Contents Introduction

2

1 Semiclassical propagator with complex trajectories

4

2 Approximation with real trajectories

5

3 Application to a 1-D potential barrier 3.1 Before the barrier: x < −a . . . . . . . 3.2 Inside the barrier: −a ≤ x ≤ a . . . . . 3.3 After the barrier: a < x . . . . . . . . 3.4 Results . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

4 Further application with 2 barriers of potential

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

7 7 8 10 10 13

Conclusion

16

Acknowledgments

16

1

Introduction During my internship, I worked in the ”Instituto de F`ısica Gleb Wataghin” in the ”Universidade Estadual de Campinas” (Brazil). It was for me the opportunity to discover my native country, and to learn Portuguese. I worked in the ”Departamento de F`ısica da Mat´eria Condensada ” (DFMC), specialized in Crystallography, Low Temperatures, Optics, Theory . . . under the supervision of Marcus A.M. de Aguiar. My research theme was the study of the propagation of a wavepacket in the context of the semiclassical approximation. During the first part of my stay, I familiarized myself with the semiclassical evolution operator (chapter 1), and calculated analytically different expressions of the propagator < x|K(T )|z >, where < x| is the final position in the coordinate representation and |z > is the initial condition in the coherent states representation (chapter 2). Then, thanks to these results, I tried to represent the time evolution of a wavepacket submitted to different potentials (chapter 3 and 4).

The success of semiclassical approximations in molecular and atomic physics or theoretical chemistry, is due to its capacity to reconcile the advantages of classical physics and quantum mechanics. It manages to retain important features which escape the classical methods such as interferences or tunnel effect, while providing an intuitive approach to quantum mechanical problems whose exact solution could be very hard. However, the study of semiclassical limit of quantum mechanics has a theoretical interest of its own, shedding light into the fuzzy boundary between the classical quantum perspectives. For example, the time evolution of a wavefunction ψ0 (x) in the semiclassical limit and in the coordinate representation can be calculated thanks to the Van Vleck approximation of the Feynman propagator Z ψ(x, T ) =< x|K(T )|ψ0 >= < x|K(T )|x0 > dx0 < x0 |ψ0 > Z (1) 0 0 0 ,→ ψ(x, T )sc = < x|K(T )|x >V V dx < x |ψ0 > One important class of initial wavefunctions |ψ0 > is that of coherent states, whose one of the great advantages is that they provide gaussian states naturally localized with a minimum uncertainty in the phase space. For this reason, in the rest of the paper we shall consider |ψ0 > as a coherent state of a harmonic oscillator of mass m and frequency ω defined by 1

2

†

|z >= e− 2 |z| ezˆa |0 >

(2)

where |0 > is the harmonic oscillator ground state, a ˆ† is the creation operator and z is 2

the eigenvalue of the annihilation operator a ˆ with respect to the eigenfunction |z > µ ¶ 1 qˆ pˆ 1 ³q p´ † a ˆ =√ −i z=√ +i , (3) c c 2 b 2 b qˆ and pˆ are operators; q and p are real numbers; z is complex. The parameters 1 1 b = (~/mω) 2 and c = (~mω) 2 are the position and momentum scales, respectively, and their product is ~. At first, we shall recall some indispensable definitions which shall enable us to write the expression of the propagator with the method of the stationary phase for complex trajectories (chapter 1). This calculation has been done in [Agu] and has been used within the context of the Initial Value Representation (IVR) with the further approximation of real trajectories. In the present paper we show that a similar but better approximation can be performed, at the expenses of the IVR (chapter 2). The resulting expression of the evolution operator is then illustrated by two examples (chapter 3 and 4), based on the study of finite potential barriers in one dimension.

3

1 Semiclassical propagator with complex trajectories Let S ≡ S(x00 , T ; x0 , 0) the action of a classical trajectory in the phase space (x, p), with x0 = x(0) and x00 = x(T ). A small initial displacement (δx0 , δp0 ) modifies the whole trajectory and leads to another displacement (δx00 , δp00 ) at time T . In the linearized approximation, the tangent matrix M connects these two vectors of the phase space    00    0 Sii c 1 δx δx0   δx − − mqq mqp  b  Sif b Sif   b         b        = µ  ≡   (4) ¶   δp00   b  δp0  Sii Sf f   δp0  mpq mpp Sif − Sf f − c Sif Sif c c c where Sii ≡ ∂ 2 S/∂x02 , Sif ≡ ∂ 2 S/∂x0 ∂x00 ≡ Sf i and Sf f ≡ ∂ 2 S/∂x002 . In the rest of the paper, it will be useful to express the derivatives of the action S in terms of the elements of the tangent matrix M Sii =

c mqq b mqp

Sif = −

c 1 b mqp

Sf f =

c mpp . b mqp

(5)

These definitions allow us to write the Van Vleck propagator [Van] in terms of the coefficients of the tangent matrix µ ¶ 1 i π 0 0 < x|K(T )|x >V V = p exp S(x, T ; x , 0) − i . (6) ~ 4 b 2πmqp Supposing some converging conditions, the stationary phase approximation allow us to perform the integral over x0 in eq.(1) (for more details, see [Agu]) µ ¶ b−1/2 π −1/4 i i (x00 − q)2 0 0 ψ(z, x, T )sc = p exp S(x, T ; x0 , 0) + p(x0 − q/2) − (7) ~ ~ 2b2 mqq + imqp where the complex stationary point x00 is given by the relation µ ¶ (x00 − q) ∂S i 0 0 (p − p0 ) − = 0 where p0 = − , ~ b2 ∂x0 x0

