Semi-classical Schrödinger equationwith random inputs:

analysis and the GWPT based method

Zhennan Zhou周珍楠1

joint work with Shi Jin, Liu Liu and Giovanni Russo

1Beijing International Center for Mathematical Research, Peking University

Shanghai Jiao Tong University, April 8, 2019

Outline

Introduction and theoretical understanding

The GWPT method

Conclusions and final remarks

Uncertainty in quantum dynamics:when UP (– principle) meets UQ (– quantification)

I In simulating physical systems, there are inevitably modeling errors,imprecise measurements of the initial data or the backgroundcoefficients.

I The solution to the Schrödinger equation is a complex valued wavefunction, whose certain nonlinear transforms lead to probabilisticmeasures of the physical observables.

Schrödinger equation with random inputs

We consider the following semiclassical Schrödinger equation with randominputs

iε∂tψε(t , x, z) = −ε

2

2∆xψ

ε(t , x, z) + V (x, z)ψε(t , x, z), (1)

ψε(0, x, z) = ψεin(x, z). (2)

I ε� 1 is the semiclassical parameter;

I V (x, z) is the scalar potential function;

I The initial condition will be assumed to be in the form of asemiclassical wave packet.

Physical observables

In many physics applications, the wave function acts as an auxiliary quantityused to compute primary physical quantities (nonlinear transforms of thewave function) such as the position density,

ρ(t , x) = |ψε(t , x)|2, (3)

the current density,

J(t , x) =12

(ψε (−iε∇x)ψε − ψε(−iε∇x)ψε

), (4)

so that the mass conservation equation is satisfied,

∂

∂tρ+∇x · J = 0. (5)

Probabilistic description of the random inputs

The uncertainty of the random inputs is described by the random variable z,which lies in the random space Iz with a probability measure π(z)dz.

We introduce the notation for the expected value of f (z) in the randomvariable z,

〈f 〉π(z) =

∫f (z)π(z)dz. (6)

I We only consider the uncertainty coming from initial data and potentialfunctions.

I For example, the external classical field, the wave packet position or thewave packet momentum is uncertain, which reflect on the uncertaintyin physical observables.

Challenges in Analysis

Semi-classical regime: ε� 1.

I the semi-classical Schrödinger equation propagates oscillation ofwavelength order O(ε) in both space and time, so that the wave functionuε does not converge in the strong sense as ε→ 0.

I the macroscopic physical quantities are nonlinear transforms of theoscillatory wave function, the classical limit for those physicalobservables are not guaranteed by the weak convergence of the wavefunction.

Micro-local analysis:H-measure, micro-local defect measure, Wigner measure...

New Question:

I O(ε) oscillations in the random variable z?

I semiclassical limit and average in randomness?

Challenges in Analysis

Semi-classical regime: ε� 1.

I the semi-classical Schrödinger equation propagates oscillation ofwavelength order O(ε) in both space and time, so that the wave functionuε does not converge in the strong sense as ε→ 0.

I the macroscopic physical quantities are nonlinear transforms of theoscillatory wave function, the classical limit for those physicalobservables are not guaranteed by the weak convergence of the wavefunction.

Micro-local analysis:H-measure, micro-local defect measure, Wigner measure...

New Question:

I O(ε) oscillations in the random variable z?

I semiclassical limit and average in randomness?

Challenges in Analysis

Semi-classical regime: ε� 1.

I the semi-classical Schrödinger equation propagates oscillation ofwavelength order O(ε) in both space and time, so that the wave functionuε does not converge in the strong sense as ε→ 0.

I the macroscopic physical quantities are nonlinear transforms of theoscillatory wave function, the classical limit for those physicalobservables are not guaranteed by the weak convergence of the wavefunction.

Micro-local analysis:H-measure, micro-local defect measure, Wigner measure...

New Question:

I O(ε) oscillations in the random variable z?

I semiclassical limit and average in randomness?

Quantum Uncertainty

I A quantum system is completely determined by ψε(t , x, z).

I 〈hε〉 = 〈ψε|h|ψε〉 is the expectation value of observable h, which is afunction of t and z.

