Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Eulerian Gaussian beam method inquantum mechanics
Xu YangProgram in Applied and Computational Mathematics
Princeton University, USA
Collaboration with Prof. Shi Jin and Mr. Hao Wu
March 5, 2009
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Outline
1 Schrdinger equation and its semiclassical limit
2 Gaussian beam method - Lagrangian formulation
3 Gaussian beam method - Eulerian formulation
4 Numerical results
5 Applications in quantum mechanics
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Schrdinger equation
The time-dependent one-body Schrdinger equation:
i
t+
2
2 V (x) = 0, x Rn ,
(t , x) = A(t , x)eiS(t ,x)/
It models: single electron in atoms, KS density functionaltheory, Molecule Orbital theory ...
Numerical difficulties:(t , x) is oscillatory of wave length O().
Methods Mesh size Time stepFinite difference 1 o() o()Time splitting spectral2 O() -indep.
1Markowich, Pietra, Pohl and Stimming
2Bao, Jin and Markowich
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Geometric optics - WKB analysis
WKB ansatz (t , x) = A(t , x)eiS(t ,x)/,
eikonal St +12|S|2 + V (x) = 0,
transport t + (S) = 0, (t , x) = |A(t , x)|2 .
Eikonal (Hamilton-Jacobi type) singularity (caustics)Figures from the review paper of Engquist and Runborg:
Caustics NOT physical Semiclassical limit
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Semiclassical limit + phase correction
Theorem 1. If V (x) is constant, by the stationary phasemethod we have, away from caustics,
(x, t) K
k=1
A0(yk )1 + tD2S0(yk ) exp(
i
S(k , yk ) +i4
k
)
where the phase
S(, y) = x y (1/2) ||2 t + S0(y),
has finitely many (K < ) stationary phases k and yk :
k = S0(yk ), yk = x tS0(yk ) ,
D2S0 is the Hessian matrix, and k = sgn(D2S(k , yk )) is theKeller-Maslov index of the k th branch.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Gaussian beam method - motivation
Problems of the semiclassical limit: invalid at caustics
1 the density (t , x) in the transport equation,2 1 + tD2S0(yk ) is singular in the stationary phase method.
Computation around caustics is important in many applica-tions, for example:
Seismic imaging Single-slit diffraction
Gaussian beam method, developed by Popov, allowsaccurate computation around caustics.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Beam-shaped ansatz
la(t , x , y0) = A(t , y)eiT (t ,x ,y)/,
T (t , x , y) = S(t , y)+p(t , y)(xy)+12(xy)>M(t , y)(xy),
beam center:dydt
= p(t , y), y(0) = y0.
Here S R, p Rn, A C, M Cnn. The imaginary part ofM will be chosen so that la has a Gaussian beam profile.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Lagrangian formulation
Apply the beam-shaped ansatz to the Schrdinger equation:
center:dydt
= p,
velocity:dpdt
= yV ,
Hessian:dMdt
= M2 2yV ,
phase:dSdt
=12|p|2 V ,
amplitude:dAdt
= 12(Tr(M)
)A.
The first two ODEs are called ray tracing equations, and theHessian M satisfies the Riccati equation.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Validity at caustics and beam summation
M, A could be solved via the dynamic ray tracing equations:
dPdt
= R,dRdt
= (2yV )P,
M = RP1, A =((det P)1A20
)1/2,
R(0) = M(0) = 2yS0(y) + iI, P(0) = I.
Ralston (82, wave-type eqn), Hagedorn (80, Schrdinger)proved the validity of the Gaussian beam solution at caustics:P complexified = P never singular = A always finite.
The Gaussian beam summation solution (Hill, Tanushev):
la(t , x) =
Rn
(1
2
) n2
r(x y(t , y0))la(t , x , y0)dy0.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Level set method
The level set method has been developed to compute thesemiclassical limit of the Schrdinger equation. (Jin-Liu-Osher-Tsai)
The idea is to build the velocity u = yS into the intersectionof zero level sets of phase-space functions j(t , y , ), i.e.
j(t , y , ) = 0, at = u(t , y), j = 1, , n.
= (1, , n) satisfies the Liouville equation:
t + yyV = 0.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Eulerian formulation I - semiclassical limit
Lagragian formulation = Eulerian formulation
ODEs = PDEs ddt
= L = t + y yV
As shown by Jin, Liu, Osher and Tsai,
velocity: L = 0,
phase: LS = 12||2 V ,
amplitude: LA = 12
Tr(()1y
)A.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Eulerian formulation II - semiclassical limit
If one introduces the new quantity
f (t , y , ) = A2(t , y , )det(),
then f (t , y , ) satisfies the Liouville equation
Lf = 0.
The level set method for the semiclassical limit still sufferscaustics where det() = 0.
Motivated by the Gaussian beam method, we need tocomplexify the Liouville equation for .
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Construct the Hessian function
y(t , y , u(t , y)) = 0 2yS = yu = y()1
Recall the Lagrangian formulation: M = R P1
Conjecture: R = y, P = .
Complex R and P = complex
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Conjecture verification
The first two lines are equivalent to each other once theyhave the same initial conditions:
0(y , ) = iy + ( yS0)
R(0) = 2yS0(y) + iI, P(0) = I.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Eulerian formulation - Gaussian beam
Step 1: L = 0, 0(y , ) = iy + ( yS0).
