1. Introduction
Imaginary potentials have the effect that the time evolution defined by the Schrödinger equation is no longer unitary; rather, they lead to gain or loss of ∣ψ∣2 weight, depending on whether the potential is positive or negative imaginary. Such a loss is desirable to model absorption or detection of particles [1–5]. Here, we are interested in detection and consider two kinds of detectors: a ‘hard' detector that registers a particle as soon as it enters the detector volume, and a ‘soft' detector that takes some time to register the particle. Imaginary potentials are suitable as models of a soft detector, as discussed in particular by Allcock [4]. For example, for a single, non-relativistic quantum particle of mass m > 0 in 1 dimension with a soft detector in the region [0, ∞ ), we consider the Schrödinger equation
$\begin{eqnarray}{\rm{i}}{\hslash }\displaystyle \frac{\partial \psi }{\partial t}=-\tfrac{{{\hslash }}^{2}}{2m}\displaystyle \frac{{\partial }^{2}\psi }{\partial {x}^{2}}-{\rm{i}}\sigma \,{\rm{\Theta }}(x)\,\psi (x),\end{eqnarray}$
where σ > 0 is a constant (ℏ/2 times the detection rate) and Θ is the Heaviside function [i.e. Θ(x) = 1 for x ≥ 0 and 0 otherwise].Let us turn to hard detectors. The question of a detecting surface is a natural one. After all, the standard Born rule ρ(x) = ∣ψ(x)∣2 can be regarded as providing detection probabilities for detectors placed along the spacelike surface $t=\mathrm{const}.$, so the question arises, what happens if we place detectors along a timelike surface such as x1 = 0? The question had been considered early on; for example, Pauli [6], p. 60 famously had an argument that there is no self-adjoint operator for time, and Aharonov and Bohm [7] came up with a proposal for the distribution of the detection time at x1 = 0, but not a convincing one. One might expect that this distribution could be obtained as a limit of a soft detector, letting the parameter σ, representing the strength of the detector, tend to ∞ . However, Allcock [4] found that in this limit, for an initial wave function in 1d concentrated in the negative half axis, with probability 1 the detector never clicks, and ψt(x) = 0 at all x ≥ 0 and t ≥ 0—a situation reminiscent of the quantum Zeno paradox [4, 8–11]. Similar results were found for other non-Hermitian Hamiltonians [12, 13]. Allcock (prematurely) concluded that a hard detector was mathematically impossible. Misra and Sudarshan [9] even wrote: ‘A watched pot never boils'—an apparently paradoxical conclusion that goes against common sense.
But a successful model of a hard detector is possible. It is provided by the ‘absorbing boundary rule' [14, 15]. According to it, the wave function ${\psi }_{t}:(-\infty ,0]\to {\mathbb{C}}$ evolves according to the free Schrödinger equation
$\begin{eqnarray}{\rm{i}}{\hslash }\displaystyle \frac{\partial \psi }{\partial t}=-\tfrac{{{\hslash }}^{2}}{2m}\displaystyle \frac{{\partial }^{2}\psi }{\partial {x}^{2}},\end{eqnarray}$
supplemented by the ‘absorbing' boundary condition
$\begin{eqnarray}\displaystyle \frac{\partial {\psi }_{t}}{\partial x}(0)={\rm{i}}\kappa {\psi }_{t}(0),\end{eqnarray}$
It is of interest now to understand all aspects of the absorbing boundary rule more closely. In particular, we would like to understand how to obtain this boundary condition as a limit of imaginary potentials. That is the topic of the present paper. It describes a relationship between two natural mathematical descriptions of detectors (soft and hard detectors, that is, imaginary potentials and absorbing boundary conditions). The obvious way of making a soft detector harder, the limit σ → ∞ that we will henceforth call ‘Allcock's limit,' failed. But we describe a different, non-obvious limiting procedure in which the soft detector model (1 ) does approach the hard detector model given by the absorbing boundary rule (2 ), (3 ). Our derivations are not mathematically rigorous. The convergence occurs for wave functions as well as for the distribution of the detection time, specified in (9 ) in section 2 .
