Splitting methods and the Magnus expansion for the
time dependent Schrödinger equation
Sergio Blanes
Instituto de Matemática Multidisciplinar
Universidad Politécnica de Valencia, SPAIN
Workshop:
Time dependent quantum mechanics – analysis and numerics
Oslo April 27
Oslo,
27-29,
29 2010
The Schrödinger
g equation
q
The system
y
can interact with other systems
y
or external fields and can have a
more involved structure
but the Hamiltonian has to be a linear Herminian operator.
After Spectral decomposition, Spatial discretization, etc.,
in general, we have to solve a linear ODE in complex variables
where c = (c1,…, cN)T ∈ N and H ∈  N×N is an Hermitian matrix.
If H is constant, the formal solution is given by
Th solution
The
l i corresponds
d to a unitary
i
transformation.
f
i
If H is of large dimension, it is important to have an efficient procedure to
compute
t or approximate
i t the
th exponential
ti l multiplying
lti l i a vector.
t
If H is time-dependent the problem can be more involved. We can be
i t
interested
t d to
t write
it the
th solution
l ti as an exponential
ti l
with W; being and skew-Hermitian operator. There is the problem, however,
g
of such operator.
p
of the existence and convergence
Depending on the problem and the techniques used, the structure of the
matrix H(t) can differ considerably, and then different numerical techniques
have to be used.
We know the system has several qualitative properties (preservation of the
energy, momentum, norm, etc.) and it is desirable the approximation
methods
h d share
h
these
h
qualitative
li i properties.
i
I this
In
thi ttalk
lk we b
briefly
i fl consider:
id
Splitting Methods
and
Magnus integrators
Splitting Methods
Consider the autonomous time dependent SE
It is separable in its kinetic and potential parts. The solution of the
discretised equation is given by
where c = (c1,…, cN)T ∈ N and H = T + V ∈ N×N Hermitian matrix.
Fourier methods are frequently used
i the
is
h ffast F
Fourier
i transform
f
(FFT)
Consider the Strang-splitting or leap-frog second order method
Notice that
the exponentials are computed only once and are stored at the beginning.
Similarly, for the kinetic part we have
This splitting was proposed in:
Feit, Fleck, and Steiger, J. Comput. Phys., 47 (1982), 412.
Higher Orders
The well known fourth-order
fourth order composition scheme
with
Higher Orders
The well known fourth-order
fourth order composition scheme
with
Or in general
with
-Creutz-Gocksh(89)
-Suzuki(90)
S
ki(90)
-Yoshida(90)
Higher Orders
The well known fourth-order
fourth order composition scheme
with
Or in general
with
-Creutz-Gocksh(89)
-Suzuki(90)
S
ki(90)
-Yoshida(90)
Excellent trick!!
In practice, they show low performance!!
Basic References
E. Hairer, C. Lubich, and G. Wanner,
Geometric Numerical Integration. Structure-Preserving Algorithms
for Ordinary Differential Equations, Springer-Verlag, (2006).
R.I. McLachlan and R. Quispel,
Splitting methods, Acta Numerica, 11 (2002), 341-434.
S. Blanes, F. Casas, and A. Murua,
Splitting and composition methods in the numerical integration of
differential equations, Boletín SEMA, 45 (2008), 89-145. (arXiv:0812.0377v1).
Polynomial Approximations: Taylor, Chebyshev and Splitting
Let us consider again the linear time dependent SE
with H real and symmetric.
This problem can be reformulated using real variables.
Consider
then
Hamiltonian system:
or, in short:
z’ = M z
Formal solution:
with
z = (q,p)T
The Taylor Method
An m-stage Taylor method for solving the linear equation can be
written as
where
is the Taylor expansion of the exponential. It is a
y
function of degree
g
m which approximates the exact
polynomial
solution up to order m
and we can advance each step by using the Horner's algorithm
This algorithm can be trivially rewritten in terms of the real vectors
vectors,
q,p, and it only requires to store two extra vector q and p.
The Taylor Method
The matrix
written as
that propagates the numerical solution can be
where the entries T1(y) and T2(y) are the Taylor
y series
expansion of cos(y) and sin(y) up to order m, i.e.
Notice that
i iis not a symplectic
it
l i transformation!
f
i ! Th
The eigenvalues
i
l
are given
i
b
by
Th scheme
The
h
iis stable
t bl if
For practical purposes, we require however
The Chebyshev Method
The Chebyshev method approximates the action of the exponential
on the initial conditions by a near-optimal polynomial given by:
where
with
Here, Tk(x) is the kth Chebyshev
polynomial generated from the recursion
and T0(x)=1, T1(x)=x. Jk(w) are the Bessel functions of the first kind
which provides a superlinear convergence for m > w or, in other
words,
d when
h
The Chebyshev Method
The Clenshaw algorithm allows to compute the action of the polynomial
by storing only two vectors
which can also be easily rewritten in term of the real vectors q,p.
The scheme can be written as
As in the Taylor case:
It is not a symplectic transformation.
The Symplectic Splitting Method
We have built splitting methods for the harmonic oscillator!!!
