euclidean reference process is Brownian motion killed at the boundary. The quantum real-time wall reflects, but killed Euclidean paths should not themselves be read as real-time bouncing trajectories. The reflected-packet estimate below proves a quantitative coherent-packet position limit through the bounce.
A subtle domain issue is important: the symmetric differential operator \(-i\hbar\,d/dq\) on the half-line with Dirichlet boundary has unequal deficiency indices and no self-adjoint extension on \(L^2(0,\infty)\). The Dirichlet Hamiltonian is nevertheless self-adjoint. A current or a quadratic-form expectation involving \(-i\hbar d/dq\) can be studied, but it should not be presented as the expectation of a self-adjoint half-line momentum operator. See Al-Hashimi and Wiese, arXiv:2103.01715 (abstract) for the boundary momentum problem.
1. The packet and its boundary value
Fix \(m,\omega,q_0>0\) and \(p_0<0\), with \(v=p_0/m\). On the full line take the normalized coherent state \[ g_{\hbar,0}(q)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp\!\left[-\frac{m\omega(q-q_0)^2}{2\hbar} +\frac{i p_0(q-q_0)}{\hbar}\right]. \] Let \(g_{\hbar,t}=e^{-itP^2/(2m\hbar)}g_{\hbar,0}\), \(a(t)=q_0+vt\), \(A(t)=1+\omega^2t^2\), and \[ \gamma_\hbar=\langle g_{\hbar,0},Rg_{\hbar,0}\rangle =\exp\!\left[-\frac{m\omega q_0^2+p_0^2/(m\omega)}{\hbar}\right], \qquad (Rg)(q)=g(-q).\tag{2} \] The odd and even extensions restrict to the normalized half-line states \[ \psi_{\hbar,D}(q,t) =\frac{g_{\hbar,t}(q)-g_{\hbar,t}(-q)}{\sqrt{1-\gamma_\hbar}}, \qquad \psi_{\hbar,N}(q,t) =\frac{g_{\hbar,t}(q)+g_{\hbar,t}(-q)}{\sqrt{1+\gamma_\hbar}}, \quad q>0.\tag{3} \] They solve the free Schrödinger equation with Dirichlet and Neumann conditions, respectively. Their regularity and decay justify the boundary calculations below.
The free Gaussian formula gives \[ |g_{\hbar,t}(0)|^2 =\sqrt{\frac{m\omega}{\pi\hbar A(t)}} \exp\!\left[-\frac{m\omega a(t)^2}{\hbar A(t)}\right],\qquad \hbar\,\frac{g'_{\hbar,t}(0)}{g_{\hbar,t}(0)} =D(t):=\frac{m\omega q_0+i p_0}{1+i\omega t}.\tag{4} \] The simplification in \(D\) follows from \(a=q_0+p_0t/m\); \(g_{\hbar,t}(0)\) is never zero. Define \(B=(m\omega q_0)^2+p_0^2\). Since \(\psi'_{\hbar,D}(0,t)=2g'_{\hbar,t}(0)/\sqrt{1-\gamma_\hbar}\), the boundary-force density is exactly \[ F_{\hbar,D}(t) =\frac{2B}{m(1-\gamma_\hbar)A(t)} \sqrt{\frac{m\omega}{\pi\hbar A(t)}} \exp\!\left[-\frac{m\omega a(t)^2}{\hbar A(t)}\right].\tag{5} \]
Position through impact and current away from impact
Write \(a(t)=q_0+p_0t/m\), and let \(g_{\hbar,t}\) be the freely evolved Gaussian. Its position variance is \[\sigma_q(t)^2=\frac{\hbar}{2m\omega}(1+\omega^2t^2).\tag{P21}\] Because \(||x|-|a||\le|x-a|\) and reflection commutes with the free evolution, \[ \left|\langle Q\rangle_{\psi_\hbar(t)}-|a(t)|\right| \le\sqrt{\frac{2}{1-\gamma_\hbar}}\,\sigma_q(t).\tag{P22} \] Indeed the half-line position expectation equals the full-line expectation of \(|X|\) in the normalized full-line odd state \(\phi_\hbar=(g_\hbar-Rg_\hbar)/\sqrt{2(1-\gamma_\hbar)}\); apply the triangle inequality to the two Gaussian terms. Thus position converges uniformly on every bounded time interval, including the impact time, to the elastic classical folded trajectory.