We proceed as follows. Both the Hamiltonian Hiσ of (1 ) with imaginary potential and the Hamiltonian Hiκ of (2 ), (3 ) with absorbing boundary condition are non-selfadjoint, and the time evolution operators ${W}_{t}=\exp (-{\rm{i}}{Ht}/{\hslash })$ they define are not unitary but are contractions (i.e. ∥Wtψ∥ ≤ ∥ψ∥). We compute the eigenfunctions and eigenvalues of the Hamiltonians. Since the Hamiltonians are not selfadjoint, either their eigenvalues are complex and not all real, or their eigenfunctions do not form a complete orthonormal system. We want to show that the eigenfunctions and eigenvalues of Hiσ approach those of Hiκ in a suitable limit. This limit involves one more modification of Hiσ: we make the detector volume a finite interval [0, L] and impose Neumann boundary conditions at L > 0
$\begin{eqnarray}\displaystyle \frac{\partial {\psi }_{t}}{\partial x}(L)=0.\end{eqnarray}$
The (non-selfadjoint) Hamiltonian in ${L}^{2}\left(\left(-\infty ,L\right]\right)$ defined by (1 ) and (4 ) will be denoted by Hiσ,L. We claim that
$\begin{eqnarray}{H}_{{\rm{i}}\sigma ,L}\to {H}^{{\rm{i}}\kappa },\end{eqnarray}$
in the $\begin{eqnarray}\begin{array}{l}^{\prime} \mathrm{hard}^{\prime} \,\mathrm{limit}\,\,\,L\to 0,\,\,\,\sigma \to \infty ,\\ \quad \sigma L\to \displaystyle \frac{{{\hslash }}^{2}\kappa }{2m}\gt 0,\end{array}\end{eqnarray}$
while keeping m (and ℏ) constant. Moreover, the distribution of the detection time converges to that of the absorbing boundary rule in the hard limit.A different way of approaching the absorbing boundary rule through a limit has been identified by Dubey, Bernardin, and Dhar [24]. Their consideration begins with a quantum particle moving on a lattice of width ϵ and repeated quantum measurements of the projection to Ω at time intervals of length τ, a setup similar to that of the quantum Zeno effect. The limit considered by Dubey, Bernardin, and Dhar involves first letting τ → 0 while assuming the transition amplitudes in the Hamiltonian between the lattice sites of ∂Ω and their neighbors in the interior of Ω diverge like τ−1/2, and then letting ϵ → 0 while rescaling time and ψ appropriately.
In section 2 , we give more detail about the models of hard and soft detectors. In section 3 , we compute their eigenvalues and eigenfunctions. In section 4 , we derive the limiting statement. In section 5 , we collect some remarks. In section 6 , we conclude.
2. Setup of equations
2.1. Imaginary potential
The Schrödinger equation (1 ) with imaginary potential leads to the continuity equation7 ), we may think of the Bohmian trajectory associated with it: it is the solutions t ↦ X(t) of the equation of motion dX/dt = j(X)/∣ψ(X)∣2 that has random initial condition X(0) with ∣ψ0∣2 distribution and ends at a random time with rate (2σ/ℏ)Θ(X(t)); that is, whenever the particle is in the detector volume, it has probability (2σ/ℏ)dt to disappear in the next dt seconds. We can think of this disappearance as an absorption due to detection. As a consequence, assuming ∥ψ0∥ = 1, the probability distribution of the time T and place X of detection (or, equivalently, of the end of the trajectory) is
$\begin{eqnarray}\displaystyle \frac{\partial | \psi {| }^{2}}{\partial t}=-\displaystyle \frac{\partial j}{\partial x}-\tfrac{2\sigma }{{\hslash }}{\rm{\Theta }}(x)\,| \psi (x){| }^{2},\end{eqnarray}$
