Arrival time from Hamiltonian with non-hermitian boundary term

We develop a new method for ﬁnding the quantum probability density of arrival at the detector. The evolution of the quantum state restricted to the region outside of the detector is described by a restricted Hamiltonian that contains a non-hermitian boundary term. The non-hermitian term is shown to be proportional to the ﬂux of the probability current operator through the boundary, which implies that the arrival probability density is equal to the ﬂux of the probability current.


I. INTRODUCTION
Consider a quantum particle described by a spatially extended wave packet impinging on the detector region D. Since different parts of the packet approach D at different times, there is an inherent quantum uncertainty about the time at which the particle arrival to D will be detected.The arrival time problem is to make a theoretical prediction for the probability distribution P arr (t) that the arrival will be detected at time t.Remarkably, there are many different theoretical approaches to that problem that make different measurable predictions, for reviews see [1][2][3][4], and it is not clear, neither theoretically nor experimentally, which approach is correct.
In general, quantum mechanics makes unambiguous probabilistic predictions for various phenomena, so why the arrival time problem is a problem at all?The point is that quantum mechanics makes unambiguous probabilistic predictions for measurements of observables represented by self-adjoint operators, while time, in the usual formulation of quantum mechanics, is not an observable in that sense.Time is a classical parameter, not a quantum operator, so, by starting from general axioms of quantum theory, it is not immediately clear how to make quantum probabilistic predictions associated with measurement of time.In particular, one does not know at which time a quantum event, such as a particle detection, will happen, so one wants to use quantum mechanics to compute a probability that the event will happen at a given time.The problem is, how to compute this probability, when the time is not an operator?The arrival time problem is the simplest version of this problem, where the quantum event is taken to be the particle arrival to the detector, or more operationally, a click in the detector which happens when the particle arrives.
One class of possibilities (see e.g.[1,4] and references therein) is to reformulate quantum mechanics such that time is treated as an operator, the problem with such approaches is that they may require a radical reformulation of general principles of quantum mechanics, which makes them rather controversial.There are also axiomatic ap- * Electronic address: tjuric@irb.hr† Electronic address: hnikolic@irb.hrproaches by Kijowski and others (see e.g.[1,4] and references therein) that postulate axioms for the arrival time distribution, the problem is that those axioms seem somewhat ad hoc because they cannot be derived from standard axioms of quanum mechanics.Another class of possibilities (see e.g.[4] and references therein) are semi-classical approaches, the problem is that they also seem too ad hoc and lack a deeper understanding of the problem.
Yet another class [5][6][7][8][9][10][11][12] of approaches to the arrival time problem predicts that P arr (t) is given by the flux of the probability current.Within this class, some approaches are based on standard quantum mechanics (QM) [5][6][7][8], while others are based on Bohmian formulation of QM in terms of particle trajectories [9][10][11][12].In this paper we present one new approach to the arrival time problem, based on standard QM, which confirms that the arrival time distribution is given by the flux of the probability current.Given that it is not generally accepted in the community that the arrival time distribution should be given by the flux of the probability current, we believe that it is valuable to present one more independent theoretical evidence that it is indeed so.
The approach in this paper is partially inspired by the approach in [8], but is motivated with a goal to avoid some mathematical subtleties that appeared in that work.The approach in [8], which arose from the development of earlier ideas in [13,14], is based on time evolution governed by a projected Hamiltonian H = πH π, where π is the projector to the region D defined as the complement of the detector region D. The mathematical subtleties appear because, in the position representation, π is represented by a characteristic function with a discontinuity at the boundary of D, which leads to ambiguities when the second-derivative operator appearing in H acts on a function with a discontinuity.The goal of this paper is to develop a formalism based on an alternative definition of H that, at the same time, captures the same physics as H in [8,13,14], but uses a different mathematical definition of H so that the mathematical difficulties appearing in [8,14] are avoided.