(8)

0

S ≡ S(x, T ; x00 , 0) is the action of the stationary classical trajectory connecting x00 to x in time T , with an initial momentum p00 . Since q and p are real, x00 and p00 are usually complex (eq.(8)). This implies that the stationary trajectories are complex as well, even if x ∈ R. 4

2 Approximation with real trajectories The goal of this chapter is to find real trajectories whose boundary conditions are as close as possible to (q, p) at time 0 and position x at time T . The whole problem is that a trajectory (or a particle to be more concrete) whose, for example, initial conditions are q and p, doesn’t have any reason to be in x after a time T ; it’s possible, but generally it’s not the case, because the third boundary condition is imposed by the two others. In [Agu], the author chose to use the IVR. He fixed the initial conditions (q, p) in order to demonstrate that we can obtain the famous Heller’s Thawed Gaussian Approximation (TGA). However, this is not the only possible approximation of eq.(7) with real trajectories. Actually, since we can fix two variables among three, there are three possibilities : §1. we fix q and p and calculate the final position and momentum respectively qf and pf (IVR) §2. we fix q and x and calculate the initial and final momenta respectively pi and pf §3. we fix p and x and calculate the initial position qi and the final momentum pf Each case gives a different approximation for x00 and for the action S(x, T ; x00 , 0), and so the expressions of the evolution operator are not identical according to the variables we chose to fix (see eq.(7)). Of course, there is the possibility not to fix any boundary conditions and to consider a trajectory beginning in (qi , pi ) and ending in (qf , pf ); even if the propagator thus obtained is more general, it wouldn’t be useful for us and its calculation would only weigh down the present paper. Since the calculation of the propagator (§1) has already been performed in [Agu], we are only going to show the path to follow for (§2) and (§3). In the hypothesis (§2), we lay at first x00 = q + ∆x0 ¶ µ ∂S 0 = pi + ∆p0 = pi − Sii ∆x0 . p0 = − ∂x0 x0

(9)

0

The complete development of p00 at the first order should be pi − Sii ∆x0 − Sif ∆x, but as x is chosen as the final position of the real trajectory, ∆x = 0. The equation (8) give us the relation between ∆x0 and ∆p0 (x00 − q) i i i = (p − p00 ) = (pi − p00 ) + (p − pi ) 2 b ~ ~ ~ 5

(10)

then we have thanks to eq.(9) and (10) ∆p0 = i~

∆x0 + (p − pi ) = −Sii ∆x0 2 b

(11)

which gives, in terms of the coefficient of the tangent matrix (eq.(5)), ∆x0 =

(pi − p) b mqp = (pi − p). i~ c mqq + imqp Sii + b2

(12)

We are now able to calculate the approximation of the term in the exponential of eq.(7) when the complex trajectory is expressed in terms of the real trajectory (§2) : i (x0 − q)2 i S(x, T ; x00 , 0) + p(x00 − q/2) − 0 2 ~µ ~µ ¶ µ 2b ¶ ¶ ∂S i i i ∆x0 2 ∂ 2S ∆x0 2 0 0 = S(x, T ; q, 0) + ∆x + + p∆x + pq − ~ ∂x0 i ∂x0 2 ii 2 ~ 2~ 2b2 µ ¶ i 1 i i ∆x0 2 S(x, T ; q, 0) − pi ∆x0 + Sii ∆x0 2 + p∆x0 + pq − = ~ 2 ~ 2~ 2b2 µ ¶ i i i 1 iSii 1 = S(x, T ; q, 0) + pq + (p − pi )∆x0 + − 2 ∆x0 2 ~ 2~ ~ 2 ~ b 2 i i imqp (pi − p) 1 imqp (pi − p)2 = S(x, T ; q, 0) + pq − + ~ 2~ mqq + imqp c2 2 mqq + imqp c2

(13)

The semiclassical propagator can be written as a function of q, p, x and pi as à ¶2 ! µ imqp b−1/2 π −1/4 i i 1 p − pi ψ(z, x, T )sc2 = p exp S(x, T ; q, 0) + pq − .(14) ~ 2~ 2 mqq + imqp c mqq + imqp In the same way, if we chose to fix p and x (§3) instead of q and x (§2), we write x00 = qi + ∆x0 µ ¶ ∂S 0 p0 = − = p + ∆p0 = p − Sii ∆x0 ∂x0 x0

(15)

0

and thanks to eq.(8) and (15), the new expression of ∆x0 is ∆x0 =

imqp (q − qi ). mqq + imqp

(16)

At the end we obtain a semiclassical wavepacket expressed in terms of this real trajectory which is different from the two others à −1/2 −1/4 i b π i exp S(x, T ; qi , 0) + pq ψ(z, x, T )sc3 = p ~ 2~ mqq + imqp (17) µ ¶2 ! 1 q − qi i imqq + p(qi − q) − . ~ 2 mqq + imqp b We now have general expressions of the semiclassical propagator and we can use them in the next chapter to see the evolution of a wavepacket through a potential barrier. 6

