Time-dependentSchrodinger Equation

• Numerical solution of the time-independent equation is straightforward

• constant energy solutions do not require us to make time discrete

• how would we solve the time-dependent equation?

• Naïve approach would be to produce a grid in the x-t plane

• tn=t0+n t ; xs=x0+s x ; (x,t) => (xs,tn)

Algorithms• One approach treats the real and imaginary parts

of separately

• this algorithm ensures that the total probability remains constant

( , ) ( , ) ( , )x t R x t i I x t • The Schrodinger equation becomes (=1)

( , )( , )

( , )( , )

op

op

dR x tH I x t

dtdI x t

H R x tdt

Algorithm• Numerical solution of these equations is based on

1( , ) ( , ) ( , )

23 1

( , ) ( , ) ( , )2 2

op

op

R x t t R x t H I x t t t

I x t t I x t t H R x t t

• The probability density is conserved if we use

2

2

1 1( , ) ( , ) ( , ) ( , )

2 21 1

( , ) ( , ) ( , ) ( , )2 2

P x t R x t I x t t I x t t

P x t t R x t t R x t I x t t

Initial Wavefunction• Consider a Gaussian wave packet

202

0 0

1/ 4 ( )( ) 4

2

1( ,0)

2

x xik x xx e e

• The expectation value of the initial velocity is <v>=p0/m= k0/m

• in the simulation set m= =1

tdse1

Random Walk Monte Carlo• We now consider a Monte Carlo approach

based on the relationship of the Schrodinger equation to a diffusion process in imaginary time

• if we substitute =it/ into the time-dependent Schrodinger equation for a free particle (V=0) we have

2 2

2

( , ) ( , )

2

x t x t

m x

Diffusion Monte Carlo

• Compare with the classical diffusion equation

2

2

( , ) ( , )P x t P x tD

t x

2 2

2

( , ) ( , )

2

x t x t

m x

• Can interpret as a probability density with a diffusion constant D=2/2m

Random Walk

• We can use a random walk algorithm to solve the diffusion equation

• how do we include the potential term V(x) ?

2 2

2

( , ) ( , )( ) ( , )

2

x xV x x

m x

• Note: x corresponds to a probability density in this analogy with random walks and NOT 2x

Algorithm• The general solution of the Schrodinger

equation in imaginary time is

( , ) ( ) nEn n

n

x t c x e • For large , the dominant term comes from

the eigenvalue of lowest energy E0

00 0( , ) ( ) Ex c x e

• Population of walkers goes to zero unless E0 is zero but is proportional to ground state wave function

Algorithm• We can measure E0 from an arbitrary reference

energy Vref and we can adjust Vref until a steady population of walkers is obtained

2 2

2

( , ) ( , )( ) ( , )

2 ref

x xV x V x

m x

Using 0( )0 0( , ) ( ) refE Vx c x e

It is easy to show

0

( ) ( , )

( , )

V x x dxE

x dx

Random Walkers

• Hence

0

( ) ( , )

( , )

V x x dxE

x dx

0

( )i i

i

nV xE V

n

• ni is the density of walkers at xi

Possible Algorithm• 1. Place N0 walkers at the initial set of positions xi

• 2. compute the reference energy Vref= Vi/N0

• 3. randomly move a walker to the right or left by fixed step length s

• s is related to by (s)2=2D • if m= =1, then D=1/2

• 4. compute V= [V(x)-Vref] and a random number r in the interval [0,1]

• if V>0 and r < V , then remove the walker• if V<0 and r < -V , then add a walker at x

• 5. Repeat 3. and 4. for all N0 walkers

Possible Algorithm• Compute the new number of walkers N

• compute <V>

• The new reference potential is

00

( )ref

aV V N N

N

• The constant a is adjusted so that N remains approximately constant

• 6. Repeat steps 3-5 until the ground state energy estimate <V> has small fluctuations

Program• Input parameters are:

• number of initial walkers N0, number of Monte Carlo steps mcs, and step size ds

• consider a harmonic oscillator potential

• V(x)= (1/2)kx2

qmwalk

N0 =50mcs=1000ds=0.1

Diffusion QuantumMonte Carlo

• Introduce the concept of a Green’s function or propagator defined by

( , ) ( , , ) ( ,0)x G x x x dx • G propagates the wave function from time t=0 to

time • similar to electrostatics:

3

0

1 ( )( )

4

rr d r

r r

Diffusion QuantumMonte Carlo

• Operate on both sides with / and then with (Hop-Vref)

• hence G satisfies

( , ) ( , , ) ( ,0)x G x x x dx

( )op ref

GH V G

• With solution ( )( ) op refH VG e

• But Hop=Top + Vop and [Top,Vop] 0

• only for short can we factor the exponential

( )( ) op refH VG e

1( )

21

( )2

/ 2 / 2

( )V Vop refop ref op

V V T

branch diff branch

G e e e

G G G

opTdiffG e

1( )

2/ 2

op refV V

branchG e

opTdiffG e

1( )

2/ 2

op refV V

branchG e

22

22diff diff

op diff

G GT G

m x

/ 2/ 2( )branch

ref op branch

GV V G

21/ 2 ( ) / 4( , , ) (4 ) x x DdiffG x x D e

1( ( ) ( ) )2( , , )

refV x V x V

branchG x x e

2

2D

m

Diffusion Quantum Monte Carlo• This approach is similar to the random walk

• 1. begin with N0 walkers but there is no lattice

• positions are continuous

• 2. chose one walker and displace it from x to x’

• the new position is chosen from a Gaussian distribution with variance 2D and zero mean

21/ 2 ( ) / 4( , , ) (4 ) x x DdiffG x x D e

Diffusion Quantum Monte Carlo• 3. Weight the configuration x by

1( ) ( )

2( , )refV x V x V

w x x e

• For example, if w~2, we should have two walkers at x where previously there was one

• to implement this weighting(branching) correctly we must make an integer number of copies that is equal on average to w

• take the integer part of w+r where r is a random number in the unit interval

Diffusion Quantum Monte Carlo• 4. Repeats steps 2 and 3 for all random walkers

(the ensemble) and create a new ensemble

• one iteration of the ensemble is equivalent to performing the integration

( , ) ( , , ) ( , )x G x x x dx • The quantity (x,) will be independent of the original ensemble (x,0) if a sufficient number of Monte Carlo steps are used.

• We must keep N(), the number of configurations at time , close to N0

qmwalk