The paper is organized as follows.In Sec.II we start by defining the notion of restricted wave function, which in D coincides with the full wave function, but vanishes outside of D where the full wave function, in general, does not vanish.This implies that the norm of the restricted wave function, in general, is not conserved in time.Then in Sec.III we define the restricted Hamiltonian H that governs the evolution of the restricted wave function.It turns out that the restricted Hamiltonian is non-trivial at the boundary of D, because it has a non-hermitian boundary term proportional to the flux of the probability current operator, which accounts for non-conservation of the norm of the restricted wave function.In Sec.IV we explain how this non-conservation of the norm implies that the arrival time distribution is equal to the flux of the probability current.The conclusions are drawn in Sec.V, and in the Appendix an alternative derivation of the non-hermitian part is presented.

II. RESTRICTED WAVE FUNCTION
Let us start with an elementary warmup needed to establish the notation.Consider a particle moving in a 3dimensional space R 3 .It is described by a wave function where |ψ(t) is the state in the Hilbert space H denoted in the Dirac's "bra-ket" formalism.The |ψ(t) and ψ(x, t) satisfy the respective Schrödinger equations where H is an abstract operator, while Ĥ is its coordinate representation given by a concrete derivative operator and we work in units = 1.The relation between H and Ĥ can be expressed as Now, after this warmup, let us divide the full space R 3 into a detector region D and its complement D, so that D ∪ D = R 3 , D ∩ D = ∅.Physically, such a division is motivated with the goal to study the arrival of the particle to the detector.We define the restricted wave function ψ D(x, t) as It can be expressed as where is the projector to D. Thus we see that the restricted wave function can also be expressed as where Since ψ D (x, t) coincides with ψ(x, t) in D, but vanishes outside of D, we have In fact, since we assume that ψ(x, t) is a travelling wave packet (rather than a stationary state), one expects that the left-hand side depends on time t, despite the fact that the right-hand side is time independent.This means that the norm of ψ D(x, t) is expected to change with time, i.e., that the norm ψ The restricted wave function can also be interpreted in terms of wave function collapse.When a part of the wave function enters the detector region D, there is a non-zero probability that the detector will detect the particle, i.e., that the wave function will collapse to the region D. But there is also a probability that the detector will not detect the particle, in which case we know that the particle is still outside of the detector region D, so the wave function collapses to the region D. Thus the restricted wave function can be interpreted as the collapsed wave function, corresponding to a negative outcome of measurement by the detector.For more details of such an interpretation see [8].
From a theoretical point of view, the exact specification of the detector region D is in our analysis still somewhat ambiguous.In principle, one could specify it by a more detailed model.But in practice, we believe that an experimental approach would be more fruitful.One could take an actual detector and impinge on it particles with wave functions which are very narrow in the position space, so that particles are effectively "classical" in the sense that their arrival time can be predicted as a classical deterministic event.In this way, one can determine the relevant detector region D experimentally, for the specific detector at hand.After that, once D is known, one can make non-trivial theoretical analysis with "truly quantum" non-narrow wave functions.

III. RESTRICTED HAMILTONIAN
The full state evolves with time as |ψ(t) = e −iHt |ψ(0) .This evolution is unitary because the Hamiltonian H is hermitian.On the other hand, since one expects that the evolution of the restricted state |ψ D(t) is not unitary, its evolution can be described as where H is expected to be some non-hermitian operator.
Our goal is to find an explicit expression for H.
Heuristically, since ψ D (x, t) and ψ(x, t) coincide for x ∈ D, the coordinate representation Ĥ must coincide with the derivative operator Ĥ in (3) for x ∈ D. Likewise, since ψ D(x, t) = 0 for x / ∈ D, the operator Ĥ can be taken as the trivial zero operator for x / ∈ D. However, a particular care should be taken about definition of Ĥ at the boundary of D, which is the only place where subtleties in the definition of Ĥ may be expected.For that purpose we find more convenient to work with the abstract H operator, rather than its coordinate representation Ĥ.Thus, since the arbitrary matrix element of H is we postulate that the arbitrary matrix element of H is which has the same form as (12), except that the integration region is restricted from R 3 to D. Hence we refer to H as the restricted Hamiltonian.The goal now is to find the explicit operator representation of H, analogous to H in (3).We first write (13) as where ) Hence partial integration and the Gauss theorem give where ∂ D is the boundary of D and dS is the area element directed outwards from D. Next we use the identities implying that ( 16) can be written as Since π commutes with V , the last term in (18) can also be written in alternative forms thus we see that H can be written in the operator form The hermitian conjugation gives thus we see that the first and the last term are hermitian operators, but that the middle term is not.This shows that the restricted Hamiltonian H is not a hermitian operator, owing to the boundary term.
