Paradoxical Quantum Scattering off a Time Dependent Potential?

Ori Reinhardt and Moshe Schwartz Department of Physics, Raymond and Beverly Sackler Faculty Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel We consider the quantum scattering off a time dependent barrier in one dimension. Our initial state is a right going eigenstate of the Hamiltonian at time t=0. It consists of a plane wave incoming from the left, a reflected plane wave on the left of the barrier and a transmitted wave on its right. We find that at later times, the evolving wave function has a finite overlap with left going eigenstates of the Hamiltonian at time t=0. For simplicity we present an exact result for a time dependent delta function potential. Then we show that our result is not an artifact of that specific choice of the potential. This surprising result does not agree with our interpretation of the eigenstates of the Hamiltonian at time t=0. A numerical study of evolving wave packets, does not find any corresponding real effect. Namely, we do not see on the right hand side of the barrier any evidence for a left going packet. Our conclusion is thus that the intriguing result mentioned above is intriguing only due to the semantics of the interpretation. PACS numbers: 03.65.-w, 03.65.Nk, 03.65.Xp, 66.35.+a

The quantum problem of an incoming plain wave The eigenstates of the Hamiltonian (Eq. 1) can be encountering a static potential barrier in one dimension is classified according to the absolute value of incoming an old canonical text book example in which the reflection momentum and to whether they are right or left going. We and the transmission of the particle are defined and define the right going eigenstates for k > 0 : calculated [1-10]. The problem of a time dependent barrier ikx+ − ikx e+ A e forx < − a has been explored within time dependent perturbation + k , (2) Ψk (x ) =  + ikx theory [6-15]and more recently some exact results appeared  Dk e forx> a in the literature [16,17] making clear that new interesting where the solution within the segment x≤ a is not results may be expected when time dependent potentials are relevant to our discussion, although it affects the involved [18]. The purpose of the present manuscript is to coefficients A and D . The first term in the region x< − a present a new intriguing result in that field. We show that is interpreted as the wave incoming from the left, the starting with a state which is traditionally interpreted as a second term is interpreted as the reflected wave and the particle travelling from left to right, partly reflected and partly transmitted through the barrier, we end up with a single term in the region x> a is interpreted as the finite probability of finding the system in a state of the transmitted wave. Similar left going eigenstates are: opposite nature. Namely, there is a finite probability to find e−ikx+ A − e ikx forx > a Ψ− (x ) = k . (3) a state which is interpreted as a particle travelling from k  − − ikx  Dk e for x< − a right to left, partly reflected back to the right and partly + − transmitted through the barrier. For a symmetric potential D= D = D and + − Consider first the quantum problem of a particle in the A= A = A . In the following paragraphs we prefer to presence of a time independent barrier (see Fig. 1). The work with a delta function potential, V0δ ( x ) , because of the Hamiltonian of the system is given by simplicity of the analysis, though ,as we will show later the
2∂ 2 intriguing result we obtain is generic. In that case H= − + V( x ) , (1) 2m∂ x 2 β ik Ak = and Dk = , (4) where V( x )= 0 for x> a . An example for such a −β + ik −β + ik mV potential is depicted in Fig. 1. where β = 0 . The transmission is given by
2 V(x) k 2 T = (5) β 2+ k 2 Now we consider a time dependent delta function σ potential, λ()t δ ( x ) . The set {Ψk (x ) | k > 0, σ =± } , which are eigenstates of the Hamiltonian at time t = 0 is x complete. The scalar products of any two states in the basis x= -a x=a is given in bra-ket notation by Fig. 1. A general potential V( x ) that vanishes for x> a . σ τ ΨΨ=k l 2πδ (k − l ) δ στ , (6)