3 Application to a 1-D potential barrier The potential V (x) is defined as follow (see fig.(1)) ( V0 if x ∈ [−a, a] where a ∈ R+ V (x) = 0 else

(18)

If the wavepacket has a low energy (E ≤ V0 ), the last two methods (except the IVR which is always wrong) give almost the same result before the barrier (x < −a) where the physics is quite the same as for the infinite wall [Agu], but for a high energy (E > V0 ) or on the other side of the wall (x > −a), while the hypothesis (§1) and (§3) are not anymore realistic, the propagator (§2) is very close to the exact one. In this chapter, we are going to express ψ(z, x, T )sc2 before, inside and after the barrier. To do so, we need to calculate the initial momentum pi , the action S ≡ S(x, T ; q, 0) and its derivatives (to obtain mqq and mqp ) for each trajectory, while keeping in our mind that we have to be cautious about the multiplicity of the possible trajectories, since (§2) is not an IVR.

3.1 Before the barrier: x < −a The specificity of this region is that there may exist two different paths connecting q to x in time T ; a direct trajectory and a reflected one (fig.(1)) whose initial momenta, respectively pi d and pi r , are not equal: © ª © ª (x−q)2 1 1 T - pi d = x−q and S = ⇒ S = , S = − ⇒ m = 1, m = d ii d if d qq d qp d T 2T T T λ © ª © ª 2 (x+q+2a) 1 1 T - pi r = − x+q+2a and S = ⇒ S = , S = ⇒ m = −1, m = − r ii r if r qq r qp r T 2T T T λ where λ = b/c. We can then write the contributions to the propagators ψd and ψr which correspond to these trajectories à µ ¶2 ! i 1 iT i (x − q)2 pT − x + q b−1/2 π −1/4 exp + pq − (19) ψd = q ~ 2T 2~ 2 λ + iT cT 1 + i Tλ à µ ¶2 ! (−i)b−1/2 π −1/4 i (x + q + 2a)2 i 1 iT pT + x + q + 2a q ψr = exp iθ + + pq − (20) ~ 2T 2~ 2 λ + iT cT T 1+i λ

In ψr , we put an extra phase θ which is due to the reflection on the wall; indeed, without this extra phase, the wave packet is not continuous when it goes through the 7

barrier. With the complex polar representation, we write the propagator inside the wall W (x)eiϕw (x) , ψd = D(x)eiϕd (x) and ψr = R(x)eiϕr (x)+iθ . When x = −a, since the trajectory inside the wall is the continuity of the directed one, we have W (−a) = D(−a) and the equations (19) and (20) show that R(−a) = D(−a). The continuity of the evolution operator imposes D(−a)eiϕd (−a) + R(−a)eiϕr (−a)+iθ = W (−a)eiϕw (−a) ⇒ eiϕd (−a) + eiϕr (−a)+iθ = eiϕw (−a) . At the end, we obtain several solutions and we choose arbitrarily θ = − π6 . The global propagator before the wall is ψ(z, x, T )sc2 = ψd + ψr and the probability density can then be written ( à µ ¶2 ! 2 1 λ 1 x − q − pT |ψ(z, x, T )sc2 |2 = √ q exp − 2 λ + T2 b b π 1 + T2 λ2 à ¶2 ! µ λ2 x + q + pT + 2a + exp − 2 λ + T2 b µ ¶ µ ¶) ¢ π λ2 (pT + q + a)2 + (x + a)2 2(x + a) ¡ 2 exp − 2 − 2 sin λ p − (q + a)T + ~(λ2 + T 2 ) 6 λ + T2 b2 (21) This is the same result as in [Agu] with a completely repulsive barrier and complex trajectories, except for the phase, because of the different boundary condition when x = −a. However an additional difficulty appears when the wall is not infinite: the reflected trajectory doesn’t always exist. If p2i r /2 > √ V0 , we can not have a reflected part. The maximum initial momentum allowed is then 2V0 and a wave packet with such a ”veloca+q . Furthermore, after tc the reflected ity” can not reach the wall before a time tc = − √ 2V0 √ x + q + 2a √ trajectory only exists if pi r = − ≤ 2V0 ⇔ |x| = −x ≤ xc = q + 2a + 2V0 T . T Therefore, if t ≥ tc and |x| ≤ xc , we can use the equation (21), but in the opposite case, we only have the direct trajectory and the propagator is simply ψd with a probability density à µ ¶2 ! 2 1 1 λ x − q − pT |ψ(z, x, T )sc2 |2 = √ q exp − 2 (22) λ + T2 b b π 1 + T2 λ2

Except for the disappearing reflected trajectory, the behavior of the wavepacket is not much modified whether it’s a finite or an infinite wall. Things become more interesting when we look at the propagation of this wavepacket through the barrier and how it splits after the collision.