Away from the impact, the corresponding probability-current momentum also converges to the incoming or reflected value. More precisely, if \(|a(t)|\ge\eta>0\), with \(\sigma_p=\sqrt{\hbar m\omega/2}\), \[ \left|\int_0^\infty\overline{\psi_\hbar} (-i\hbar\partial_q)\psi_\hbar\,dq -p_0\operatorname{sgn}a(t)\right| \le\sqrt{\frac2{1-\gamma_\hbar}} \left(\sigma_p+\frac{2|p_0|\sigma_q(t)}{\eta}\right).\tag{P23} \] For the proof, the half-line integral is the full-line expectation of \(\operatorname{sgn}(X)P\) in the normalized full-line odd state \(\phi_\hbar=(g_\hbar-Rg_\hbar)/\sqrt{2(1-\gamma_\hbar)}\). On the Gaussian, \(\|(\operatorname{sgn}X\,P-\operatorname{sgn}a\,p_0)g_t\|\) is at most \(\sigma_p+2|p_0|\mathbb P(aX<0)^{1/2}\); Chebyshev bounds the last probability by \(\sigma_q(t)^2/a(t)^2\). The reflected Gaussian obeys the same estimate. For each fixed \(T,\eta>0\) independent of \(\hbar\), (P23) proves uniform convergence over \(|t|\le T\) with \(|a(t)|\ge\eta\). It gives no uniform bound in an \(\hbar\)-dependent impact layer \(|a(t)|\sim\sqrt\hbar\) or momentum value at \(t_*\). The expression in (P23) is a real current form on this domain, not an expectation of a self-adjoint half-line momentum.
Position converges uniformly through impact; current cannot converge uniformly to a discontinuous momentum jump. The next calculation resolves that layer in the weak measure topology.
2. The pulse and the limiting impulse
Put \(A_*=A(t_*)\) and \[\Lambda=|v|\sqrt{\frac{m\omega}{A_*}}>0.\tag{6}\] Because \(m\omega q_0=-p_0\omega t_*\), one has \(B=p_0^2A_*\). At every fixed \(s\in\mathbb R\), (5) gives \[ \sqrt\hbar\,F_{\hbar,D}(t_*+\sqrt\hbar\,s) \longrightarrow \frac{2p_0^2}{m}\sqrt{\frac{m\omega}{\pi A_*}}\, e^{-\Lambda^2s^2}.\tag{7} \] The integral of the right side is \(2|p_0|\).
This pointwise calculation proves (1) with the following tail control. For any fixed \(\delta>0\), \[ c_\delta:=\inf_{|t-t_*|\ge\delta}\frac{m\omega a(t)^2}{A(t)}>0: \] the expression is continuous, has only the one zero \(t_*\), and tends to \(mv^2/\omega>0\) at both infinities. Equation (5) bounds the total force outside this neighbourhood by \(C_\delta\hbar^{-1/2}e^{-c_\delta/\hbar}\), since \(A(t)^{-3/2}\) is integrable. Inside the neighbourhood, after \(t=t_*+\sqrt\hbar s\), the profiles have a common integrable Gaussian majorant. Dominated convergence proves convergence against every bounded continuous test function and gives the total mass \(2|p_0|\). This is weak convergence of measures, not total-variation convergence to an atom.