thus the state at a general time can be written as σ dc k 1 σ ∂V + ∞ ≈ Ψ(x ,0) Ψ . (12) iE t   −
k q Ψ=()tdl exp −l ctτ () Ψ τ . (7) tdt2π i∂ t ∫ 
  ∑ l l 0    τ =+ σ∂V τ Since the matrix elements Ψ(x ,0) Ψ do not The equation for the coefficients is readily obtained k∂t l from the time dependent Schrödinger equation. vanish identically for σ≠ τ , σ ∞ − dc1  iEEt(− )  −1 −∂V + 2 k =Dt* λ()(0) − λ dlD expk l ctτ () ct()= Ψ (,0) xt Ψ≠ 0 . (13)
k[]∫ l
 ∑ l k
k q dt2π i 0   τ =+ 4πi∂ t − (8) Once ck (0) does not vanish at short times it will not do dc σ so at later times. It is interesting to compare c− ( t ) with It is thus clear that k does not depend on σ . Consider k + dt ck ( t ) for k≠ q and very short time t . To keep matters next the initial condition simple let us think about a smooth symmetric potential and + Ψ(0) = Ψ q . (9) Eq smaller than the typical height of the barrier. The wave Since the potential changes in time it is expected that + + functions Ψ q (x ) and Ψ k (x ) decay within the barrier from the particle will exchange energy with the potential and left to right with rates, γ and γ respectively and the wave thus at later times the expansion of the potential in terms of q k the original eigenstates will contain contributions with functions Ψ − (x ) and Ψ − (x ) decay from right to left with q k l≠ q . The intuitive feeling based on the interpretation of the same rates, γ q and γ k . Therefore, we may expect that + the state Ψ as a wave coming from the left partly − q c the very short time ratio k decays as a function of the reflected back and partly transmitted through the barrier, c+ suggests ,however, that we should have zero overlap at any k width of the potential as time with states having negative σ , which correspond −   following the same interpretation to waves coming from the c( t ) 2sinh (γk− γ q )a k ∝  exp  −γ + γ a  , (14) σ + ()k q  dc k c( t ) γ− γ γ + γ right. The exact result that does not depend on σ k ()()k q k q dt when for both γ 's, γ a 1 . Thus as the barrier tends to proves the opposite: become macroscopic the ratio above tends exponentially to ct−()= ct + () −δ ( kq − ) . (10) k k zero and the strange waves incoming from the right To show that this result is not an artifact of our choice disappear relative to the expected ones as a response to the of the delta potential, we repeat the same calculation for a time dependence of the potential. Can we say something time dependent potential, V( x , t ) , which vanishes for about the time dependence of that ratio? Equation (11) can x> a . be taken as a starting point for a time dependent perturbation expansion. It is clear that for a potential of the The equations for the coefficients are given in the general form, say, case by Vxt(,)= λ ()() tVx , (15) σ ∞ − dc1  iEEt( − )   σ τ τ k k l − =dlexp Ψ∆Ψk Vct l l () ,
∫ 
  ∑ ck ( t ) dt2π i 0    τ =+ and for k≠ q ,the ratio does not depend on time , c+ ( t ) (11) k where ∆V = Vxt(,) − Vx (,0) . within first order time dependent perturbation theory. This σ τ implies that although there is a finite transition rate into the For the δ function, the matrix element Ψk ∆V Ψ l strange waves incoming from the right it is exponentially does not depend on σ (and on τ ) and that is the reason small in the width of the barrier. dc σ The result presented here implies that it is necessary to that k does not depend on σ . Now the situation is consider wave packets in order to verify or disprove the dt existence of a real effect. Therefore we solve numerically dc σ different. To get oriented consider k at very short times the Schrödinger equation for the time dependent delta dt potential barrier. This solution gives the time dependent + solution for every initial wave function. For simplification, for the initial state Ψq . We will approximate we suppress in the following, the parameters m and
in ∂V ∆V = ( x ,0) t . For very short times t and for k 's such the Schrödinger equation by inserting a general diffusion ∂t coefficient η . We consider an initial state which is a Ek− E q t normalized Gaussian of the form that 1,

1 2 ()x− x 4 − 0   2 1 4σ 0ik 0 x ψ (,xt= 0) = 2  ee . (16) 2πσ 0 

For x0 < 0 and k0 > 0 this is a wave packet starting on the left and travelling from left to right. Consider first the evolution of the wave packet above in the presence of a time independent delta barrier. The probability density and probability current of the evolving wave function are given in Fig. 2.

Fig. 2. The evolution of a wave packet encountering a delta barrier. The initial state is a normalized Gaussian with

x0=−3,σ 0 = 0.2, k 0 = 50 . The delta barrier has an amplitude of λ = 6 . The diffusion coefficient is η = 0.2 . Fig. 3. The evolution of a wave function encountering an 2 oscillating delta barrier. (a) Probability density (b) It can be seen that similar to the traditional ψ (x , t ) interpretation, where we talked about the reflected plain Probability current J( x , t ) . The initial state is a Gaussian wave wave and the transmitted one, we obtain here also after the packet with x0=−3,σ 0 = 0.2, k 0 = 50 . The parameters of the time arrival of the initial packet at the barrier a transmitted dependent barrier (Eq. 17) are λαω=5, = 1,0 =⋅ 7( 2 π /0.3 ) . The packet moving to the right and a reflected packet moving to diffusion coefficient is η = 0.2 . the left, which look very much Gaussian. It is interesting to note that the total probability to find the particle on the right of the barrier, as obtained by numerical integration, is To check on the numerical procedure and to obtain an equal, indeed, to the transmission calculated for plane analytic expression for the evolving wave packet, it is also interesting to look at this problem using first order waves, with k replacing k . 0 perturbation theory. An analytical expression has been Now the interesting question is what will be the effect derived from equation (8) for the first order c± coefficients of a time dependent potential? Is it possible to see any k earmark of the seemingly paradoxical result we have in the case of an initial state of the form given by equation mentioned previously? An example for an evolving wave (16) with a time dependent potential given by equation function in the presence of a time dependent potential is (17), shown in Fig 3, where the time dependent part of the potential was taken as