For example

〈qε〉 = 〈qε〉(t , z) = 〈ψε|q|ψε〉 =

∫ψεxψε dx,

〈pε〉 = 〈pε〉(t , z) = 〈ψε|p|ψε〉 = −iε∫ψε∇xψ

ε dx,

where q = x and p = −iε∇x denote, respectively, the position andmomentum operators when the wave function ψε is in the spacerepresentation.

Classical uncertainty

In classical mechanics, position and momentum of the particle follow

q =∂H∂p

, p = −∂H∂q

,

with q(0, z) = q0(z), p(0, z) = p0(z). And the Hamiltonian is

H =|p|2

2m+ V (q, z),

where the random parameter z is distributed with a given density π(z).

Given q(t , z), we find the probability distribution of the spatial variable

Pq(x, t) =

∫δ(x− q(t , z))π(z) dz. (7)

Partially open question: In what sense does 〈qε〉 → q and 〈pε〉 → p?

Wigner transform and Semiclassical limit

The potential function V (x, z) can be decomposed as

V (x, z) = V (x) + N(x, z), (8)

such that〈V 〉π = 〈V 〉π, 〈N〉π = 0.

The Wigner transform is defined as a phase-space function

wε(f , g) (t , x, ξ) =1

(2π)d

∫Rd

eiy·ξ f(

x +ε

2y)

g(

x− ε

2y)

dy. (9)

Denote wε(t , x, ξ, z) = wε(ψε, ψε), which satisfies the Wigner equation

∂twε + ξ · ∇xwε + Θ[V ]wε = 0, (10)

in which Θ[V ]wε is a pseudo-differential operator.

Wigner transform and Semiclassical limit (cont’d)

By Weyl’s Calculus, as ε→ 0, the Wigner measure

w0(t , x, ξ, z) = limε→0

wε(ψε, ψε)

satisfies the classical Liouville equation

w0t + ξ · ∇xw0 −∇xV · ∇ξw0 −∇xN · ∇ξw0 = 0. (11)

The bi-characteristics of the Liouville equation are{x = ξ,

ξ = −∇xV−∇xN.(12)

Note that, −∇xN can be viewed as a random force, and

〈−∇xN〉π = −∇x〈N〉π = 0.

Oscillations in the random variable z

Consider the following averaged norm

||f ||Γ :=

(∫Iz

∫R3|f (t , x, z)|2 dxπ(z)dz

) 12

. (13)

Unfortunately, we have in general, for k = (k1, k2, · · · , kn) ∈ Nn, denote|k| =

∑nj=1 kj , we conclude that

‖∂kzψ

ε‖Γ = O(ε−|k|

). (14)

For example, with some ODEs for the parameters,

Φ(t , x , z) = exp[

iε

(α(t , z)

(x − q(t , z)

)2 − p(t , z)(x − q(t , z)

)+ γ(t , z)

)]is an exact solution to the Schrödinger equation, when potential V (x , z) isquadratic in x .

Oscillations in the random variable z

Consider the following averaged norm

||f ||Γ :=

(∫Iz

∫R3|f (t , x, z)|2 dxπ(z)dz

) 12

. (13)

Unfortunately, we have in general, for k = (k1, k2, · · · , kn) ∈ Nn, denote|k| =

∑nj=1 kj , we conclude that

‖∂kzψ

ε‖Γ = O(ε−|k|

). (14)

For example, with some ODEs for the parameters,

Φ(t , x , z) = exp[

iε

(α(t , z)

(x − q(t , z)

)2 − p(t , z)(x − q(t , z)

)+ γ(t , z)

)]is an exact solution to the Schrödinger equation, when potential V (x , z) isquadratic in x .

Challenges in Numerics

Direct simulation: Even for unconditionally stable methods, unresolved

mesh may lead to wrong physical observables.

Challenges in Numerics (continued)Previous results for

iε∂tψε = −ε

2

2∆xψ

ε + Vψε.

1. Finite difference approximation:

I correct physical observables ∆x = o(ε), ∆t = o(ε);

I correct L2 approximation of wave functions, even more restrictive.