3.2 Inside the barrier: −a ≤ x ≤ a In this region, there is only one real classical trajectory which is the direct one (see fig.(1)) ,because a reflection on the other side √ of the wall (x = a) can not be considered without quantum mechanics. If p1 = pi > 2V0 and p2 are respectively the momenta before and inside the wall, we have the energy conservation p21 /2 = p22 /2 + V0 . It is the 8

first equation relating p1 and p2 , but we need a second one which is imposed by the time T = t1 + t2 where: - t1 = − a+q is the time necessary to go from q to −a with a momentum p1 p1 - t2 =

x+a p2

is the time necessary to go from −a to x with a momentum p2

The combination of these two equations gives a+q x+a +p 2 p1 p1 − 2V0 µ ¶2 a+q (x + a)2 ⇔ T+ = 2 p1 p1 − 2V0 2 ⇔ (p1 − 2V0 )(p1 T + a + q)2 = (x + a)2 p21 T =−

(23)

Like in the section 3.1, to express the propagator, we need pi and the coefficients of the tangent matrix, but eq.(23) is a quartic polynomial which can not be solved analytically (actually, there is an analytical solution but it is too much complicate to be of any help). Fortunately, p1 can be easily obtained numerically. We have four solutions; two of them are sometimes complex, a third one is always negative (which is impossible since we impose the initial position q on the left of the wall) and the last one is always a real positive number, which tends to x−q when V0 is negligible (like for a free particle). But T even with the correct value of p1 , it is impossible to calculate the second derivatives of the action S with respect to q and x (and so mqq and mqp ), because S is a function of p1 ¶ Z Tµ 2 p21 p2 S(z, x, T ) = dt + − V0 dt 2 2 0 t1 µ 2 ¶ p2 p21 t1 + − V0 t2 = 2 2 q V0 (x + a) 1 1 = − (a + q)p1 + (x + a) p21 − 2V0 − p 2 2 2 p1 − 2V0 Z

t1

(24)

In fact, we are obliged to calculate S (and so p1 ) numerically with the initial conditions ∂S S(z, x + dx, T ) − S(z, x, T ) (q, x), (q + dq, x), (q, x + dx) . . . and to do (z, x, T ) = , ∂x dx ∂S (z, x, T ) = . . . in order to obtain mqq and mqp . ∂q Finally, the propagator inside the wall is simply the equation (14) and the probability density is à µ ¶2 ! 2 m p − p 1 1 1 qp exp − 2 |ψ(z, x, T )sc2 |2 = √ p 2 (25) mqq + m2qp c b π mqq + m2qp with the correct values of p1 , mqq and mqp calculated numerically. We can now follow the propagation of the wavepacket on the other side of the barrier, whose study is very similar to this one.

9

3.3 After the barrier: a < x Henceforth, the trajectory has to be divided into 3 parts : before (index 1), inside (index 2) and after (index 3) the barrier. The precedent reasoning works in the same way: the energy conservation is p21 /2 = p22 /2 + V0 = p23 /2 and the different times are t1 = − a+q , p1 2a x−a t2 = p2 and t3 = p3 , which gives a+q 2a x−a + +p 2 p1 p1 p1 − 2V0 ¶2 µ (2a)2 2a + q − x = 2 ⇔ T+ p1 p1 − 2V0 2 ⇔ (p1 − 2V0 )(p1 T + 2a + q − x)2 = (2a)2 p21 T =−

whereas the action S becomes ¶ Z t1 2 Z t1 +t2 µ 2 Z T p23 p1 p2 S(z, x, T ) = dt + − V0 dt + dt 2 2 0 t1 t1 +t2 2 q 2a V0 1 = (x − q − 2a)p1 + a p21 − 2V0 − p 2 2 p1 − 2V0

(26)

(27)

The argument about the quartic polynomial is also valid and the probability density is the same as eq.(25) with the new values of p1 , mqq and mqp .

3.4 Results √ In the figures (2), we fixed V0 = 0.5 (and so 2V0 = 1) , q = −10, a = 1, b = 1 and ~ = 1. On the left, we can see very well the interferences between the reflected and the direct trajectories, whereas the part of the wavepacket which goes through the wall becomes broader and smaller with the time. The first remark we can make is that the bigger pi , the better the semiclassical approximation. For pi = 3, the exact propagator and the semiclassical one superimpose themselves perfectly, whereas for pi = 0.5 the approximation is good but not perfect. If we look closely at the problem (see fig.(3)), we can find three principal sources of differences: - when x > a, the semiclassical wavepacket after the barrier is a little bit more asymmetrical than the quantum one. The mathematical reason is that the probability density (eq.(25)) is not really a gaussian but a multiplication between the 2 prefactor √ 2 1 2 which increases with x and the exponential e−(p−pi ) which mqq +mqp

decreases (the bigger x gets, the further the wavepacket must go and the higher pi is); the exponential term is not always able to compensate the increase of the prefactor. There is a physical reason for this asymmetry due to the semiclassical approximation that we are going to explain in the rest of this section. - when x < −a, the interferences between the direct and the reflected trajectories are very similar but not identical. The most√ visible difference is the discontinuity of the propagator at −x = xc = q + 2a + 2V0 T (see the argument in section 3.1) but it’s in fact more a mathematical problem than a physical one, because 10

the edges of a real barrier of potential should be smoother and the discontinuity shouldn’t be so abrupt. The other point is that for p ∼ 2, the interferences of the semiclassical wavepacket are a bit higher than the ones of the exact propagator, as if we should add a attenuation coefficient to the reflected trajectory. - we can not see it on the figures, but in some cases, there is a small peak inside the barrier. Even if the physical arguments are the same inside and after the wall (see sections 3.2 and 3.3), it appears that the results are very good in one case (x > a) but more approximate in the other one (−a ≤ x ≤ a), as the propagator is sometimes discontinuous at the edges of the barrier. The reason comes certainly from the way of computing the derivatives of the action. Thanks to this propagator, we can now calculate some physical observables, like for example the propagation time < T (z, x) > for the wavepacket centered at (q, p) to reach x, whose definition was taken in [Xav] Z +∞ Z +∞ −1 2 < T (z, x) >= N T | < x|K(T )|z > | dT where N = | < x|K(T )|z > |2 dT (28) 0

