numerical solution of the schrödinger equation on...
TRANSCRIPT
Numerical solution of the Schrödinger
equation on unbounded domains
Maike Schulte
Institute for Numerical and Applied Mathematics
Westfälische Wilhelms-Universität Münster
Cetraro, September 2006
1
OUTLINE
Transparent boundary conditions (TBC)
OUTLINE 2
OUTLINE
Transparent boundary conditions (TBC)
Analytical transparent boundary conditions for the Schrödinger
equation
OUTLINE 2-A
OUTLINE
Transparent boundary conditions (TBC)
Analytical transparent boundary conditions for the Schrödinger
equation
Derivation of discrete transparent boundary conditions (DTBC) for theSchrödinger equation for a higher order compact 9-point scheme
OUTLINE 2-B
OUTLINE
Transparent boundary conditions (TBC)
Analytical transparent boundary conditions for the Schrödinger
equation
Derivation of discrete transparent boundary conditions (DTBC) for theSchrödinger equation for a higher order compact 9-point scheme
Approximation of DTBC
OUTLINE 2-C
OUTLINE
Transparent boundary conditions (TBC)
Analytical transparent boundary conditions for the Schrödinger
equation
Derivation of discrete transparent boundary conditions (DTBC) for theSchrödinger equation for a higher order compact 9-point scheme
Approximation of DTBC
Simulation of quantum wave guides
OUTLINE 2-D
OUTLINE
Transparent boundary conditions (TBC)
Analytical transparent boundary conditions for the Schrödinger
equation
Derivation of discrete transparent boundary conditions (DTBC) for theSchrödinger equation for a higher order compact 9-point scheme
Approximation of DTBC
Simulation of quantum wave guides
Extension of the DTBC to (nearly) arbitrary potentials
OUTLINE 2-E
OUTLINE
Transparent boundary conditions (TBC)
Analytical transparent boundary conditions for the Schrödinger
equation
Derivation of discrete transparent boundary conditions (DTBC) for theSchrödinger equation for a higher order compact 9-point scheme
Approximation of DTBC
Simulation of quantum wave guides
Extension of the DTBC to (nearly) arbitrary potentials
more realistic simulations
OUTLINE 2-G
OUTLINE
Transparent boundary conditions (TBC)
Analytical transparent boundary conditions for the Schrödinger
equation
Derivation of discrete transparent boundary conditions (DTBC) for theSchrödinger equation for a higher order compact 9-point scheme
Approximation of DTBC
Simulation of quantum wave guides
Extension of the DTBC to (nearly) arbitrary potentials
more realistic simulations
solving the Schrödinger equation on circular domains
OUTLINE 2-H
TRANSPARENT BOUNDARY CONDITIONS (IN GENERAL )
Rn
ψ
Γ
Ω
Definition (TBC) :Consider a given whole-space initial value problem (IVP) on Rn
and Ω ⊂ Rn, Γ = ∂Ω. We are interested in the solution of the IVPon Ω. Therefore we need new artificial boundary conditions on Γ.We call these artificial BC transparent, if the solution of the IVBPon Ω corresponds to the whole-space solution of the IVP restrictedon Ω.
TRANSPARENT BOUNDARY CONDITIONS (IN GENERAL) 3
ANALYTICAL TBC FOR THE SCHRÖDINGER EQUATION
IVP: time-dependent Schrödinger equation (here: 1D)
i~∂
∂tψ(x, t) =
„
− ~2
2m∗
∂2
∂x2+ V (x, t)
«
ψ(x, t), x ∈ R, t > 0
ψ(x, 0) = ψI(x) ∈ L
2(R)
on a domain of interest Ω = x ∈ R|0 < x < X.
ANALYTICAL TBC FOR THE SCHRÖDINGER EQUATION 4
ANALYTICAL TBC FOR THE SCHRÖDINGER EQUATION
IVP: time-dependent Schrödinger equation (here: 1D)
i~∂
∂tψ(x, t) =
„
− ~2
2m∗
∂2
∂x2+ V (x, t)
«
ψ(x, t), x ∈ R, t > 0
ψ(x, 0) = ψI(x) ∈ L
2(R)
on a domain of interest Ω = x ∈ R|0 < x < X.