Exact solution (ortogonal and symplectic)
We consider the composition
Notice that
The Symplectic Splitting Method
We have
which are polynomials of degree twice higher as in the previous cases
for the same number of stages.
The algorithm: a generalisation of the Horner
Horner's
s algorithm or the
Clenshav algorithm
It only requires to store one additional real vector of dimension
The Symplectic Splitting Method
The methods preserve symplecticity by construction:
Stability: M is stable if |Tr K|<2, i.e.
Theorem:Any composition method is conjugate to an orthogonal method,
and unitarity is preserved up to conjugacy.
There is a recursive p
procedure to g
get the coefficients of the splitting
p
g
methods from the coefficients of the matrix K.
We can build different matrices with:
- Large stability domain
- Accurate approximation to the solution in the whole interval (like
Chebyshev)
- Methods with different orders of accuracy and very large number of
satges.
Taylor
order
tol=inf
tol=10-8
tol=10-4
10.
0.
0.
0.
15. 0.111249 0.111249 0.111249
20. 0.164515 0.164515 0.164515
25.
0.
0.065246 0.065246
30.
0.
0.108088 0.108088
35. 0.0461259 0.138361 0.322661
40 0.0804521
40.
0 0804521 0
0.160884
160884 0.321618
0 321618
45.
0.
0.178294 0.320804
50.
0.
0.192154 0.32015
Chebyshev
order
tol=inf
tol=10-8
tol=10-4
20. 0.00362818 0.321723 0.643217
25
25.
0 00233368 0.384289
0.00233368
0 384289 0.64029
0 64029
30. 0.00162599 0.425648 0.638324
35. 0.00119743 0.45502 0.636912
40 0
40.
0.000918402
000918402 0.476957
0 476957 0.63585
0 63585
45. 0.000726646 0.635021 0.635021
50. 0.000589228 0.634356 0.634356
55
55.
0
0.
0 633812 0.633812
0.633812
0 633812
60.
0.
0.633357 0.633357
Splitting methods
Error bounds
Schrödinger equation with a Poschl-Teller potential
with
Initial conditions
Poschl-Teller potential
Taylor: 10-40
0
-2
LOG(E
ERROR)
-4
-6
-8
-10
-12
-14
14
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
LOG(FFT CALLS)
3.9
4
4.1
Poschl-Teller potential
Split: 4-12
0
-2
LOG(E
ERROR)
-4
-6
-8
-10
-12
-14
14
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
LOG(FFT CALLS)
3.9
4
4.1
Poschl-Teller potential
Chebyshev: 20-100
0
-2
LOG(E
ERROR)
-4
-6
-8
-10
-12
-14
14
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
LOG(FFT CALLS)
3.9
4
4.1
Poschl-Teller potential
New Split
0
-2
LOG(E
ERROR)
-4
-6
-8
-10
-12
-14
14
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
LOG(FFT CALLS)
3.9
4
4.1
References
S. Blanes, F. Casas, and A. Murua, Work in Progress.
Group webpage: http://www.gicas.uji.es/
S. Blanes, F. Casas and A. Murua, On the linear stabylity of splitting
methods, Found. Comp. Math., 8 (2008), pp. 357-393.
S. Blanes, F. Casas and A. Murua, Symplectic splitting operator
methods for the time-dependent Schrödinger equation, J. Chem.
Phys. 124 (2006) 234105.
Th Magnus
The
M
series
i expansion
i
Let us now consider the time-dependent problem
or, equivalently
Th Magnus
The
M
series
i expansion
i
Let us now consider the time-dependent problem
or, equivalently
We can use the time-dependent
p
Perturbation Theory:
y The Dyson
y
p
perturbative
series
where
Wh th
When
the series
i iis ttruncated,
t d th
the unitarity
it it iis llost!
t!
Th Magnus
The
M
series
i expansion
i
with A(t) ∈ d×d
Th Magnus
The
M
series
i expansion
i
Theorem (Magnus 1954). Let
A(t) be a known function of t (in
general, in an associative ring), and let Y(t) be an unknown function
satisfying
y g Y’ = A(t)
( ) Y, with Y(0)
( ) = I. Then, if certain unspecified
p
conditions
of convergence are satisfied, Y(t) can be written in the form
where
and Bn are the Bernouilli numbers.
The convergence is asured if
The Magnus expansion
which can be obtained by Picard iteration or using a recursive formula
where
Convergence condition:
This is a sufficient but
NOT necessary condition
F Casas,
F.
C
S
Sufficient
ffi i t conditions
diti
ffor th
the convergence off th
the M
Magnus
expansion, J. Phys. A: Math. Theor. 40 (2007), 15001-15017
This theory allows us to build relatively simple methods at different orders:
A second order method is given by:
with
ith
This theory allows us to build relatively simple methods at different orders:
A second order method is given by:
with
ith
A fourth order method is given by:
with
and
This theory allows us to build relatively simple methods at different orders:
A second order method is given by:
with
ith
A fourth order method is given by:
with
and
Methods
M
th d using
i different
diff
t quadratures
d t
and
d higher
hi h order
d exist.
i t Th
These methods
th d
have been used for the SE beyond the convergence domain:
M. Hochbruck,
M
H hb k C.