P. A. Markowich, P. Pietra, and C. Pohl (1999).

2. Time splitting spectral approximation:

I correct physical observable ∆x = O(ε), ∆t = O(1);

I correct L2 approximation of wave functions, ∆x ∼ O(ε), ∆t = O(ε).

W. Bao, S. Jin, and P. A. Markowich (2002).S. Jin, and Z. Zhou (2013). Z. Ma, Y. Zhang and Z. Zhou (2017)

Approximate approaches

I Geometric optics and WKB methods

u = aε(t , x)eiS(t,x)/ε = (a0(t , x) + εa1(t , x) + · · · ) eiS(t,x)/ε.

I Wigner transform

W ε(t , x , k) =

∫dy

(2π)d eik·y u(

t , x − ε

2y)

u(

x +ε

2y).

I Gaussian beam approach

u = a(t ; y)eiT (t,x ;y)/ε,

T (t , x ; y) = S(t ; y) + p(t ; y) · (x − y) +12

(x − y) ·M(t ; y)(x − y).

Approximate approaches

I Geometric optics and WKB methods

u = aε(t , x)eiS(t,x)/ε = (a0(t , x) + εa1(t , x) + · · · ) eiS(t,x)/ε.

I Wigner transform

W ε(t , x , k) =

∫dy

(2π)d eik·y u(

t , x − ε

2y)

u(

x +ε

2y).

I Gaussian beam approach

u = a(t ; y)eiT (t,x ;y)/ε,

T (t , x ; y) = S(t ; y) + p(t ; y) · (x − y) +12

(x − y) ·M(t ; y)(x − y).

Approximate approaches

I Geometric optics and WKB methods

u = aε(t , x)eiS(t,x)/ε = (a0(t , x) + εa1(t , x) + · · · ) eiS(t,x)/ε.

I Wigner transform

W ε(t , x , k) =

∫dy

(2π)d eik·y u(

t , x − ε

2y)

u(

x +ε

2y).

I Gaussian beam approach

u = a(t ; y)eiT (t,x ;y)/ε,

T (t , x ; y) = S(t ; y) + p(t ; y) · (x − y) +12

(x − y) ·M(t ; y)(x − y).

More comments on Gaussian beams

I Leading order approximation: error O(√ε).

I Higher order approximation: error O(εk/2)

u = aε(t , x ; y)eiT (t,x ;y)/ε,

T (t , x ; y) = S(y) +p · (x−y) +12

(x−y) ·M(x−y) +O(

(x − y)3)

+ · · · ,

aε(t , x ; y) = a0(t ; y) + O(ε) + · · · .

Drawback: non-constant cut-off functions needed to maintain Gaussianprofile.

Alternative: Hagedorn wave packets approach .

u ≈∑

n

Hn

(x − qε1/2

)× "Gaussian wave packet"

G. Hagedorn, C. Lubich, Z. Zhou...

More comments on Gaussian beams

I Leading order approximation: error O(√ε).

I Higher order approximation: error O(εk/2)

u = aε(t , x ; y)eiT (t,x ;y)/ε,

T (t , x ; y) = S(y) +p · (x−y) +12

(x−y) ·M(x−y) +O(

(x − y)3)

+ · · · ,

aε(t , x ; y) = a0(t ; y) + O(ε) + · · · .

Drawback: non-constant cut-off functions needed to maintain Gaussianprofile.

Alternative: Hagedorn wave packets approach .

u ≈∑

n

Hn

(x − qε1/2

)× "Gaussian wave packet"

G. Hagedorn, C. Lubich, Z. Zhou...

More comments on Gaussian beams

I Leading order approximation: error O(√ε).

I Higher order approximation: error O(εk/2)

u = aε(t , x ; y)eiT (t,x ;y)/ε,

T (t , x ; y) = S(y) +p · (x−y) +12

(x−y) ·M(x−y) +O(

(x − y)3)

+ · · · ,

aε(t , x ; y) = a0(t ; y) + O(ε) + · · · .

Drawback: non-constant cut-off functions needed to maintain Gaussianprofile.