λ()t= λ( 1 + α sin ( ω 0 t )) . (17) We can see from Fig. 3. that the arriving, reflected and transmitted wave packets are seen very clearly, though they are not pure Gaussians any more but have undergone some modulation. Most importantly, it can be seen that there is no evidence of anything travelling from right to left on the right hand side of the barrier. This was studied rather carefully running many solutions, finding no real effect of the intriguing result mentioned earlier.

1 side, as seemingly implied by the former mentioned λασ π 1  4 ik∞ l 2 ct±(1) = 0 eik0 x 0 dl ⋅ analytical result. This was shown numerically for a delta k () 2 2  ∫ 2 2 ()2π 2πσ0  β+ik0 β + l potential and joins our previous statement that the effect disappears when the potential is wide (14).  2 2    −σ 0(k 0 − l ) − ilx 0 x0 This seeming contradiction most probably relates to the ⋅eβ ⋅−− erf i σ 0( k 0 −++ l )  il  2σ  fact that the traditional interpretation of the non-localized  0   states as a sum of the original incoming wave, a transmitted  −σ 2(k + l ) 2 + ilx x   and a reflected wave, is still classical. +e0 0 0  −⋅−−β erf0 iklil σ ( ++⋅ )   0 0   2σ 0   

2 2 2 2 i()ω0 + η () klt − −i()ω0 − η () klt −  e −1 e −1 [1] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum ⋅+  2 2 2 2  mechanics (Wiley, New York, 1977) Vol. 1. ω0 + η ()k − lω0 − η () k − l   . [2] S. Gasiorowicz, Quantum physics (Wiley, New York, (18) 1974). The first order correction to the time dependent wave [3] R. M. Eisberg and R. Resnick, Quantum physics function due to the oscillating part is (Wiley, New York, 1974).

∞ −1k  [4] A. S. Davydov, Quantum mechanics (Pergamon, 2 2i tan    ψ (1)()()xt, = dke ⋅−iη k t c ± (1) te−ikx + eβ  e ikx  . Oxford, U.K., 1976). ∫ k   0   [5] E. Zaarur and R. Pnini, Schaum’s Outline of (19) Quantum Mechanics (1998).

In the following graph we compare the analytic first [6] A. Messiah, Quantum mechanics (North-Holland, order correction to the wave function with the Amsterdam ,1961). corresponding numerical result for a specific time. We [7] E. Merzbacher, Quantum Mechanics (John Wiley and consider a relatively small coupling and we see good Sons, New York, 1961). agreement. [8] D. J. Griffiths and G. Dick, Introduction to quantum mechanics (Prentice Hall, Englewood Cliffs, New Jersey, 1995). [9] F. Schwabl, Quantum Mechanics (Springer, 2007) [10] R. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1998). [11] K. Gottfried, Quantum mechanics (W.A. Benjamin, New York, 1966) Vol. 1. [12] A. Galindo and P. Pascual, Quantum mechanics (Springer, Berlin, 1990), Vol. 1 and 2. [13] P. W. Atkins and R. S. Friedman, Molecular quantum mechanics (Oxford University Press, Oxford, 1997), Vol. 3. [14] A. K. Ghatak and S. Lokanathan, Quantum mechanics: theory and applications (Macmillan India Limited, 1994).

[15]R. Shankar, R. Shankar, and R. Shankar, Principles Fig. 4. The first order perturbation theory wave function. of quantum mechanics (Plenum Press, New York, Continuous line: analytical expression, dashed line: numerical 1994). solution expression. The initial state is a Gaussian wave packet [16] E. Eisenberg and Y. Ashkenazy, Found. Phys. 27 , with x0=−3,σ 0 = 0.2, k 0 = 50 . The delta barrier has the 191 (1997).

parameters λα=3, = 0.1, ω0 =⋅ 7() 2 π /0.3 . The diffusion [17]T. J. Park, Bull. Korean Chem. Soc. 23 , 1733 (2002).

coefficient is η = 0.2 . [18]M. Y. Azbel, Solid State Commun. 91 , 439 (1994)

To conclude, we have shown that a time dependent barrier on which particles are scattered from the left generates particles that seem to arrive from the right and being scattered off the barrier. By numerically investigating the time dependent Schrödinger equation with the initial state being a wave packet travelling from left to right, we found no evidence of wave packets coming from the other