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Absorbing boundary conditions for Schrodinger equation in a time-dependent interval
O. Karpova1'2, K. Sabirov1'3, D. Otajanov1, A. Ruzmetov1, A. A. Saidov1
1 Turin Polytechnic University in Tashkent, 17 Kichik Halka Yuli, Tashkent, 100095, Uzbekistan 2 National University of Uzbekistan, Tashkent, Uzbekistan 3 Tashkent University of Information Technologies, Tashkent, Uzbekistan
ola_july@mail.ru
PACS 02.60.Lj; 02.70.Bf DOI 10.17586/2220-8054-2017-8-1-13-19
We consider the reflection-transmission of the Gaussian wave packet through the moving wall with absorbing boundary conditions based on the time dependent one-dimensional Schrodinger equation. The reflection coefficient is calculated for the case when the walls are fixed, and probability density is calculated for the case when the wall is moving linearly.
Keywords: absorbing boundary conditions, Gaussian wave packet, one-dimensional Schrodinger equation.
Received: 10 August 2016 Revised: 5 September 2016
1. Introduction
Absorbing boundary conditions (ABC) for different wave equations have attracted much attention in different practically important contexts (see, e.g., papers [1-10] for review). Such boundary conditions describe absorption of particles and waves in their transition from one domain to another one. From the physical viewpoint particle absorption can occur in dissipative media. In addition, such type of boundary condition appears in reflectionless transmission of particles (waves) through the boundary of a given domain. Therefore one uses similar terminology for the boundary conditions of both types of processes, calling them absorbing or transparent boundary conditions. Strict mathematical treatment of ABC for different wave equations, including quantum mechanical Schrodinger equation can be found in the Refs. [11-16]. For both processes the boundary conditions can be derived by factorization of the differential operator corresponding to a wave equation which in general lead to complicated equations for the boundary conditions The explicit form of such boundary conditions is much more complicated than those of Dirichlet, Neumann and Robin conditions.
For absorbing boundary conditions the wave equation cannot be solved analytically and always requires using numerical methods. Depending on the type of the wave equation, they require different discretization schemes, which vary also for the process types. Finding an effective numerical scheme is a complicated task, especially, for quantum mechanical wave equations. Therefore, in some cases it is reasonable to use approximate methods enabling simplification of the numerical solution. One such method, based on the use of dispersion relation was introduced in the Ref. [11] for solving time-dependent Schrodinger equation with ABC on a finite interval. Later, it was improved by Kuska [11] with the use of alternative approximation for dispersion relation. In this paper we extended the results of Kuska to the case of moving boundaries. In particular, using the approach developed in the Ref. [12], we reduce time-dependent Schrodinger equation with moving boundary conditions into that of fixed ones. Imposing the ABC for the problem with moving boundaries, we derive the corresponding boundary conditions for the Schrodinger equation on a finite interval with unit length which is then solved numerically.
2. Absorbing boundary conditions for fixed interval
Here, following the Refs. [10,11] we briefly recall derivation of the approximate ABC for one dimensional Schrodinger equation on a finite interval. Consider the one-dimensional time-dependent Schrodinger equation:
ih^t (x,t) = -2m J0 + V (x) 0 (x,t), (1)
describing the motion of a quantum particle with the mass m in the interval x under the influence of
the potential V. Taking a plane wave of the form:
0 (x, t) = exp [—i(wt — kx)], (2)
from Eqs. (1) and (2) one gets the dispersion relation for the wave vector k [11]:
h2k2 = 2m [hw - V]. (3)
This relation can be solved for k and yields (in the units m = h = 1):
k = ±^2 [w - V], (4)
where the plus sign describes waves moving to x = and the minus sign means waves moving to x = —to. The left boundary has to be transparent for waves travelling to the left and the right boundary must be transparent for those heading right.
The square root function can be approximated using the rational function approximation method following [11]:
,- 1 + 3z/z0
Vz — zo ~Vzo^;—1—. (5)
3 + z/zo
With the approximation (5) for the square root in the dispersion relation (4) one gets:
with
12 (w — V1 + 3z k2
2(w — V)
k = ±k V k0 "±ko^T7, (6)
k02
We write (6) as:
k (3 + z) «±k0 (1 + 3z). Putting expression for z into the above equation, one gets:
k^3ko2 + w — v) «±k„(f + 3w — 3v).