Step 2: compute y and M = y()1.
Step 3: solve S either by LS = 12||2 V or
path integral S(t , x) = x
au(t , s)ds + Constant.
Step 4: Lf = 0, f0(t , y , ) = A0(y)2,
Step 5: A = (det()1f )1/2.
Parallel to Ralstons proofs, complexified non-degenerate A never blows up
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Eulerian Gaussian beam summation
Define
eu(t , x , y , ) = A(t , y , )eiT (t ,x ,y ,)/,
where
T = S + (x y) + 12(x y)>M(x y),
Eulerian Gaussian beam summation formula:
eu(t , x) =
Rn
Rn
(1
2
) n2
r(x y)eunj=1(Re[j ])ddy ,
r is a truncation function with r 1 in a ball of radius > 0about the origin.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Computing the summation integral
Method 1: Discretized delta function integral (Wen, in 1D).
Method 2: Integrate out first:
eu(t , x) =
Rn
(1
2
) n2
r(x y)
k
eu(t , x , y , uk )|det(Re[]=uk )|
dy ,
where uk , k = 1, , K are the velocity branches.Problem: det
(Re[]
)= 0 at caustics.
Solution: Split the integral into two parts:
L1 ={
y det(Re[p](t , y , pj)) }
L2 ={
y det(Re[p](t , y , pj)) < }
The integration on L1 is regular; the integration on L2 couldbe solved by the semi-Lagrangian method (Leung-Qian-Osher).
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Efficiency and accuracy
Efficiency:
Methods Mesh size Time stepFinite difference o() o()Time splitting spectral O() -indep.
Gaussian beam O(
) O(2p )
p: numerical orders of accuracy in time.
Accuracy: O(
) in caustic case, O() in no caustic case.
It could be easily generalized to higher order Gaussian beammethods by including more terms in the asymptotic ansatz.Tanushev-Runborg-Motamed
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
1D example
Free motion particles with zero potential V (x) = 0. The initialconditions for the Schrdinger equation are given by
A0(x) = e25x2, S0(x) =
15
log(2 cosh(5x)).
which implies that the initial density and velocity are
0(x) = |A0(x)|2 = exp(50x2),
u0(x) = xS0(x) = tanh(5x).
This allows for the appearance of cusp caustics.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Velocity contour
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
circle: Schrdinger square: Geometric optics cross: Phase correction star: Gaussian beam
GBmovie.mpegMedia File (video/mpeg)
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Convergence rate and mesh size
Convergence orders: 0.9082 in `1 norm, 0.8799 in `2 norm and0.7654 in ` norm.
Mesh size: y O(
)
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
2D example
Take the potential V (x1, x2) = 10 and the initial conditions ofthe Schrdinger equation as
A0(x1, x2) = e25(x21 +x
22 ),
S0(x1, x2) = 15(log(2 cosh(5x1)) + log(2 cosh(5x2))).
then the initial density and two components of the velocity are
0(x1, x2) = exp(50(x21 + x22 )),
u0(x1, x2) = tanh (5x1)
v0(x1, x2) = tanh (5x2).
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Amplitude at = 0.001 and Tfinal = 0.5
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Schrdinger equation with periodic structure
i
t=
2
22
x2 + V(
x) + U(x), x R ,
It models: electrons in the perfect crystalsBloch band decomposition:
H(k , z) :=12(iz + k)2 + V(z), z =
x
H(k , z)m(k , z) = Em(k)m(k , z),
m(k , z + 2) = m(k , z), z R, k (1/2, 1/2).
Modified WKB ansatz:
(t , x) =
m=1
am(t , x)m(xSm,x)eiSm(t ,x)/.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Equations in the m-th band
Eikonal-transport equations:
tSm + Em(xSm) + U(x) = 0 ,
tam + E m(xSm)xam +12
amx(E m(xSm)
)+ mam = 0.
Liouville-type equations:
Lm = t + E m() y U (y),Lmm = 0,LmSm = E m() Em() U(y),
Lmam =12
ymm
am mam.
m, m are constants related to m.
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Band structure for V(z) = cos(z)
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Numerical simulation for = 1/512
Initial conditions:
A0(x , z) = e50(x+0.5)2
cos z, S0(x) = 0.3x + 0.1 sin x .
External potential: U(x) = 0
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Schrdinger-Poisson equations
it =
2
2xx + V
(x),
xxV = K(
210 |
(x , t)|2)
, E =V
x.
A simple model of the radiation-matter interaction system, forexample, in nano-optics, mean field theory...
K = +1 Focusing potentialK = 1 Defocusing potential
Initialization:
A0(x) = e25x2, S0(x) =
1
cos(x).
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Convergence results
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Numerical simulation = 1/4096
Gaussianbeam
method
Xu Yang
Schrdingerequation
Gaussianbeammethod -Lagrangianformulation
Gaussianbeammethod -Eulerianformulation
Numericalresults
Applicationsin quantummechanics
Thank You!
Questions?
Schrdinger equation and its semiclassical limitGaussian beam method - Lagrangian formulationGaussian beam method - Eulerian formulationNumerical resultsApplications in quantum mechanics
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