with probability current $\begin{eqnarray}j=\tfrac{{\hslash }}{m}\mathrm{Im}\left[{\psi }^{* }\displaystyle \frac{\partial \psi }{\partial x}\right].\end{eqnarray}$
To visualize the physical meaning of ( $\begin{eqnarray}\begin{array}{rcl}\mathrm{Prob}\left({t}_{1}\leqslant T\leqslant {t}_{2},X\in B\right) & = & {\int }_{{t}_{1}}^{{t}_{2}}{\rm{d}}t\\ & & \times \,{\int }_{B}{\rm{d}}x\,\tfrac{2\sigma }{{\hslash }}| {\psi }_{t}(x){| }^{2},\end{array}\end{eqnarray}$
for any set $B\subseteq [0,\infty ),$ along with the probability $\begin{eqnarray}\mathrm{Prob}(T=\infty )=\mathop{\mathrm{lim}}\limits_{t\to \infty }{\int }_{{\mathbb{R}}}{\rm{d}}x\,| {\psi }_{t}(x){| }^{2},\end{eqnarray}$
that the particle never gets detected (as could happen, for example, if the particle wanders off to −∞ without ever entering the detector volume). The quantity ∣ψt(x)∣2 dx represents the probability that the particle is located in [x, x + dx] at time t (and, in particular, has not been absorbed yet). It follows that $\begin{eqnarray}\parallel {\psi }_{t}{\parallel }^{2}=\mathrm{Prob}(T\gt t)\end{eqnarray}$
is the ‘survival probability,' and since Prob(t < T < t + dt) = Prob(T > t) − Prob(T > t + dt), that the probability density of T is $\begin{eqnarray}{\rho }_{T}(t)=-\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}\parallel {\psi }_{t}{\parallel }^{2}.\end{eqnarray}$
Should the experiment be terminated at a time t before the detector clicks, then the collapsed wave function is ψt/∥ψt∥.2.2. Absorbing boundary rule
It is known [14, 25], that the system (2 )–(3 ) possesses a unique solution ψt(x) for every initial datum ${\psi }_{0}\in {L}^{2}\left((-\infty ,0]\right);$ we will assume ∥ψ0∥ = 1. The probability distribution of the random time T at which the detector clicks has density ρT(t) given by the probability current jt at x = 0, that is
$\begin{eqnarray}{\rho }_{T}(t)=\tfrac{{\hslash }}{m}\mathrm{Im}\left[{\psi }_{t}^{* }(0)\displaystyle \frac{\partial {\psi }_{t}}{\partial x}(0)\right].\end{eqnarray}$
By virtue of the boundary condition (3 ), this quantity can equivalently be expressed as
$\begin{eqnarray}{\rho }_{T}(t)=\tfrac{{\hslash }\kappa }{m}| {\psi }_{t}(0){| }^{2},\end{eqnarray}$
which is clearly non-negative, as a probability density must be; we see in particular that the current jt(0) is always pointing outward. Again, ∣ψt(x)∣2dx is the probability for the presence of the particle in [x, x + dx] at time t, and the distribution of T can equivalently be rewritten as $\begin{eqnarray}{\rho }_{T}(t)=-\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}\parallel {\psi }_{t}{\parallel }^{2}.\end{eqnarray}$
Moreover, again, should the experiment be terminated at a time t before the detector clicks, then the collapsed wave function is ψt/∥ψt∥.
Extensions of the absorbing boundary rule to moving detectors and to several particles are described in [26], and to the Dirac equation in [27]. Note also that the theory still works in the same way if we replace the boundary condition (3 ) by16 ).
$\begin{eqnarray}\displaystyle \frac{\partial {\psi }_{t}}{\partial x}(0)=(\sigma +{\rm{i}}\kappa ){\psi }_{t}(0),\end{eqnarray}$
with arbitrary $\sigma \in {\mathbb{R}}$ (and still κ > 0). The Hamiltonian Hσ+iκ is then defined as −(ℏ2/2m)∂2/∂x2 with boundary condition (3. Eigenvalues and eigenfunctions
3.1. Hamiltonian with imaginary potential
We aim at finding the eigenvalues and (non-normalizable) eigenfunctions of Hiσ,L defined by (1 ) and (4 ).