Assumptions:
• supp ψI ⊆ Ω
• potential V (., t) ∈ L∞(R), V (x, .) is piecewise continuous
• V constant on R\Ω (here: V (x, t) = 0 for x ≤ 0 and V (x, t) = VX forx ≥ X)
Goal:
Calculate the solution ψ(x, t) ∈ C on Ω with TBC at x = 0 and x = X.
ANALYTICAL TBC FOR THE SCHRÖDINGER EQUATION 4-A
DERIVATION OF TBC
0
ψ
x
V = const V = const
X
G G21
ψ I
Ω
x ∈ G : x ∈ G2 :
i~ψt = − ~2
2m∗ψxx + V (x, t)ψ i~ψt = − ~
2
2m∗ψxx + V (x, t)ψ
ψ(x, 0) = ψI(x) v(x, 0) = 0
ψx(0, t) = (T0ψ)(0, t) v(X, t) = Φ(t) t > 0, Φ(0) = 0
ψx(X, t) = (TXψ)(X, t) limx→∞
v(x, t) = 0
vx(0, t) = (TXΦ)(t)
DERIVATION OF TBC 5
Laplace-transformation on the exterior domains :
vxx(x, s) +2im∗
~
“
s+iVX
~
”
v(x, s) = 0 x > X
v(X, s) = Φ(s)
limx→∞
v(x, s) = 0
vx(X, s) = (TXΦ)(s)
solution: v(x, s) = e−i
+r
2im∗
~
`s+
iVX~
´(x−X)
Φ(s)
⇒ (TXΦ)(s) = −r
2m∗
~e−
iπ4
+
r
s+iVX
~Φ(s)
With the inverse Laplace-Transformation follows the analytical TBC
ψx(X, t) = −r
~
2πm∗e−i π
4 e−i
VX~
t d
dt
tZ
0
ψ(X, τ)eiVX τ
~
√τ − t
dτ.
[J. S. Papadakis (1982)]
DERIVATION OF TBC 6
Application of DTBC for the Schrödinger equation:
• Simulation of quantum transistors in quantum waveguides (withinhomogeneous DTBC for the 2D Schrödinger equation)
• Analyse steady states and transient behaviour
x0 X
0
Y
Vcontrol
ψ inc
y
ΩΓ Γ1 2
x0 X
0
Yinc
Γ Γ2
ψ
1
y
Ω
Vcontrol ψ pw
DERIVATION OF TBC 7
FORMER STRATEGIES :
• Discretization of the analytic TBCs with an numericalapproximation of the convolution integral [e.g. B. Mayfield(1989)]
⇒ only conditionally stable, not transparent!
FORMER STRATEGIES: 8
FORMER STRATEGIES :
• Discretization of the analytic TBCs with an numericalapproximation of the convolution integral [e.g. B. Mayfield(1989)]
⇒ only conditionally stable, not transparent!
• Create a buffer zone Θ of the length d with a complex potentialV (X) = W − iA around the computational domain Ω withDirichlet 0-BC at ∂Θ and absorbing boundary conditions on ∂Ω
[e.g. L. Burgnies (1997)]
ψ=0ψ=00 X
Ω ΘΘ
⇒ unconditionally stable, unphysical reflections at theboundary, huge numerical costs
FORMER STRATEGIES: 8-A
SUCCESSFUL STRATEGIES
• Family of absorbing BCs (also for the non-linear Schrödingerequation, wave equation)
[J. Szeftel (2005)]
• discretize the whole space problem with an unconditionally stablescheme (e.g. Crank-Nicolson finite difference scheme) and calculatenew discrete transparent boundary conditions for the full discretizedSchrödinger equation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