0

Each time T is level-headed by the probability of the wavepacket of being in x after a time T , and N is the normalization factor, because we only take in account the part of the wavepacket which goes through the barrier. Of course, the integral can not be carried out analytically and we must suppose the approximation < T (z, x) >≈ N

−1

¶ N µ N τ X iτ τ X iτ | < x|K( N | < x|K( iτ )|z > |2 and N ≈ )|z > |2(29) N N i=0 N N i=0

where τ = Inf{T ∈ R+ / ∀ T > T, | < x|K(T )|z > |2 < 10−5 }. In short, the contribution of the propagator after the time τ is negligible. The behavior of < T (z, x) >≡< Twall > with respect to x is linear, which is comparable to the behavior of a classical free particle where Tclass = x−q . If we define < Tf ree > in the same way as in eq.(28), but where p the propagator is the semiclassical one for a free particle, it’s surprising to see that the Wigner time TW ≡< Tf ree > − < Twall > is always positive (fig.(5)), that is to say that in a quantum point of view, the wavepacket is faster when it goes through the barrier! This behavior can be understood if we compare the propagator with and without the wall. As we can see in figure (4),√the barrier is like a filter, which cuts the contributions of the trajectories with pi ≤ 2V0 . For example if pi = 3, the part with small initial momenta of the free wavepacket is already negligible and the two propagators superimpose themselves, but if pi = 0.5, the barrier let only pass the not reflected end of √ the wavepacket. We can easily verify it; with T = 50, q = −10 and p > 2V0 = 1, a trajectory goes at least till x = q + pT = 40, and that’s approximately the point where the wavepacket begins in fig.(4). It implies that the impulsion of the part after the √ barrier is at least 2V0 : its average position seems to be indeed further than the center of the free wavepacket. It explains not only that the small part which goes through the barrier is quicker than a free particle (even if it tends to the same ”velocity” for high p), but also the asymmetrical shape of the semiclassical propagator which is in fact a ”cut gaussian”. The figure (5) compares Tclass and the average times < Twall > and < Tf ree > in function 11

of p. We can see that, as expected, these three curves superimpose themselves for high p, where the wavepackets are the same according to fig.(4). On the other hand, for small p, < Twall > is the only one which stays finite (the wavepacket is faster with a barrier). But since < Twall > is at first below the classical time, why does he become bigger when p increases? To explain that, we need to introduce a new ”statistical” time. In the classical phase space, the initial conditions of the particle are simply (q, p), but in quantum mechanics, if we consider the coherent state |z >, the probability density is a 2-D gaussian with a maximum in (q, p) and whose widths are respectively b and c. It is then possible to define a ”statistical” time which takes in account all the possible trajectories with different initial conditions and then steps forward in the quantum world Z +∞ Z +∞ 2 q−q0 2 1 x − q ( p−p 0 c ) e( b ) dq dp(30) e < Tstat (x, q0 , p0 ) >= R 2 R +∞ q−q0 2 +∞ ( p−p0 ) p −∞ e c dp −∞ e( b ) dq 0 0 We considered a free particle, because the integral is simpler, but it doesn’t change the physical interest. The important point is that < Tstat > tends to Tclass and is always bigger (see fig.(5)). It is a more intuitive way of understanding the behavior of < Twall > for high p. The semiclassical limit with the approximation of real trajectories provides us a propagation time in agreement with the classical physics for high p (the wavepacket tends to be like a free particle, because the barrier becomes negligible) and stays finite when p decreases, which is a purely quantum effect. Another very interesting quantity is the√ necessary time interval to cross the barrier. For a trajectory of initial momentum p1 > 2V0 , it is ttrav (T ) = p22a . In = √ 2 2a (T ) the same way as before, we can define the average traversal time Z +∞ −1 ttrav (T ) | < x|K(T )|z > |2 dT < Ttrav >= N

p1 (T )−2V0

(31)

0

where p1 is calculated numerically thanks to the equation (26). An important remark is that this time is independent on the final position x (except for small fluctuations due to the computation): when the wavepacket crosses the barrier, it doesn’t ”know” where we are going to look at it and so x doesn’t affect the time it spends inside the wall. On the other hand, as we can see in fig.(6), < Ttrav > depends on p: the reasons of its shape are the same as above, about < Twall >. The other curve is the time that a quantum free particle to go from −a to a when there isn’t any barrier; R +∞ spends 2aT < Tf ree (−a→a) >= N−1 | < x|Kf ree (T )|z > |2 dT . For p & 2, the wavepacket f ree 0 x−q is slower than a free particle, that is to say that it is actually ”trapped” inside the wall. Then, the semiclassical √ limit enables us to define a tunnelling time, because < Ttrav > stays finite for p ≤ 2V0 . The semiclassical method with real trajectories gives thus a very correct approximation of the behavior of a wavepacket against a finite barrier. We find classical results for high initial momenta p and we can describe some quantum phenomena like the interferences and the tunnelling effect. We are now going to extend this study with 2 barriers of potential and see what happens inside the ”cavity” and on each side.