Focus first on x < 0 (‘region I'); being an eigenfunction of −(ℏ2/2m)∂2/∂x2 means to solve an ODE in x whose general solution has the form
$\begin{eqnarray}{f}_{{\rm{I}}}(x)={d}_{k}\,{{\rm{e}}}^{{\rm{i}}{kx}}+{c}_{k}\,{{\rm{e}}}^{-{\rm{i}}{kx}},\end{eqnarray}$
possibly with complex k. It seems plausible that, although Hiσ,L will not be self-adjoint, only real values of k are relevant to the eigenfunctions, as $\exp ({\rm{i}}{kx})$ with k > 0 then represents an incoming wave from the left. It follows that $\begin{eqnarray}E=\displaystyle \frac{{{\hslash }}^{2}{k}^{2}}{2m},\end{eqnarray}$
and we can choose without loss of generality that k > 0, so that $k=\sqrt{2{mE}}/{\hslash };$ we can and will also choose dk = 1, so $\begin{eqnarray}{f}_{{\rm{I}}}(x)={{\rm{e}}}^{{\rm{i}}{kx}}+{c}_{k}\,{{\rm{e}}}^{-{\rm{i}}{kx}}.\end{eqnarray}$
Let us turn to 0 < x < L (‘region II'). Any eigenfunction must then have the form
$\begin{eqnarray}{f}_{\mathrm{II}}(x)={a}_{k}\,{{\rm{e}}}^{{\rm{i}}\lambda x}+{b}_{k}\,{{\rm{e}}}^{-{\rm{i}}\lambda x},\end{eqnarray}$
with complex λ satisfying $\begin{eqnarray}{\lambda }^{2}={k}^{2}+{\rm{i}}\displaystyle \frac{2{mv}}{{{\hslash }}^{2}},\end{eqnarray}$
and, say, $\mathrm{Re}\,\lambda \gt 0$ (to define which of the two square roots is called λ and which −λ). The contribution that shrinks exponentially may seem plausible in view of the absorption taking place in region II; the contribution that grows exponentially may be thought of as reflected at L.It also seems plausible that the eigenfunctions should satisfy matching conditions4 ) at L implies that23 ), (24 ) together, we obtain that
$\begin{eqnarray}{f}_{{\rm{I}}}(0)={f}_{\mathrm{II}}(0)\quad \mathrm{and}\quad \frac{\partial {f}_{{\rm{I}}}}{\partial x}(0)=\frac{\partial {f}_{\mathrm{II}}}{\partial x}(0),\end{eqnarray}$
which imply that $\begin{eqnarray}1+{c}_{k}={a}_{k}+{b}_{k}\,\,\,\mathrm{and}\,\,\,k(1-{c}_{k})=\lambda ({a}_{k}-{b}_{k}).\end{eqnarray}$
The Neumann boundary condition ( $\begin{eqnarray}{b}_{k}={{\rm{e}}}^{{\rm{i}}2\lambda L}{a}_{k}.\end{eqnarray}$
From these three relations ( $\begin{eqnarray}{c}_{k}=\displaystyle \frac{(k-\lambda )+(k+\lambda ){{\rm{e}}}^{{\rm{i}}2\lambda L}}{(k+\lambda )+(k-\lambda ){{\rm{e}}}^{{\rm{i}}2\lambda L}}.\end{eqnarray}$
We also note for later use the explicit expression for ak, $\begin{eqnarray}{a}_{k}=\displaystyle \frac{2k}{(k+\lambda )+(k-\lambda ){{\rm{e}}}^{{\rm{i}}2\lambda L}}.\end{eqnarray}$
3.2. Hamiltonian with absorbing boundary condition
We now aim at finding the eigenvalues and eigenfunctions of Hσ+iκ, defined by the Schrödinger equation (2 ) and the absorbing boundary condition in the more general version (16 ).