|ψ|
DTBC: dx=0.00625, dt=2e−05, t=0.014
[A. Arnold, M. Ehrhardt (since 1995)]
SUCCESSFUL STRATEGIES 9
DERIVATION OF DISCRETE TRB
Discretize 2D Schrödinger equation:
• Crank-Nicolson scheme in time, tn = n∆t, n ∈ N, H := − 12∆ + V
Hamilton-Operator, Ω = [0,X] × [0, Y ]
„
1 +iH∆t
2
«
ψ(x, y, t+ ∆t) =
„
1 − iH∆t
2
«
ψ(x, y, t)
⇒„∂2
∂x2+∂2
∂y2
«`ψ
n+1(x, y) + ψn(x, y)
´
= gn+ 1
2 (x, y)`ψ
n+1(x, y) + ψn(x, y)
´+Wψ
n(x, y)
• compact 9-point scheme in space, xj = j∆x, yk = k∆y withj ∈ Z, 0 ≤ k ≤ K
• DTBC at x0 = 0 and xJ = X = J∆x with J ∈ Z
• 0-BC at y = 0 and y = Y = K∆y with K ∈ N
DERIVATION OF DISCRETE TRB 10
9-point discretization scheme:
(j,k)
(j+1,k+1)
(j+1,k)
“
D2x +D
2y +
∆x2 + ∆y2
12D
2x D
2y
”
ψn+ 1
2j,k
=
„
I +∆x2
12D
2x +
∆y2
12D
2y
« h
2Vn+ 1
2j,k ψ
n+ 12
j,k − 2iD+t ψ
nj,k
i
with
ψn+ 1
2j,k =
1
2
`ψ
n+1j,k + ψ
nj,k
´
D+t ψ
nj,k =
ψn+1j,k − ψn
j,k
∆t, n ≥ 0
D2x ψ
nj,k =
ψnj−1,k − 2ψn
j,k + ψnj+1,k
∆x2, j ∈ Z
D2y ψ
nj,k =
ψnj,k−1 − 2ψn
j,k + ψnj,k+1
∆y2, k ∈ N
DERIVATION OF DISCRETE TRB 11
Discrete Sine-Transformation in y-direction:
ψnj,m :=
1
K
K−1X
k=1
ψnj,k sin
„πkm
K
«
m = 0, . . . ,K
Motivation:Solve discrete stationary Schrödinger equation in 1D:
−1
2∆2
y χmj,k = E
mχ
mj,k, k = 0, . . . ,K
χmj,0 = χ
mj,K = 0.
The eigenfunctions χmj,k = sin
`πkm
K
´provide the energies
Em =
1
∆y2
“
1 − cos“πm
K
””
.
Hence follows for m = 0, . . . ,K
⇒ − 1
2∆y2
“
ψnj,k−1 − 2ψn
j,k + ψnj,k+1
”b
m=
1
∆y2
“
1 − cos“πm
K
” ”
ψnj,m.
DERIVATION OF DISCRETE TRB 12
Sine-Transformation of the discrete Schrödinger equation on theexterior domains j ≤ 0, j ≥ J yields for the modes m = 0, . . . , K:
Cmj+1ψ
n+1j+1,m + Cm
j−1ψn+1j−1,m +Rm
j ψn+1j,m
= (D − Cmj+1)ψ
nj+1,m + (D − Cm
j−1)ψnj−1,m + (Bm
j −Rmj )ψn
j,m.
DERIVATION OF DISCRETE TRB 13
Sine-Transformation of the discrete Schrödinger equation on theexterior domains j ≤ 0, j ≥ J yields for the modes m = 0, . . . , K:
Cmj+1ψ
n+1j+1,m + Cm
j−1ψn+1j−1,m +Rm
j ψn+1j,m
= (D − Cmj+1)ψ
nj+1,m + (D − Cm
j−1)ψnj−1,m + (Bm
j −Rmj )ψn
j,m.
Definition [Z - Transformation ]:The Z-Transformation of a sequence (ψn)n∈N is given by
Z ψn = Ψ(z) :=
∞∑
n=0
ψnz−n z ∈ C, |z| > 1.
DERIVATION OF DISCRETE TRB 13-A
One can show:
• Z“
ψn+1j,m
”
= −zψ0j,m + zΨm
j (z)
• ψ0J+1,k = ψ0
J−1,k = ψ0J,k = 0 for k = 0, . . . ,K
• Vj constant for j ≤ 1, j ≥ J − 1 ⇒ Cmj = Cm, Rm
j = Rm, Bmj = Bm
⇒ ΨJ+1(z) +
»Rz + R−B
C z + C −D
–
ΨJ(z) + ΨJ−1(z) = 0.