12

4 Further application with 2 barriers of potential The new potential is ( V0 V (x) = 0

if x ∈ [−d − a, −d] ∪ [d, d + a] where d, a ∈ R+ else

(32)

On one hand, if we launch the wavepacket from the left (q < −d − a), nothing interesting happens because the first wall behaves like a filter: the part of the wavepacket with low initial momentum pi is reflected whereas the other part goes through the wall, but can not anymore be reflected by the second barrier since pi is already above √ 2V0 . On the other hand, if the initial position q is between the two walls, it is then possible to study the multi-reflections inside the cavity. With V0 = ∞, there would be an infinite number of reflected trajectories, and the propagator couldn’t be calculate √ exactly. Fortunately, in our case, only the part of the wavepacket with small pi ≤ 2V0 can stay between the two√barriers, which implies that the total ”length” of a trajectory must be less or equal to 2V0 T (same argument as at the end of the section 3.1). When x 6∈ [−d, d], the way of computing the propagator is the same as in the sections 3.2 and 3.3: we must calculate numerically pi and the derivatives of the action S. Of 2 course, since the expression of the propagator is proportional to e−(p−pi ) , the part of the wavepacket which goes to the left (with pi < 0) is very low, but not negligible for small p if we compare it with the part which goes to the right (as we can see in fig.(9)). However, when x ∈ [−d, d], we must be very cautious because of all the trajectories which have to be taken in account and whose evolution operators are different according to wether it comes from the left or the right, whether pi is positive or negative. At first, it is necessary to know the moments of the appearance of each reflected trajectory. For example, if pi > 0, the first one appears at time t1 = √d−q , the second 2V0 3d−q one at time t2 = √2V0 . . . It is easy to generalize  (2j − 1)d − q   √ t = j   2V0  t0 = 0 and

  (2j − 1)d + q   tj = √ 2V0

(∀j > 1) if pi > 0 (33) (∀j > 1) if pi < 0

If T is between the times tj and tj+1 , a trajectory can at the most has j reflections

13

and in this case the distance l covered by a particle from q to x is (see fig.(7)) ( l = 2jd − x − q (if pi > 0) ⇒ pi = 2jd−x−q T if j is odd 2jd+x+q l = 2jd + x + q (if pi < 0) ⇒ pi = − T ( l = 2jd + x − q (if pi > 0) ⇒ pi = 2jd+x−q T if j is even 2jd−x+q l = 2jd − x + q (if pi < 0) ⇒ pi = − T p2 T

The formula of the action is always the same S = i2 = definition (4), we obtain for tj ≤ T ≤ tj+1 © ª if j is odd ⇒ mqq = −1 , mqp = − Tλ © ª if j is even ⇒ mqq = 1 , mqp = Tλ

l2 , 2T

(34)

and thanks to the

(35)

We fix q = 0, because as we can see in eq.(33), the different times tj are then the same for pi positive and negative, which is going to simplify a lot the computation. For T between tj and tj+1 , the global expression of the propagator comes from the contribution of (2j + 1) trajectories at the most: the first one is direct, and each ”slice” of time adds two others reflected trajectories (one from the left and one from the right). Therefore, for a given time T , our first work is to calculate tn such as tn < T ≤ tn+1 , which gives the number of trajectories we have to consider. The contribution to the propagator of the trajectories which are present for all x inside the cavity can be calculated thanks to (14), (34) and (35), with q = 0 Ã µ ¶2 ! b−1/2 π −1/4 i x2 pT − x 1 1 + i Tλ ψd = q exp − (36) ~ 2T 2 λ2 + T 2 c 1 + iT λ

ψodd

" Ã ¶ µ ¶2 ! n−1 µ X 1 − (−1)j (−i)b−1/2 π −1/4 pT − 2jd + x i (2jd − x)2 1 1 + i Tλ q = exp − 2 ~ 2T 2 λ2 + T 2 c 1 + i Tλ j=1 (37) Ã µ ¶2 ! # pT + 2jd + x i (2jd + x)2 1 1 + i Tλ + exp − ~ 2T 2 λ2 + T 2 c

ψeven

à " µ ¶ ¶2 ! n−1 µ X 1 − (−1)j+1 b−1/2 π −1/4 i (2jd + x)2 1 1 + i Tλ pT − 2jd − x q = exp − 2 ~ 2T 2 λ2 + T 2 c 1 + i Tλ j=1 (38) à µ ¶2 ! # pT + 2jd − x i (2jd − x)2 1 1 + i Tλ + exp − ~ 2T 2 λ2 + T 2 c