The real current-momentum form is \[J_\hbar(t)=\hbar\operatorname{Im} \int_0^\infty\overline{\psi_\hbar}\,\partial_q\psi_\hbar\,dq.\tag{8}\] The differential operator \(-i\hbar\partial_q\) on a half-line is not a self-adjoint momentum operator; (8) is a real current form. For a sufficiently regular free half-line wave, \[ \frac{dJ_\hbar}{dt} =\frac{\hbar^2}{2m}\left( |\psi_\hbar'(0,t)|^2 -\operatorname{Re}[\overline{\psi_\hbar(0,t)} \psi_\hbar''(0,t)]\right).\tag{9} \] To see this, differentiate (8), insert \(\partial_t\psi=i\hbar\psi''/(2m)\), and integrate by parts. For Dirichlet the second term vanishes, so \(dJ_{\hbar,D}/dt=F_{\hbar,D}\). Using the off-impact current estimate above together with (7), \[ J_{\hbar,D}(t_*+\sqrt\hbar\,s) \longrightarrow |p_0|\,\operatorname{erf}(\Lambda s).\tag{10} \] The incoming value is \(p_0=-|p_0|\), the outgoing value is \(|p_0|\), and their difference is the mass in (1). The transition is continuous for each \(\hbar>0\) and becomes discontinuous only in the limit.
3. Neumann and Robin boundary information
For \(\psi_{\hbar,N}'(0,t)=0\), the expression in (1) is identically zero. Equation (9), rather than the Dirichlet-only term, is the appropriate boundary stress. Since \[ \frac{g_{\hbar,t}''(0)}{g_{\hbar,t}(0)} =\frac{D(t)^2}{\hbar^2} -\frac{m\omega}{\hbar(1+i\omega t)}, \] it is explicitly \[ F_{\hbar,N}(t)=\frac{2|g_{\hbar,t}(0)|^2}{m(1+\gamma_\hbar)} \left[-\operatorname{Re}D(t)^2+\frac{\hbar m\omega}{A(t)}\right]. \tag{11} \] This density need not be pointwise nonnegative. At the impact, \(D(t_*)=ip_0\), and therefore the rescaled Neumann density has the same limit (7). Moreover \[ |F_{\hbar,N}(t)| \le\frac{2|g_{\hbar,t}(0)|^2}{m(1+\gamma_\hbar)} \left[\frac{B}{A(t)}+\frac{\hbar m\omega}{A(t)}\right], \] so the same argument after (7) controls the total variation outside every impact neighbourhood. Consequently the finite signed measures \(F_{\hbar,N}(t)dt\) converge weakly to the same \(2|p_0|\delta_{t_*}\). Their impact-layer current has the same limit as (10). The even-reflection argument also gives the same folded classical position \(|a(t)|\). Thus the limiting position and impulse do not recover the Dirichlet/Neumann reflection phase.
All self-adjoint realizations of the free half-line Laplacian use Dirichlet or a real Robin condition \(\psi'(0)=\kappa\psi(0)\), including Neumann at \(\kappa=0\). For an incoming stationary wave \(e^{-ikq}+r_\kappa(k)e^{ikq}\), \[ r_\kappa(k)=\frac{ik+\kappa}{ik-\kappa},\qquad |r_\kappa(k)|=1,\quad r_0=1,\quad r_D=-1.\tag{12} \] At fixed real \(\kappa\) and fixed nonzero physical momentum, \(k=|p|/\hbar\to\infty\) and \(r_\kappa(k)\to1\). This is a stationary scattering comparison, not a proof of the impulse limit for every Robin packet. For \(\kappa<0\) there is additionally a bound state \(\sqrt{-2\kappa}e^{\kappa q}\) of energy \(-\hbar^2\kappa^2/(2m)\), showing another quantum distinction that the free incoming bounce does not encode.
4. Consequence for STATE
For a specified coherent packet, the Dirichlet force is a positive boundary measure converging to Newton’s impulsive momentum jump; the full Neumann boundary stress has the same limit. A classical bounce does not identify the boundary condition or reflection phase. This constructs neither a general boundary-groupoid theorem nor a positive-action-necessity result.