DERIVATION OF DISCRETE TRB 14
One can show:
• Z“
ψn+1j,m
”
= −zψ0j,m + zΨm
j (z)
• ψ0J+1,k = ψ0
J−1,k = ψ0J,k = 0 for k = 0, . . . ,K
• Vj constant for j ≤ 1, j ≥ J − 1 ⇒ Cmj = Cm, Rm
j = Rm, Bmj = Bm
⇒ ΨJ+1(z) +
»Rz + R−B
C z + C −D
–
ΨJ(z) + ΨJ−1(z) = 0.
This difference equation with constant coefficients is solved byΨj(z) = νj(z):
ν2(z) +
»Rz +R−B
C z + C −D
–
ν(z) + 1 = 0,
Physical background forces decay of the solution for j → ∞
⇒ |ν(z)| > 1 und ν(z)ΨJ(z) = ΨJ−1(z) → Z-transformed DTBC
DERIVATION OF DISCRETE TRB 14-A
Theorem [ DTBC for the 2D Schrödinger equation ]: Discretizethe 2D Schrödinger-Equation with the compact 9-point differencescheme in space and with the Crank-Nicolson scheme in time.Then the DTBC at xJ = J∆x and x0 = 0 for n ≥ 1 read
ψn1,m − s
(0)0,mψ
n0,m =
n−1∑
ν=1
s(n−ν)0,m ψν
0,m − am1 ψ
n−11,m ,
ψnJ−1,m − s
(0)J,mψ
nJ,m =
n−1∑
ν=1
s(n−ν)J,m ψν
J,m − amJ−1ψ
n−1J−1,m.
The convolution coefficients s(n)j,m can be calculated by
s(n)j,m = αm
j
(λmj )−n
2n− 1
[
Pn(µmj ) − Pn−2(µ
mj )
]
with the Legendre-Polynomials Pn ( P−1 ≡ P−2 ≡ 0).
DERIVATION OF DISCRETE TRB 15
Advantages of the new DTBC :
• no numerical reflections
• 3-point recursion for s(n)
• these DTBC have exactly the same structure like the DTBCcalculated with the 5-point scheme [A. Arnold, M. Ehrhardt]
• convergence: O(∆x4 + ∆y4 + ∆t2)
• same numerical effort like discretized analytical TRB
• s(n)j,m = O(n−3/2)
• CN-FD scheme with DTBC is unconditionally stable, withA := I + ∆x2
12 D2x + ∆y2
12 D2y follows:
D+t ||ψ||2A = 0 with ||ψ||2A := 〈ψ,Aψ〉. (1)
DERIVATION OF DISCRETE TRB 16
some drawbacks :
• DTBC are non-local in time→ high memory costs: the solution ψn
j,m has to be saved in
x0 and xJ for all time steps n = 1, 2, . . .
→ in each time step n = 1, 2, . . . you have to calculate K
convolutions of the length n
(FFT not useable, ⇒ O(Kn2))
DERIVATION OF DISCRETE TRB 17
some drawbacks :
• DTBC are non-local in time→ high memory costs: the solution ψn
j,m has to be saved in
x0 and xJ for all time steps n = 1, 2, . . .