Again, eigenfunctions must be of the form (17 ), and again, k should be real, without loss of generality positive, and dk can be taken to be 1, so the eigenfunction is again of the form (19 ) with real eigenvalue given by (18 ). This time, the coefficient ${c}_{k}\in {\mathbb{C}}$ must be chosen so that (16 ) is satisfied, i.e. ik(1 − ck) = (σ + iκ)(1 + ck) or
$\begin{eqnarray}{c}_{k}=\displaystyle \frac{k-\kappa +{\rm{i}}\sigma }{k+\kappa -{\rm{i}}\sigma }.\end{eqnarray}$
In this case, there is no region II.3.3. Reflection coefficient
The eigenfunctions f(x) contain a right-moving wave eikx (k > 0) coming from − ∞ and a reflected wave cke−ikx coming from the right boundary (at 0) and moving to the left. The absolute square of ck provides the reflection coefficient [28]27 ) this occurs for Hσ+iκ whenever
$\begin{eqnarray}{R}_{k}=| {c}_{k}{| }^{2},\end{eqnarray}$
or idealized probability of reflection at this value of k; the absorption coefficient is Ak = 1 − Rk. As discussed in [15], an absorbing boundary means that the particle gets absorbed there, but not necessarily (or not completely) the wave. Perfect absorption, Ak = 1, is reached when Rk = 0 or ck = 0, and by ( $\begin{eqnarray}k-\kappa +{\rm{i}}\sigma =0.\end{eqnarray}$
Since k is real, this situation can only occur when σ = 0, and that is why an ideal detector was assumed to have σ = 0 in [15].
4. Limiting cases
4.1. Allcock's limit
The simplest situation in which we can consider Allcock's limit has no right boundary, L = ∞ . The eigenvalues and eigenfunctions for this situation can actually be obtained from the formulas of section 3.1 in the limit L → ∞ . Fix σ > 0 and k > 0; since λ2 has phase between 0 and π/2 and thus λ between 0 and π/4, we know that λ has a positive imaginary part, with the consequence that as L → ∞ , eiλL → 0 and ei2λL → 0. Thus, bk → 0, so the exponentially growing contribution to fII disappears and, in the limit L → ∞
$\begin{eqnarray}{c}_{k}=\displaystyle \frac{k-\lambda }{k+\lambda }.\end{eqnarray}$
Now take Allcock's limit σ → ∞ . Since ∣λ2∣ → ∞ , also ∣λ∣ → ∞, so
$\begin{eqnarray}{c}_{k}\to -1.\end{eqnarray}$
In particular, the reflection coefficient is 1 and the absorption coefficient 0. As Allcock found, the probability that the particle ever gets detected (and thus absorbed) is 0.In fact, in the limit σ → ∞ also ak → 0, so fII(x) → 0 for every x > 0, so the probability of the particle ever entering the detector volume is 0. Since ck → −1
$\begin{eqnarray}{f}_{I}(x)\to {{\rm{e}}}^{{\rm{i}}{kx}}-{{\rm{e}}}^{-{\rm{i}}{kx}},\end{eqnarray}$
which are the eigenfunctions of the Schrödinger equation on ( −∞ , 0] with a Dirichlet boundary condition $\begin{eqnarray}{\psi }_{t}(0)=0.\end{eqnarray}$
4.2. Hard limit
We now consider the hard limit (6 ) and show that Hiσ,L → Hiκ with σ = 0 in the sense that the eigenfunctions and eigenvalues converge.
Our first claim is that in this limit, for any k > 034 ).
$\begin{eqnarray}\displaystyle \frac{1-{{\rm{e}}}^{{\rm{i}}2\lambda L}}{1+{{\rm{e}}}^{{\rm{i}}2\lambda L}}\lambda \to \kappa \gt 0.\end{eqnarray}$
Indeed, λ2 → i ∞ , λ → (1 + i) ∞ , λ2L → iκ, λL → 0, ei2λL → 1, and (1 − ei2λL)λ ≈ (−i2λL)λ = −i2λ2L → 2κ, which implies (Now (25 ) can be rewritten as34 ) it follows that27 ), the ck of the absorbing boundary condition (16 ) with σ = 0. Thus, fI converges to the fI of Hiκ.