By using the correspondence between quantities k and w with their operator definitions:
k ^ — i — and w i—, dx dt
we obtain the following partial differential equation for the ABC:
—1( 3 f — v) £ (x,t) + d^tdx (x,t)= ±ko ( f — (x,t) ± 3iko ^ (x,t). (7)
The plus sign in (7) corresponds to the boundary conditions for the right side wall of the interval, while the minus sign in (7) corresponds to the boundary conditions for the left wall of the interval.
2.1. Numerical results
For the numerical experiments, we consider the transport of Gaussian wave packet:
f (x — £)2 \
-0(x, 0) = i/iff exp--exp(ik0x), (8)
V /
through the wall located at the right side. We consider that the particle is initially located near the right wall (£ = 3L/4) in order to minimize the calculation time of the reflection-transmission process through this wall. The discrete one dimensional space is chosen between —2L < x < 4L, which is large enough to satisfy the normalization condition:
/«x,t)*M)dx = 1, (9)
during numerical computation.
In Fig. 1, the evolution of the Gaussian wave packet described by the time-dependent Schrodinger equation with ABC given by Eq. (7) is presented. As it can be seen from this plot, for given initial conditions of the Gaussian wave packet, the reflectionless transmission through the boundary of the domain occurs.
In Fig. 2, we plot the dependence of the reflection coefficient on the initial mean wave vector of the wave packet. As it can be seen from the plot, there is an energy range for which the reflection coefficient becomes minimal. For these energies the wall becomes almost transparent. In order to calculate the reflection coefficient, R, we follow the evolution of the initially normalized probability of the quantum particle (wave packet) inside the interval (x G [0, L]) and continue this until that moment when the scattering starts through the domain wall
Fig. 1. (Color online) The spatiotemporal evolution of the probability density in logarithmic scale for the Gaussian wave packet described by Eq. (1), for which absorbing boundary conditions (7) are imposed. Initial wave packet width a = 0.05L, the length of the interval L =1, and the initial position of quantum particle (wave packet) is £ = 3L/4. Numerical calculations are carried out with the discretization parameters: one dimensional space grid size Ax = 10-3 and time step size At = 10-4. The mean value of the wave vector of the initial wave packet is k0 = 25.
(at x = L) with ABC, and to the time when the scattering (reflection and transmission) processes are totally completed. Normalization condition implies that the summation of the reflection and transmission coefficient must be equal to 1, until the domain (discretized 1D space) is taken long enough (-2L < x < 4L) where the tail of probability density of the wave packet will not reach to the edge of the domain (at x = 4L) due to dispersion during simulation.
For the normalized wave function by Eq. (9), one can easily track of the transmitted quantum particle (wave packet) position by computing the following integral:
4L
/.rtMWtMfc 00)
L
which is necessary to verify that the transmission was completed totally.
Easy quantification of the wave packet reflection through the ABC can be done by checking the probability of the wave packet inside the domain —2L < x < 4L. During the transmission process, the integral (10) will be increasing and when it reaches to some value and stabilizes at this value, one can be sure that the process is completed.
3. Absorbing boundary conditions for time-dependent interval
Here, we extend the treatment of the previous section to the case of moving boundaries, by considering time-dependent interval with one moving end which is given as:
x = 0 and x = L(t) . (11)
Earlier, the Schrodinger equation with time-dependent boundary conditions given in a one-dimensional box has been studied by many authors (see, e.g., [12,17] and references therein) The Schrodinger equation with time-dependent boundary conditions is an ill posed problem unless the moving boundaries are not mapped onto fixed ones [12,17]. This can be done by introducing a new coordinate as [12,17]:
Fig. 2. Dependence of the reflection coefficient on the initial wave vector of wave packet is computed. Initial wave packet width a = 0.05L, the length of the interval L =1 and the initial position of quantum particle (wave packet) £ = 3L/4. Numerical calculations are carried out with the discretization parameters: one dimensional space grid size Ax = 10-3 and time step size At = 10-4.
y=Lt), (12)
which leads to Eq. (1) in the following form (V = 0):
M (y,t) 1 dV (y,t) ,L (y,t)
dT~ = - 2L dy2 + iLy~d^, (13)
with
L- dL = ~dt'
And for the partial differential equation (13), one can apply the new boundary conditions at y = 0 and y = 1. The normalization condition for the wave function ^(y,t) is given by:
J ^*(y,t)^(y,t)dy = const. (14)
To construct ABC, the boundary should be almost transparent for a plane wave of the form:
^(y, t) = exp(-iwt + ikLy). (15)
Then, from Eqs. (15) and (13), one gets the dispersion relation for the wave vector k:
k2
w = T.