→ in each time step n = 1, 2, . . . you have to calculate K
convolutions of the length n
(FFT not useable, ⇒ O(Kn2))
• These DTBC are given in the Sine-transformed form: BC ofone mode is a linear combination of all other boundary points
→ diagonal structure of the system matrix is destroyed
DERIVATION OF DISCRETE TRB 17-A
0 20 40 60 80 100
0
10
20
30
40
50
60
70
80
90
100
Sparsity pattern of the system matrix for J = K = 10
DERIVATION OF DISCRETE TRB 18
Example 1 :
free 2D Schrödinger equation on Ω = [0, 2] × [0, 2] with the initialdata:
ψI(x, y) = eikxx+ikyy−60
(
(x− 12 )
2+(y− 1
2 )2
)
, (x, y) ∈ Ω
0
50
100
020
4060
80100
120
0
0.2
0.4
0.6
0.8
1
x
it = 10
y
T = 10
0
50
100
020
4060
80100
120
0
0.2
0.4
0.6
0.8
1
x
it = 60
y
T = 60∆t
0
50
100
020
4060
80100
120
0
0.01
0.02
0.03
0.04
0.05
x
it = 160
y
T = 80∆t
DERIVATION OF DISCRETE TRB 19
*
APPROXIMATION OF DTBC
• Idea: Approximate the convolution coefficients s(n)j,m by a sum of
exponentials:
s(n) ≈ s(n) =L
∑
l=1
blq−nl , n ∈ N, |ql| > 1, L ≤ 40
APPROXIMATION OF DTBC 20
*
APPROXIMATION OF DTBC
• Idea: Approximate the convolution coefficients s(n)j,m by a sum of
exponentials:
s(n) ≈ s(n) =L
∑
l=1
blq−nl , n ∈ N, |ql| > 1, L ≤ 40
• bl, ql are calculated by the Padé - Approximation of
f(x) =2L−1∑
n=0
s(n)xn, |x| ≤ 1
[A.Arnold, M. Ehrhardt, I. Sofronov (2003)]
APPROXIMATION OF DTBC 20-A
*
Recursion formula for the convolution coefficients:n−1X
t=0
s(n−t)
ψt =
LX
l=1
c(n)l
with
c(n)l = q
−1l c
(n−1)l + blq
−1l ψ
n−1, n = 1, . . . , N
c(0)l = 0
APPROXIMATION OF DTBC 21
*
Recursion formula for the convolution coefficients:n−1X
t=0
s(n−t)
ψt =
LX
l=1
c(n)l
with
c(n)l = q
−1l c
(n−1)l + blq
−1l ψ
n−1, n = 1, . . . , N
c(0)l = 0
Advantages:
If you have calculated bl, ql once for a set ∆x,∆y,∆t, V , you’ll easilyderive b∗l , q
∗
l for any ∆x∗,∆y∗,∆t∗, V ∗ by
q∗
l =qla− b
a− qlb
b∗
l = blql
aa− bb
(a− qlb)(qla− b)
1 + q∗l1 + ql
Numerical effort: O(Kn2) → O(KLn)
Memory: O(Kn) → O(KL)
APPROXIMATION OF DTBC 21-A
*
Example 2 :
free 2D Schrödinger-Equation in Ω = [0, 1] × [0, 1]
Initial function:
ψI(x, y) = sin(πy)eikxx−60(x− 1
2 )2 , (x, y) ∈ Ω
00.2
0.40.6
0.81
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
y
x
IT=150
|psi
(x,y
)|, t
=T
L = 5 : T = 150∆t
00.2
0.40.6
0.81
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
y
x
IT=300
|psi
(x,y
)|, t
=T
T = 300∆t
00.2
0.40.6
0.81
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
y
x
IT=500
|psi
(x,y
)|, t
=T
T = 500∆t
00.2
0.40.6
0.81
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
y
x
IT=150
|psi
(x,y
)|, t
=T
L = 20 : T = 150∆t
00.2
0.40.6
0.81
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
y
x
IT=300
|psi
(x,y
)|, t
=T
T = 300∆t
00.2
0.40.6
0.81
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
y
x
IT=500
|psi
(x,y
)|, t
=T
T = 500∆t
APPROXIMATION OF DTBC 22
SIMULATION OF QUANTUM WAVEGUIDES
x0 X
0
Y
Vcontrol
ψ inc
y
ΩΓ Γ1 2
40.5nm32nm
20nm
20nm
• Incoming wave at x = 0:
ψinc(0, y, t) = sin(πy)e
−iEt~ , E = 29.9meV
• inhomogeneous DTBC at x = 0, DTBC at x = X
SIMULATION OF QUANTUM WAVEGUIDES 23
Little trick to suppress oscillations in time:
ψinc oscillates like e−iEt
~ in time.
E big:
fast oscillation of the solution in time
small time step size is necessary
high numerical effort for the analysis of steady state and long-timebehaviour
Define
ϕ(x, y, t) := e−iωt
ψ(x, y, t) mit ω = −E~.