C Lubich,
L bi h On
O Magnus
M
integrators
i t
t
for
f time-dependent
ti
d
d t
Schrödinger equations, SIAM J. Numer. Anal. 41 (2003) 945-963.
A two level quantum system
The Rosen-Zenner Model
with
If we consider
id the
h iinteraction
i picture
i
whch depends on two parameters:
The transition probability from
to
is given by
ξ=0.3
=0.3
Trans
sition
Prrobabilit
robabilit
ty
Trans
ition Pr
obabilitty
y
1
1
1111
0.9
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0
0
0
0
11
22
33
γγ
44
ξξ=1
=1
55
66
0000
0000
RK4
E1
E1
M4
M2
1111
2222
3333
γγ
4444
5555
6666
Numerical integration for
ξ=0.3
1
1
Trans
sition
Prrobabilit
robabilitty
ty
Trans
ition Pr
using 50 evaluations
111
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0
0
0
0
11
22
33
γγ
ξξ=1
=1
44
55
66
000
000
RK4
E1
E1
M4
M2
111
222
333
γγ
444
555
666
Numerical integration for
using 50 evaluations
Transsition Prrobabilitty
ξ=0.3
ξ=1
1
11
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
1
2
3
γ
4
5
6
00
00
RK4
E1
M4
M2
11
22
33
γ
44
55
66
Numerical integration for
using 50 evaluations
Transsition Prrobabilitty
ξ=0.3
ξ=1
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
γ
4
5
6
0
RK4
E1
M4
M2
0
1
2
3
γ
4
5
6
Numerical integration for
using 50 evaluations
Transsition Prrobabilitty
ξ=0.3
ξ=1
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
γ
4
5
6
0
RK4
E1
M4
M2
0
1
2
3
γ
4
5
6
(ξ=0.3,γ=10)
(ξ
0.3,γ 10)
(ξ=0.3,γ=100)
(ξ
0.3,γ 100)
0
0
-1
-2
Log(Error)
-2
Log(Error)
-1
Euler
RK4
RK6
Magnus-2
Magnus-4
Magnus
4
Magnus-6
-3
-4
-3
-4
-5
5
-5
5
-6
-6
-7
2.5
3
Log(Evaluations)
3.5
-7
3
3.2
3.4
3.6
3.8
Log(Evaluations)
4
(ξ=0.3,γ=10)
(ξ
0.3,γ 10)
(ξ=0.3,γ=100)
(ξ
0.3,γ 100)
0
0
-1
-2
Log(Error)
-2
Log(Error)
-1
Euler
RK4
RK6
Magnus-2
Magnus-4
Magnus
4
Magnus-6
-3
-4
-3
-4
-5
5
-5
5
-6
-6
-7
2.5
3
Log(Evaluations)
3.5
-7
3
3.2
3.4
3.6
3.8
Log(Evaluations)
4
Splitting-Magnus
Splitting
Magnus methods
with
Splitting-Magnus
Splitting
Magnus methods
with
We can approximate the solution by the composition
Splitting-Magnus
Splitting
Magnus methods
with
We can approximate the solution by the composition
with the averages
S. Blanes,
S
Blanes F
F. Casas and A
A. Murua
Murua, Splitting methods for non
non-autonomous
autonomous
Linear systems, Int. Jour. Comp. Math., 6 (2007), 713-727
Conclusions
Magnus and splitting methods are powerful tools for
numerically solving the Schrödinger equation
Conclusions
Magnus and splitting methods are powerful tools for
numerically solving the Schrödinger equation
But, the performance strongly dependes on the
choice of the appropriate methods for each problem
Conclusions
Magnus and splitting methods are powerful tools for
numerically solving the Schrödinger equation
But, the performance strongly dependes on the
choice of the appropriate methods for each problem
Alternatively, one can build methods tailored for
particular problems
p
p
Basic References
S. Blanes, F. Casas, J.A. Oteo, and J. Ros, The Magnus expansion and
some of its applications, Phys. Rep. 470 (2009), 151-238.
S. Blanes, F. Casas, and A. Murua, Splitting and composition methods in
the numerical integration of differential equations,
(
), 89-145. (arXiv:0812.0377v1)).
Boletín SEMA,, 45 (2008),
C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced
Models and Numerical Analysis,
Analysis Zurich Lectures in Adv
Adv. Math
Math., (2008).
E. Hairer,
E
Hairer C
C. Lubich
Lubich, and G
G. Wanner
Wanner,
Geometric Numerical Integration. Structure-Preserving Algorithms
for Ordinary Differential Equations, Springer-Verlag, (2006).
A. Iserles, H.Z. Munthe-Kaas, S.P. Nörsett and A. Zanna,
group
p methods,, Acta Numerica,, 9 (2000),
(
), 215-365.
Lie g
R.I. McLachlan and R. Quispel,
Splitting methods, Acta Numerica, 11 (2002), 341-434.