ψd is the propagator of the direct trajectory, ψodd and ψeven are the contributions of the trajectories which appears at time tj where j is respectively odd or even. As we can see in the sum, the indices go from 1 to n − 1, which means that we haven’t included yet the propagator which appears at time tn , because this last one doesn’t exist for all x ∈ [−d, d]. For example, we can decide that for a time T and a potential V0 , the 14

√ trajectories drawn on the figure (8) are the longest possible, that is to say l = 2V0 T (see the discussion at the beginning of this √ section). A longer trajectory would need a bigger pi which would become greater than 2V0 and it wouldn’t be reflected. We have then three possibilities - from q to x1 : 6 trajectories, the last one of which with p1 < 0 - from q to x2 : 5 trajectories - from q to x3 : 6 trajectories, the last one of which with p1 > 0 The eq.(34) gives the length l in each case and we can write ( √ l = 2jd − x ≤ 2V0 then we add ψ1 √ if n is odd and l = 2jd + x ≤ 2V0 then we add ψ2 ( √ l = 2jd + x ≤ 2V0 then we add ψ3 √ if n is even and l = 2jd − x ≤ 2V0 then we add ψ4 where, thanks to (14), (34) and (35) Ã i (2nd − x)2 (−i)b−1/2 π −1/4 q exp ψ1 = ~ 2T 1 + i Tλ Ã (−i)b−1/2 π −1/4 i (2nd + x)2 q ψ2 = exp ~ 2T 1 + i Tλ Ã −1/2 −1/4 b π i (2nd + x)2 q ψ3 = exp ~ 2T 1 + i Tλ Ã b−1/2 π −1/4 i (2nd − x)2 q ψ4 = exp ~ 2T 1 + iT

1 1 + i Tλ − 2 λ2 + T 2 1 1 + i Tλ − 2 λ2 + T 2 1 1 + i Tλ − 2 λ2 + T 2 1 1 + i Tλ − 2 λ2 + T 2

µ

µ

µ

µ

(pi > 0) (pi < 0)

(39)

(pi > 0) (pi < 0)

pT − 2nd + x c pT + 2nd + x c pT − 2nd − x c pT + 2nd − x c

¶2 ! ¶2 ! ¶2 !

(40)

¶2 !

λ

P The global propagator inside the cavity is then ψsc = ψd + ψodd + ψeven + 4k=1 αk ψk , where αk ∈ {0, 1} depending on the conditions given above. With V0 = 0.5, q = 0, d = 5, a = 1, b = 1 and ~ = 1, we obtain the figures (9) which are in fact quite different from the exact solution; it seems that the small differences we remarked with a single barrier become more important if we consider several reflections. √ For example, the discontinuity of the wavepacket which is due to the limited length ( 2V0 T ) of the trajectories inside the cavity, cuts some interferences in fig.(9(c)). The semiclassical approximation can not describe the interferences outside the cavity, because after the first rebound against the barriers, the wavepacket is already √ 2V between the two barriers and the other split into three parts; one with |p | ≤ 0 i √ ones with |pi | > 2V0 outside. There is only one contribution (or trajectory) towards the propagator for x > d + a (and another one single as well for x < −d − a) and the interferences are then impossible. The other limit of this method is that we can’t impose the boundary conditions in x = −d and x = d. We have two equations of continuity and then we need to fix two variables, instead of a single extra phase θ like in the section 3.1, but it wouldn’t be physical to put different extra phases at x = −d and x = d. Thus, the semiclassical approximation doesn’t take in account the phenomenon of resonance. 15

Conclusion Thanks to the Van Vleck formula and the stationary phase method, we obtained a semiclassical approximation of the evolution operator (chapter 1), and applied this for real classical trajectories (chapter 2). Considering a finite potential barrier, the equation (14) gave us the time evolution of a wavepacket with initial conditions (q, p) and final position x, in very good agreement with the exact solution. Thus, we saw that this propagator is in fact the part of the free wavepacket whose energy is above the barrier (E = p2i /2 > V0 ). Furthermore, we were able to calculate different typical times, like the traversal time, and to compare them with a free particle; the semiclassical approximation managed to reconcile the classical limit for high p, when the barrier is negligible, and to provide us some quantum data, like the tunnelling time when the energy of the wavepacket is below the barrier. For example, it appears that the part which goes through the barrier can be ”faster” than a free particle (section 3.4). In the last chapter, we extended the study in the case of two finite barriers with multi-reflections. The outstanding point of this paper is the possibility to obtain results very close to the exact ones with real trajectories, which are much simpler to find than the complex ones, and to describe some quantum effects in the context of the semiclassical limit. Some interesting perspectives of the semiclassical propagation of a wavepacket would be, for example, to make the barrier ”smoother” (which is more realistic), to expand the study for higher dimension, or to consider chaotic systems, with at least two degrees of freedom, with a time dependant barrier for example... Of course it will provide much more complicated calculations and computations, but the results, especially for the two finite barriers, would certainly be even more accurate.

