Eulerian Gaussian beam method in quantum mechanics · Schrödinger equation Gaussian beam method -...

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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


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



Numericalresults


Velocity contour

Gaussianbeam

method

Xu Yang

Schrdingerequation



Numericalresults


circle: Schrdinger square: Geometric optics cross: Phase correction star: Gaussian beam

GBmovie.mpegMedia File (video/mpeg)

Gaussianbeam

method

Xu Yang

Schrdingerequation



Numericalresults


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



Numericalresults


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



Numericalresults


Amplitude at = 0.001 and Tfinal = 0.5

Gaussianbeam

method

Xu Yang

Schrdingerequation



Numericalresults


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



Numericalresults


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



Numericalresults


Band structure for V(z) = cos(z)

Gaussianbeam

method

Xu Yang

Schrdingerequation



Numericalresults


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



Numericalresults


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



Numericalresults


Convergence results

Gaussianbeam

method

Xu Yang

Schrdingerequation



Numericalresults


Numerical simulation = 1/4096

Gaussianbeam

method

Xu Yang

Schrdingerequation



Numericalresults


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Schrdinger equation and its semiclassical limitGaussian beam method - Lagrangian formulationGaussian beam method - Eulerian formulationNumerical resultsApplications in quantum mechanics

Eulerian Gaussian beam method in quantum mechanics · Schrödinger equation Gaussian beam method -...

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