$\begin{eqnarray}\begin{array}{rcl}{c}_{k} & = & \displaystyle \frac{(1+{{\rm{e}}}^{{\rm{i}}2\lambda L})k-(1-{{\rm{e}}}^{{\rm{i}}2\lambda L})\lambda }{(1+{{\rm{e}}}^{{\rm{i}}2\lambda L})k+(1-{{\rm{e}}}^{{\rm{i}}2\lambda L})\lambda }\\ & = & \displaystyle \frac{k-\tfrac{1-{{\rm{e}}}^{{\rm{i}}2\lambda L}}{1+{{\rm{e}}}^{{\rm{i}}2\lambda L}}\lambda }{k+\tfrac{1-{{\rm{e}}}^{{\rm{i}}2\lambda L}}{1+{{\rm{e}}}^{{\rm{i}}2\lambda L}}\lambda },\end{array}\end{eqnarray}$
and from ( $\begin{eqnarray}{c}_{k}\to \displaystyle \frac{k-\kappa }{k+\kappa },\end{eqnarray}$
which agrees with (In order to show that for any given k > 0, the eigenfunction of Hiσ,L converges to that of Hiκ, we need to verify that fII disappears in the hard limit. While the interval [0, L] shrinks to a point, it is not as obvious that26 ) and the relations mentioned between (34 ) and (35 )
$\begin{eqnarray}\parallel {f}_{\mathrm{II}}{\parallel }^{2}={\int }_{0}^{L}{\rm{d}}x\,| {f}_{\mathrm{II}}(x){| }^{2}\to 0.\end{eqnarray}$
To see that this is indeed the case, note that, by ( $\begin{eqnarray}\begin{array}{rcl}{a}_{k} & = & \displaystyle \frac{2k}{(1+{{\rm{e}}}^{{\rm{i}}2\lambda L})k+(1-{{\rm{e}}}^{{\rm{i}}2\lambda L})\lambda }\\ & & \to \displaystyle \frac{k}{k+\kappa },\end{array}\end{eqnarray}$
as well as, by (24 ), bk → k/(k + κ). Hence
$\begin{eqnarray}\parallel {b}_{k}{{\rm{e}}}^{-{\rm{i}}\lambda x}{\parallel }^{2}=| {b}_{k}{| }^{2}{\int }_{0}^{L}{\rm{d}}x\,| {{\rm{e}}}^{-{\rm{i}}\lambda x}{| }^{2}\end{eqnarray}$
$\begin{eqnarray}=| {b}_{k}{| }^{2}{\int }_{0}^{L}{\rm{d}}x\,{{\rm{e}}}^{2\mathrm{Im}\lambda x}\end{eqnarray}$
$\begin{eqnarray}=| {b}_{k}{| }^{2}{\int }_{0}^{L}{\rm{d}}x\,{{\rm{e}}}^{4{mvx}/{{\hslash }}^{2}}\end{eqnarray}$
$\begin{eqnarray}=| {b}_{k}{| }^{2}\,\displaystyle \frac{{{\rm{e}}}^{4{mvL}/{{\hslash }}^{2}}-1}{4{mv}/{{\hslash }}^{2}}\end{eqnarray}$
$\begin{eqnarray}\to 0,\end{eqnarray}$
since bk stays bounded, vL → ℏ2κ/2m stays bounded, and σ → ∞ . In a similar way, one can see that also ∥akeiλx∥ → 0, so that ∥fII∥ → 0, as claimed.We thus have that the eigenfunctions converge, while the eigenvalue is the same, viz., (18 ). That is, Hiσ,L → Hiκ in the hard limit. It is thus also plausible that $\exp (-{\rm{i}}{H}_{{\rm{i}}\sigma ,L}t/{\hslash })\to \exp (-{\rm{i}}{H}^{{\rm{i}}\kappa }t/{\hslash })$, and that ψt converges accordingly for every fixed t.
Our further claim that the distribution density ρT of the detection time T converges to that of the absorbing boundary rule is then suggested by the fact that in both settings (the imaginary potential and the absorbing boundary), ρT(t) = −d∥ψt∥2/dt, see (12 ) and (15 ).