This relation can be solved for k and yields:
k2 - = 0,
with the solution:
ki,2 = ±\/2~U).
t
Fig. 3. The spatiotemporal evolution of the probability density in logarithmic scale for the Gaussian wave packet described by Eq. (13) for linearly moving (with velocity b = 10-6) boundary given by Eq. (17). The mean value of wave vector of the initial wave packet k0 = 70. Initial value of the wave packet width a = 0.001L0, the initial length of the interval L0 = 1 and the initial position of quantum particle (wave packet) at £ = 3L0/4. Numerical calculations are carried out with the discretization parameters: one dimensional fixed space grid size Ay = 10-3, and time step size At = 10-4.
Using the rational function approximation as in the above equation (5) we write:
k 4. k /2W 1 + 3z
k = ±k0V k2 "±k0^T7'
with
_ 2w
ko
Replacing k and w by corresponding differential operators as:
d d
k — i —, w i—,
dy dt
we obtain adsorbing boundary conditions the same as in the case of a fixed interval Eq. (7):
3ik2 d d2 k3 d
—"2° ay^(y,t) + dddy^(y,t) = ± ^(y,t) ± 3iko ^(y,t). (16)
Here, the equation with the positive sign corresponds to the ABC at the right boundary, while the negative sign corresponds to the ABC at the left boundary.
Fig. 4. (Color online) Profile of the Gaussian WP at different time moments for fixed (green) and for time-dependent (red) intervals. The parameters of the packet are chosen as £ = 3/4; a = 0.001; k0 = 20.
Furthermore, using the following discretization scheme [11]:
1
4 vTj+i + v +
0 (y, t) « - j1 + vn+1 + vn+1 + j ,
dy (y,t) « j - vr1 + vn+1 - j, # (y,t) « .A j1 + vr1 - V+1 - j,
dt dtdy
(y, t)
2At 1
AyAt
(vn+11 - v?+1 - v?+1 + v?).
with
v? = v(yj ,tn).
We have the following discretization for the boundary condition:
-3ik2 .A ( v+11 - v?+1+- v?)+AyAt ( j1 - v?+1 - +v?)
2Ay 1
AyAt 1
=±k3 4 j+v?+1++j ± ^ ^ (v^1+v?+1 - - j
or
b2v?+1 + b1v?+11 = + Mj+1, for y = 0, (j = 1), ^j1 + ^j! = bsv? + b4v?+1, for y =1, (j = yAy),
where the coefficients are given:
3ik2
bi = —
1 kg 3ik0
+---+ — +--0,
2Ay AyAt 4 2At '
3ik0 1 kg 3ik0 63 = -7 1
+ -o -
62 =
64 =
3ik2
2A
3ik,
Ay AyAt 4
2 1 3
kg 3ik0
+ -0 +--0,
+ ' + 2At,
+
k3 3«ko +—:--
2Ay AyAt ' 4 2At' "4 2Ay AyAt 4 2At'
In Fig. 3, the evolution of the Gaussian wave packet described by Eq. (13) is presented for the linearly moving boundary given by:
L(t) = Lo + bt. (17)
1
The plot shows reflectionless transmission of the packet through the boundary of the interval.
Figure 4 presents the profile at the GWP at different time moments both for constant and time-dependent intervals. The initial position of the packet is chosen as £ = 3/4. As is shown in the plot, in case of time-dependent BCs dispersion of the packet occurs more quickly than that for the fixed interval.
4. Conclusions
Thus, we have formulated and solved the problem of absorbing boundary conditions for the time-dependent interval by considering time-dependent Schrodinger equation. The time-dependence of the boundary is taken as linear. We found that unlike the case of fixed boundaries, for time-dependent absorbing boundary conditions provide more extensive (reflectionless) transmission of the particle through the domain boundaries and during considerably shorter time. The results of this work can be useful for modeling of quantum systems touching to absorbing or transparent media.
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