Acknowledgments I want to thank my supervisor Marcus A.M. de Aguiar for having accepted me as a ”summer student” (even if it was winter in Brazil !). He proposed to me a very good research theme, which was understandable in a short interval of time and which gave very interesting results in three months. Thank you for the time he managed to find in order to explain me the obscure aspects of the propagation of semiclassical wavepackets, and for his help during my administrative problems. This internship was really enriching, because I could face the whole progress of the theoretical research, from the discovery of the subject to the writing of an article. Thank you to Marcel and Ricardo for their daily help, as well as for little computer problems, as for the life in Brazil. And at last, I would like to thank again the people I have already mentioned, as well as Fernando, Marcus, Cabelo... for their welcome and their kindness! 16

Bibliography [Agu] Semiclassical propagation of wavepackets with complex trajectories, M.A.M. de Aguiar, A.D. Riberio, F. Parisio, M. Baranger submitting soon (2004) [Xav] Phase-Space Approach to the Tunnel Effect: a New Semiclassical Traversal Time, A.L. Xavier Jr. and M.A.M. de Aguiar, Phys. Rev. Lett. 79 3323 (1997). [Van] The correspondence principle in the statistical interpretation of quantum mechanics J.H. Van Vleck, Proc. Nat. Acad. Sci. USA 14 178 (1928) [BAKKS] Semiclassical approximations in phase-space with coherent states, M. Baranger, M.A.M. de Aguiar, F. Keck H.J. Korsch and B. Schellaas, J. Phys. A 34 7227-7286 (2001). [Par] A Semiclassical Coherent-State Propagator via Path Integrals with Intermediate States of Variable Width, F. Parisio and M.A.M. de Aguiar, Phys. Rev. A 68 062112 (2003).

17

q

x1

−a

x2

a

x3

Figure 1: Direct and reflected trajectories

0,12

0,12

p = 0.5

0,10

0,12

p = 0.5

0,10

T = 5

2

0,08

0,06

|ψ|

2

|ψ|

2

|ψ|

T = 40

0,08

0,08

0,06

0,06

0,04

0,04

0,04

0,02

0,02

0,02

0,00

0,00 -60

-40

-20

0

20

40

60

0,00 -60

80

-40

-20

0

20

40

60

80

-60

0,14

0,14

0,12

0,12

T = 5

T = 20

p = 1.5 T = 40

2

|y|

0,06

0,04

0,04

0,02

0,02

0,02

0,00

0,00 125

0,00 -50

-25

0

25

x

50

75

100

125

-50

-25

0

25

x

0,12

75

100

125

0,14

0,12

p = 3

0,10

T = 5

T = 20

0,08

50

x

0,12

p = 3

0,10

80

|y|

2

2

|y|

0,06

100

60

0,08

0,04

75

40

0,10

0,06

50

20

0,12

p = 1.5

0,08

25

0

0,14

0,10

0,08

0

-20

x

p = 1.5 0,10

-25

-40

x

x

-50

p = 0.5

0,10

T = 20

p = 3 T = 40

0,10

0,08

2

|y|

2

|ψ|

|ψ|

2

0,08 0,06

0,06

0,06

0,04

0,04

0,02

0,02

0,00

0,00

0,04

-30

0

30

60

90

x

120

150

180

210

0,02

0,00 -30

0

30

60

90

x

120

150

180

210

-30

0

30

60

90

120

150

180

210

x

Figure 2: Wavepacket at time T for the exact (doted) and the semiclassical (solid) solution

18

0,016

p = 1.5 T = 40

|y|

2

0,012

0,008

0,004

0,000 -50

-25

0

25

50

75

100

125

x

Figure 3: Zoom on one of the wavepacket

0,012 0,012

0,012

p = 0.5 0,010

T = 50

p = 1.5

0,010

p = 3

0,010

T = 50

T = 50

0,008

2

2

0,008

0,006

0,004

0,002

0,000

|y|

|y|

|y|

2

0,008

0,006

0,006

0,004

0,004

0,002

0,002

0,000

0

30

60

90

x

120

150

0,000 0

40

80

120

x

160

200

0

50

100

150

200

250

300

x

Figure 4: After the barrier, the semiclassical wavepacket (solid) is the wavepacket of a free particle (doted) without the part with low initial momenta

19

50

time

40

x = 50

30

20

10 0

1

2

3

4

5

6

p

Figure 5: classical free time (dashed), statistical time (doted) and average time with the semiclassical approximation for a free particle (solid) and with the barrier (thick line) to go from q to x

5

4

time

3

2

1

0

0

1

2

3

4

5

p

Figure 6: average semiclassical time to go from −a to a with (thick line) and without (solid) the wall

20

−d

q

x

d

−d

q

(a) t2 ≤ T ≤ t3

x

d

(b) t3 ≤ T ≤ t4

Figure 7: trajectories from q to x, with pi positive (solid) or negative (dashed)

−d

x1

q=0 x 2

x3

d

Figure 8: The length of the trajectories is at the most

0,5

0,5

0,4

0,4

0,3

0,3

√

2V0 T

0,30

0,25

|y|2

|y|2

|y|2

0,20

0,2

0,2

0,1

0,1

0,0

0,0

0,15

0,10

0,05

-30

-20

-10

0

10

20

x

(a) p = 0.05 and T = 10

30

0,00 -30

-20

-10

0

10

20

x

(b) p = 1 and T = 10

30

-4

-2

0

4

(c) inside the cavity for p = 1.5 and T = 37

Figure 9: Wavepacket at time T for two finite potential barriers

21

2

x