5. Remarks
| 1. | 1. Higher dimension. Our analysis of the hard limit carries over directly to the case in which the detector volume is {(x1,…,xd): 0 < x1 < L}, the particle is restricted to x1 < L, and a Neumann boundary condition is imposed at x1 = L. The question then arises whether also the probability distribution of the detection place X (and the joint distribution of T and X) obtained from the imaginary potential model converges to that obtained from the absorbing boundary rule [15]. That this should be so is visible in the Bohmian picture: If ψt for the imaginary potential converges to ψt for the absorbing boundary, then also the Bohmian trajectories should converge. Since in the imaginary potential case, the particle can only be absorbed if X(t) > 0, in the limit the particle can only be absorbed when reaching x = 0; but then it must be absorbed since there is no Bohmian trajectory leading from the boundary to the left. So in the limit all trajectories must end exactly when they reach the boundary, so the distribution of the detection events must agree with the distribution of the arrival events [29], which coincides with the distribution ( |
| 2. | 2. General shapes in higher dimension. It seems plausible that the hard limit still yields the absorbing boundary rule for more general shapes of the detecting surface, as the limit L → 0 focuses on small length scales, on which a curved surface looks flat; also surfaces with edges (such as that of a cube) seem unproblematical since the probability current into the edges will be negligible. It also seems very plausible that the joint probability distribution of the detection time T and the detection location X approaches that of the absorbing boundary rule. |
| 3. | 3. Robin condition. The hard limit still agrees with the absorbing boundary rule if we replace the Neumann condition ( $\begin{eqnarray}\displaystyle \frac{\partial \psi }{\partial x}(L)=\alpha \,\psi (L),\end{eqnarray}$ with arbitrary constant $\alpha \in {\mathbb{R}}$, but not if we replace it with a Dirichlet condition ψ(L) = 0. That is because, for a Robin condition, the factor $\exp ({\rm{i}}2\lambda L)$ gets replaced by $\tfrac{{\rm{i}}\lambda -\alpha }{{\rm{i}}\lambda +\alpha }\exp ({\rm{i}}2\lambda L)$, and any limit involving σ → ∞ entails that λ → (1 + i) ∞ , so $\tfrac{{\rm{i}}\lambda -\alpha }{{\rm{i}}\lambda +\alpha }\to 1$, and the limiting behavior of ( |
| 4. | 4. Finite interval. Consider now a finite interval, which it will be convenient to take to be [ −ℓ, 0], with the absorbing boundary condition ( $\begin{eqnarray}\displaystyle \frac{k-\kappa +{\rm{i}}\sigma }{k+\kappa -{\rm{i}}\sigma }=-{{\rm{e}}}^{-{\rm{i}}2k{\ell }},\end{eqnarray}$ which restricts the possible k values to a discrete set and forces them to become complex, resulting in complex eigenvalues ( $\begin{eqnarray}\begin{array}{rcl}\parallel {f}_{t}{\parallel }^{2} & = & {\parallel \exp (-{\rm{i}}Ht/\hslash )f\parallel }^{2}\\ & = & {\parallel \exp (-{\rm{i}}\omega t/\hslash )f\parallel }^{2}\\ & = & \exp (-2\mu t/\hslash )\,\parallel f{\parallel }^{2},\end{array}\end{eqnarray}$ shrinks with time. I expect that also in this situation the Hamiltonian with imaginary potential converges in the hard limit ( |
6. Conclusions
The question of how to mathematically represent a detecting surface has long been mysterious, and attempts to define it through a limit of imaginary potentials originally failed. Now that such a mathematical representation is available in the form of an absorbing boundary condition, we have identified in this paper a particular limit of an imaginary potential that leads to this boundary condition: let the thickness L of a detecting layer tend to 0 and the strength σ of the imaginary potential in this layer to infinity in such a way that the product vL remains finite, and impose a Neumann or Robin boundary condition on the back wall of the layer. We have derived (non-rigorously) that in this limit, the Hamiltonian Hiσ,L with this imaginary potential converges to the Hamiltonian Hiκ with an absorbing boundary condition, in the sense that the eigenvalues and eigenfunctions of Hiσ,L converge to those of Hiκ. The fact that neither Hiσ,L nor Hiκ is Hermitian means that both allow the particle to be absorbed.