Alternative: Hagedorn wave packets approach .

u ≈∑

n

Hn

(x − qε1/2

)× "Gaussian wave packet"

G. Hagedorn, C. Lubich, Z. Zhou...

The Gaussian wave packet transformation (GWPT)

How about a multiscale transform instead of a multiscale expansion?

We start by considering the following ansatz

ψ(x, t) = W (ξ, t) exp (f (ξ, t))

:= W (ξ, t) exp(

i(ξTαRξ + pT ξ + γ2

)/ε), (15)

where ξ = x− q, αR is a real-valued symmetric matrix and γ2 is acomplex-valued scalar.

As is discussed by Russo and Smereka, we have to require that αR is realso that, we can eventually arrive at a well-posed equantion.

The Gaussian wave packet transformation (GWPT)

How about a multiscale transform instead of a multiscale expansion?

We start by considering the following ansatz

ψ(x, t) = W (ξ, t) exp (f (ξ, t))

:= W (ξ, t) exp(

i(ξTαRξ + pT ξ + γ2

)/ε), (15)

where ξ = x− q, αR is a real-valued symmetric matrix and γ2 is acomplex-valued scalar.

As is discussed by Russo and Smereka, we have to require that αR is realso that, we can eventually arrive at a well-posed equantion.

GWT applied to the Schrödinger equation

So, direct calculation with the bi-characteristic equations reads

Wt = − iε

W ξT(αR + 2α2

R +12∇2U(q)

)ξ

+W(γ2 −

12

p2 + U(q)− iεTr(αR)

)−2ξTαR∇ξW +

iε2

∆ξW−iε

Ur W ,

whereUr = U(ξ + q)− U(q)− ξT∇U(q)− 1

2ξT∇2U(q)ξ.

GWT applied to the Schrödinger equation (II)

Then, we take

γ2 =12

p2 − U(q) + iεTr(αR),

and enforceα = −2α2 − 1

2∇2U(q),

with its real and imaginery parts are respectively

αR = 2α2I − 2α2

R −12∇2U(q),

αI = −2αIαR − 2αRαI .

GWT (III): the change of variable

Then the equation reduces to

Wt = −2ξTαR∇ξW +iε2

∆ξW−iε

(Ur +2ξTα2

I ξ)

W .

At last, introduce the change of variables, W (t , ξ) = w(t , η), where

η = ξB/√ε, (16)

where B will be chosen specifically,

B = −2BαR .

It can be shown that if initialized properly, B is always invertible.

GWT (III): the change of variable

Then the equation reduces to

Wt = −2ξTαR∇ξW +iε2

∆ξW−iε

(Ur +2ξTα2

I ξ)

W .

At last, introduce the change of variables, W (t , ξ) = w(t , η), where

η = ξB/√ε, (16)

where B will be chosen specifically,

B = −2BαR .

It can be shown that if initialized properly, B is always invertible.

The non-oscillatory w equation

With this choice, the equation reduces to

wt =i2

Tr(

BT∇2ηwB

)−2iηT (BT )−1α2

I B−1ηw+1iε

Ur w . (17)

Note that1iε

Ur = O(√ε),

so the w equation does not generate small scale oscillations in η or t .

What about small oscillations in the random variable z?

The non-oscillatory w equation

With this choice, the equation reduces to

wt =i2

Tr(

BT∇2ηwB

)−2iηT (BT )−1α2

I B−1ηw+1iε

Ur w . (17)

Note that1iε

Ur = O(√ε),

so the w equation does not generate small scale oscillations in η or t .

What about small oscillations in the random variable z?

Improved regularity result

We assume, for m ∈ N and k = (k1, k2, · · · , kn) ∈ Nn, there exists a constantCm,k such that,

|∂mx ∂

kz V | 6 Cm,k, ‖∂m

η ∂kz win‖Γ 6 Cm,k, (18)

where ∂kz = ∂

k1z1· · · ∂kn

zn .