ϕ solve the modified Schrödinger equation
i~ϕt = − ~2
2m∗(ϕxx + ϕyy) + (V − ω~)
| z
=:V
ϕ
SIMULATION OF QUANTUM WAVEGUIDES 24
0 1 2 3 4 5 6 7 8 9 100
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
t [10−12 s]
V=0V=−E
f1(t) = ||ψ1(x, y, t) − ψref (x, y, t)||2f2(t) = ||ψ2(x, y, t) − ψref (x, y, t)||2
ψ1 is calculated with V = 0, ψ2 with V = −E and ψref is a numerical
reference solution, which has been calculated with high accuracy.
SIMULATION OF QUANTUM WAVEGUIDES 25
0
20
40
60
0 10 20 30 40 50 60
0
0.5
1
1.5
2
2.5
3
y
Initial function: X=1, Y=0.86667, dx=0.016667, dy=0.016667, dt=0.0002, V=0
x
|psi
(x,y
)|, t
=0
T = 0∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
2.5
3
y [nm]
Quantenwellen.m, solution after T = −1.7998 ps, it = 1000
|ψ(x
,y,T
)|
T = 1000∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
2.5
3
y [nm]
Quantenwellen.m, solution after T = −0.1998 ps, it = 9000
|ψ(x
,y,T
)|
T = 9000∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
2.5
3
y [nm]
Quantenwellen.m, solution after T = 0.0202 ps, it = 10100
|ψ(x
,y,T
)|
T = 10100∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
2.5
3
y [nm]
Quantenwellen.m, solution after T = 0.0802 ps, it = 10400
|ψ(x
,y,T
)|
T = 10400∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
2.5
3
y [nm]
Quantenwellen.m, solution after T = 1.9002 ps, it = 19500
|ψ(x
,y,T
)|
T = 19500∆t
SIMULATION OF QUANTUM WAVEGUIDES 26
EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS
Drawbacks of the simulations:
• Hard walls (zero Dirichlet BCs) and edges are not practicable inindustry!
→ Geometry of computational domain shall be realized by
potentials also in the simulations.
→ How to choose the incoming wave and the initial function then?
• Potentials are NOT constant in the exterior domains!→ Decoupling of the modes for V (x, y) 6= const. after Sine-
Transformation is not possible
EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 27
EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS
Drawbacks of the simulations:
• Hard walls (zero Dirichlet BCs) and edges are not practicable inindustry!
→ Geometry of computational domain shall be realized by
potentials also in the simulations.
→ How to choose the incoming wave and the initial function then?
• Potentials are NOT constant in the exterior domains!→ Decoupling of the modes for V (x, y) 6= const. after Sine-
Transformation is not possible
Discrete Schrödinger equation:5-point scheme in space, Crank-Nicolson in time:
i~D+t ψ
nj,k = − ~
2
2m∗
`D
2x +D
2y
´ψ
n+ 12
j,k + Vn+ 1
2j,k ψ
n+ 12
j,k
Calculate new Eigenfunctions, which take the potential into account!
EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 27-A
Solve the eigenvalue equation on the exterior domains (j ≤ 0, j ≥ J):
− 1
2∆y2
`χ
nj,k−1,m − 2χn
j,k,m + χnj,k+1,m
´+ V
n+ 12
k χnj,k,m = E
nj,mχ
nj,k,m
with
∆y
KX
k=0
|χnj,k,m|2 = 1 and χ
nj,0,m = χ
nj,K,m = 0
for 0 ≤ k, m ≤ K, n > 0.
Transformation w.r.t. the eigenfunctions
ψnj,m = ∆y
K−1X
k=1
χnj,k,mψ
nj,k, 0 ≤ m ≤ K.