To better isolate the source of non-hermiticity it is useful to write H as which is convenient because the first term is manifestly hermitian and the second term manifestly anti-hermitian.From ( 20) and ( 21) we see that the two terms in ( 22) can be written as where are hermitian operators, [A, B] = AB − BA denotes a commutator, and {A, B} = AB + BA denotes an anticommutator.Thus (22) can be written in the final form The first term (namely, the term in square brackets) is hermitian and does not depend on the boundary.It is non-negative, provided that V is non-negative.The second term (namely, the term involving K) is a hermitian, but not non-negative, boundary term.The last term (namely, the term involving J) is an anti-hermitian boundary term.Eq. ( 25) is the main new result of this paper, so let us discuss its significance qualitatively.While the full Hamiltonian (3) describes evolution of the full wave function everywhere in the full 3-dimensional space, (25) is the restricted Hamiltonian describing evolution of the restricted wave function, namely the part of wave function defined only on the 3-dimensional region D. The π is the projector to the region D, so the term in square brackets in ( 25) is just the projected version of (3).The projector π commutes with V = V (x), but does not commute with p. Hence the potential energy term πV π can also be written as πV or V π, but the kinetic energy term proportional to pπp must we written in that form, and not e.g. as πpπpπ or pπpπ.The commutator [π, p], in the x-representation, is proportional to a Dirac δ-function on the boundary of D, so replacing pπp with πpπpπ or pπpπ would produce spurious boundary terms.In (25) all boundary terms are represented explicitly and unambiguously, without δ-functions, as surface integrals over the boundary ∂ D of D. The most important feature of the boundary term is the fact that it contains an antihermitian part involving J, the physical significance of which we discuss in more detail in the rest of the paper.
Note that the mean value of J is (see also [15]) which is the standard probability current in quantum mechanics.For that reason, we refer to J as the probability current operator.
As we said, the fact that H in ( 11) is not hermitian implies that the norm is not conserved in time.Since [18] ∂ where in the last equality we used (23), we see that ( 27) implies Note that the wave functions in ( 16) are full wave functions, not restricted wave functions.The full wave functions are assumed to be twice differentiable at the boundary of D, because only in this case the volume integral can be turned into the surface integral via the Gauss theorem.On the other hand, the current j D in (29) is expressed in terms of the restricted wave function ψ D, which has a discontinuity at D, implying that it is not differentiable.To avoid this apparent inconsistency, we must be more careful in specifying what we mean by integral over the "boundary".This really means that the surface of integration ∂ D is put infinitesimally away from the boundary towards the interior of D, where ψ D coincides with ψ.The consequence is that j D in (29) coincides with j defined by (26), implying that (29) can finally be written as This shows that the rate of change of norm of the state restricted to the region D is given by the flux of the probability current through the boundary of D.
Physically, the most important consequence of evolution governed by the restricted Hamiltonian (25) with an anti-hermitian boundary term is the change of norm of the restricted wave function, as described by (30).The result (30) is rather intuitive, it can be visualized as a leak of wave function from the region D, where the flux of the probability current quantifies how much of the wave function leaks through the boundary of D.

IV. ARRIVAL TIME DISTRIBUTION
Suppose that at the initial time t = 0, the particle is out of the detector region D. This means that i.e. the initial full state is equal to the initial state restricted to D. Then (27) is the probability P (t) that, at time t, the particle is in D Hence the probability that the particle is in the detector region D is Now suppose that, during a time interval [0, T ], the probability P (t) increases with time.Then there is a positive function P(t) such that for any t ∈ [0, T ].This, together with (33), implies Using (32) and (30), this finally gives Mathematically, the final formula (36) is rather compact and general.The same formula has also been obtained in [8] by different methods, while here we derived it through the use of the restricted Hamiltonian (25) with the anti-hermitian boundary term.
Since P (t) is a probability, it follows that P(t) in ( 34) is a probability density.In other words, P(t) is a probability distribution.But a probability distribution of what?We shall present two independent arguments, one heuristic and the other more rigorous, that P(t) is the probability distribution of arrival times to the detector.