定理 (Jin, Liu, Russo and Zhou)The w equation preserves the regularity in the following sense: for a fixedT > 0, k = (k1, k2, · · · , kn) ∈ Nn, there exists an ε-independent constantMT ,k, such that for 0 6 t 6 T ,

||∂kz w ||T 6 MT ,k . (19)

GWPT in nutshell

To summarize, with the Gaussian wave packet transform, the Schrödingerequation

iε∂tψε = −ε

2

2∆xψ

ε + Vψε.

equivalently transforms to the w equation

wt =i2

Tr(

BT∇2ηwB

)−2iηT (BT )−1α2

I B−1ηw+1iε

Ur w . (20)

together with the ODE system of the parameters

q = p,p = −∇U(q),

α = −2α2 − 12∇

2U(q),

γ2 = 12 p2 − U(q) + iεTr(αR),

B = −2BαR .

(21)

Stochastic Collocation or Stochastic Galerkin

In numerical simulation, we need to sample

I the Gaussian wave packet parameters;

I the w equation (who coefficients depend on the parameters).

In solution construction, we are interested in

I the wave equation,

I physical observables which are nonlinear transforms of the waveequation.

The stochastic collocation method offers great flexibility in computingaverages of various quantities in the random space!

With samples {zk} and corresponding weights {wk} chosen from thequadrature rule, the integrals in z are approximated by∫

Izf (t , x , z)π(z)dz ≈

Nz∑k=1

fk (t , x)wk . (22)

Features of Gaussian wave packet transformation

GWPT provides a multiscale frame to handle the high frequency waves.

1. The Time Splitting Spectral method applies to the w equation.(spectrally accuracy in space, high order accuracy in time.) Withmeshing strategies:

∆η = O(1), ∆t = O(1).

2. can use different sizes of samples in z for the GWPT parameters, the wequation and the wave function, respectively.

In particular, we can use ε independent number of samples for the wequation.

3. can handle more general initial conditions.

4. can use the w function directly to compute physical observables.

GWPT as a multiscale formulation

We have introduced three closely related quantities:

u(x, t), W (x− q, t), w(

(x− q)B√ε

, t).

Numerical Test: various ∆t and ∆η

Convergence test:

second order in time and spectral accuracy in space.

Oscillations and smoothness in z

Oscillations in z: wave function and physical observables

Implementation Details

Three sets of collocation points in z are used in our numerical tests:

I number of points to solve the ODEs for the wave packet parametersgiven by the ODE system : Nz,1,

I number of points to solve the w equation: Nz,2,

I number of points to reconstruct ψ: Nz,3.

We check the relative error for the wave function

Er[ψ] =||ψG − ψD||Γ||ψD||Γ

, (23)

and the errors in mean and standard deviation of the current density

Er1[j] =

∣∣∣∣∣E(jG − jD)

E(jD)

∣∣∣∣∣ , Er2[j] =

∣∣∣∣∣SD(jG − jD)

SD(jD)

∣∣∣∣∣ , (24)

Computing the w equation with O(1) samples in z

Fixed number of z samples for the w equation with various εNz,1 = O(ε−1), Nz,2 = O(1), Nz,3 = O(ε−1).

Fixed ε with various number of samples in the w equation.Nz,1 = O(ε−1), Nz,3 = O(ε−1).

Some additional numerical experiments

Computing physical observables with even less samples?

Nz,1 = O(1), Nz,2 = O(1), Nz,3 = O(1). O(√ε) randomness in z.

Nz,1 = O(1), Nz,2 = O(1), Nz,3 = O(1). O(1) randomness in z.

Approaching the semiclassical limit?

Recall that, we wonder if 〈hε〉(z) = 〈ψε|h|ψε〉 → h(z) .

We test the weak convergence (in distribution) in the following

Future directions

In modeling,

I random fields?.

I many-body simulations?

I computing observables with more savings?

In numerical implementation,

I what is the optimal way to choose time steps?

In analysis,

I provide a better way to do initial condition decomposition?

and many fresh ideas...

Thanks for your attention and questions!

Future directions

In modeling,

I random fields?.

I many-body simulations?

I computing observables with more savings?

In numerical implementation,

I what is the optimal way to choose time steps?

In analysis,

I provide a better way to do initial condition decomposition?

and many fresh ideas...

Thanks for your attention and questions!