yields
i~D+t ψ
nj,m = − ~
2
2m∗D
2x ψ
n+ 12
j,m + Enj,mψ
n+ 12
j,m
same structure as the sine-transformed Schrödinger equation![N. Ben Abdallah, M.S. (2005)]
EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 28
Example 3: 2D channel
0 10 20 30 40 500
10
20
30
40
50
60
70
80
90
100
110Potential V(0,y)
yk
V(0
,yk)
potential V (y) = 400y(1 − y)
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Eigenfunction χ for j=0, |χ j,k,m
|2 of the m = 1st mode with the Eigenvalue λ = 65.3211
yk
|χ j,
k,m
|2
eigenfunction χ of m = 1
initial function:
ψIj,k = e
ikxj∆xχ
0j,k,m
incoming wave:
ψincj,k,n = e
ikxj∆xχ
0j,k,me
−iExn∆t with Ex =1 − cos(kx∆x)
∆x2
EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 29
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
y
t = 0
x
|ψ(x
,y,t)
|2
T = 0∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
y
t = 50 ⋅ ∆ t = 0.01
x
|ψ(x
,y,t)
|2
T = 50∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
y
t = 150 ⋅ ∆ t = 0.03
x
|ψ(x
,y,t)
|2
T = 150∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
y
t = 250 ⋅ ∆ t = 0.05
x
|ψ(x
,y,t)
|2
T = 250∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
y
t = 500 ⋅ ∆ t = 0.1
x
|ψ(x
,y,t)
|2
T = 500∆t
0
0.5
1
00.2
0.40.6
0.81
0
0.5
1
1.5
2
y
t = 1700 ⋅ ∆ t = 0.34
x
|ψ(x
,y,t)
|2
T = 1700∆t
EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 30
SCHRÖDINGER EQUATION ON CIRCULAR DOMAINS
Solve the Schrödinger equation (in polar coordinates)
iψt = −1
2
„1
r(rψr)r +
1
r2ψθθ
«
+V (r, θ, t)ψ, r > 0, 0 ≤ θ ≤ 2π, t > 0
on a circular domain Ω = [0, R] × [0, 2π] with TBC at x = R.
Problems:
• solve a second order difference equation with varying coefficients:
ajΨJ+1(z) + bj(z)ΨJ(z) + cjΨJ−1(z) = 0
• calculation of the convolution coefficients for the DTBC
→ "recursion from infinity"
• singularity at r = 0 → not equidistant offset-grid rj = rj+ 12
• approximation of the convolution coefficients and the -sum
[A.Arnold, M. Ehrhardt, M. S., I. Sofronov (2006)]
SCHRÖDINGER EQUATION ON CIRCULAR DOMAINS 31
Example 4:free Schrödinger equation on unit disc Ω1 = [0, 1] × [0, 2π]
ψI(r, θ) =
1√αxαy
e2ikxr cos θ+2ikyr sin θ−
(r cos θ)2
2αx−
(r sin θ)2
2αy
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
0
1
0
5
10
15
20
initial function on Ω1 = [0,1]x[0,2π], ∆r = 1/128, ∆θ = 2π /128
|ψ(r
,θ,0
)|
initial function
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
0
1
0
1
2
3
4
5
6
7
solution on Ω1 = [0,1]x[0,2π] at time t= 0.125, ∆t = ∆r = 1/128, ∆θ = 2π /128
|ψ(r
,θ,t)
|
T = 0.125
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
0
1
0
0.5
1
1.5
2
2.5
3
3.5
solution on Ω1 = [0,1]x[0,2π] at time t= 0.25, ∆t = ∆r = 1/128, ∆θ = 2π /128
|ψ(r
,θ,t)
|
T = 0.25
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
initial function on Ω1 = [0,1]x[0,2π], ∆r = 1/128, ∆θ = 2π /128
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
solution on Ω1 = [0,1]x[0,2π] at time t= 0.125, ∆t = ∆r = 1/128, ∆θ = 2π /128
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
solution on Ω1 = [0,1]x[0,2π] at time t= 0.25, ∆t = ∆r = 1/128, ∆θ = 2π /128
SCHRÖDINGER EQUATION ON CIRCULAR DOMAINS 32
Error due to the scheme/TBC:
L(ψ, ϕ, tn,Ω) :=||ψ(rj , θk, tn) − ϕ(rj , θk, tn)||Ω,2
||ϕ(rj , θk, tn)||Ω,2
ψ: numerical solution
ϕ: exact solution or numerical reference solution
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510
−4
10−3
10−2
10−1
100
t
relative L2 error due to the scheme
J=K=64, ∆t=1/64J=K=128, ∆t=1/128J=K=256, ∆t=1/256
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510
−16
10−15
10−14
10−13
10−12
10−11
10−10
t
relative L2−error due to the TBC
J=K=64, ∆t=1/64
J=K=128, ∆t=1/128
J=K=256, ∆t=1/256
SCHRÖDINGER EQUATION ON CIRCULAR DOMAINS 33