For a heuristic argument, consider first an analogous quantum equation for spatial distributions.In a formula of the form P = d 3 x |ψ(x)| 2 , the quantity |ψ(x)| 2 is the probability density that the particle will appear at the position x, rather than at any other position x ′ .By analogy, P(t) in (34) is the probability density that the particle will appear at the time t, rather than at any other time t ′ .More precisely, since P (t) is the probability that the particle is in the detector region D, it follows that P(t) is the probability density that the particle will appear at time t in the detector region D. The appearance of a particle in the detector at time t means that the particle was not there immediately before t, so we can say that the particle arrives to detector at time t.Hence we conclude that (36) is the arrival time distribution.
We repeat that this interpretation is only valid when P (t) increases with time, i.e. when P(t) is positive.The formula (36) then says that the arrival time distribution is given by the flux of the probability current through the boundary of the detector, when the flux is positive.But what if the flux is negative?In that case P (t) decreases with time, rather than increases, so the particle departs from the detector, rather than arrives to it.Hence, for a negative flux, the arrival probability density is zero.In this case the − ∂ D dS • j(x, t) is positive and naturally interpreted as departure probability density [8].Now let us confirm the conclusion above, that P(t) is arrival probability density, by a more rigorous analysis.We first split the time interval [0, t] into k intervals, each of the small size δt = t/k, and imagine that particle can only arrive at one of the times from the discrete set t 1 = δt, t 2 = 2δt, . . ., t k = kδt = t.At the end we shall let δt → 0. Let w(t j ) be the conditional probability density that the particle is in the detector at time t j , given that it was not in the detector immediately before, at time t j−1 .Then the probability that it will arrive at time t = t k is where P (t − δt) is the probability that, at time t − δt, the particle was not in the detector region D (see (33)).But the probability P (t − δt) is itself a joint probability that the particle was not in D at time t − δt given that it was not there at t − 2δt, and that it was not there at t − 2δt given that it was not there at t − 3δt, etc. Hence where 1 − w(t j )δt is the conditional probability that the particle is not in D at time t j , given that it was not there at t j−1 .Since we are interested in the limit δt → 0, we can first write (38) as and then take the limit δt → 0, which gives Thus (37) in the limit δt → 0 can be written as where in the last equality we used (35).This shows that (36) is indeed the arrival probability density, provided that it is positive.The measurable predictions of the arrival time distribution based on flux of the probability current can in principle be distinguished experimentally from predictions of the arrival time distribution based on other approaches.It is beyond the scope of the present paper to discuss such measurable differences in detail, but they have been studied elsewhere [4].

V. SUMMARY AND CONCLUSION
The results of this paper can be summarized as follows.As the wave function of a particle approaches the detector, a part of the full wave function leaks into the detector region D, so the other part of wave function, that remains outside of D, diminishes with time.Since the norm of full wave function ψ(x, t) does not depend on time, the norm of its restriction ψ D (x, t) to the region D outside of the detector depends on time.Therefore the "Hamiltonian" H governing the time evolution of the restricted wave function ψ D(x, t) be a non-hermitian operator.In this paper we have found an explicit representation of H and found that its non-hermitian part can be written as a boundary term, proportional to the flux of the probability current operator through the boundary ∂ D of D. The explicit representation of H is given by Eq. (25).From the time-dependent norm of ψ D(x, t) we have computed the arrival time probability density, namely, the probability that the particle will be detected to arrive to the detector between the times t and t + dt, and found that this arrival probability density is equal to the flux of the probability current through the boundary ∂ D.
Our final result, that the arrival probability density is equal to the flux of the probability current, has also been obtained by other approaches, based on standard QM [5][6][7][8], as well as on Bohmian particle trajectories [9][10][11][12].The approach of the present paper, also based on standard QM, is complementary to the existing approaches, because we arrived to the same conclusion by using different methods.However, we stress that it is not generally accepted in the literature that the arrival probability density should be equal to the flux of the probability current, see [1][2][3][4] for reviews of other proposals, so we believe that the result of this paper is a valuable contribution towards a resolution of an important problem in physics.
1/2 is timedependent.Consequently, one expects that the evolution of |ψ D(t) is not unitary.By contrast, the evolution of the full state |ψ(t) is unitary.