1. Time-dependent Schrodinger 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 t n =t 0 +n  t ; x s =x 0 +s  x ;  (x,t) =>  (x s,t n )

2. Algorithms One approach treats the real and imaginary parts of  separately this algorithm ensures that the total probability remains constant The Schrodinger equation becomes (  =1)

3. Algorithm Numerical solution of these equations is based on The probability density is conserved if we use

4. Initial Wavefunction Consider a Gaussian wave packet The expectation value of the initial velocity is =p 0 /m=  k 0 /m in the simulation set m=  =1 tdse1

5. 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

6. Diffusion Monte Carlo Compare with the classical diffusion equation Can interpret  as a probability density with a diffusion constant D=  2 /2m

7. Random Walk We can use a random walk algorithm to solve the diffusion equation how do we include the potential term V(x) ? Note:  x corresponds to a probability density in this analogy with random walks and NOT  2  x

8. Algorithm The general solution of the Schrodinger equation in imaginary time is For large, the dominant term comes from the eigenvalue of lowest energy E 0 Population of walkers goes to zero unless E 0 is zero but is proportional to ground state wave function

9. Algorithm We can measure E 0 from an arbitrary reference energy V ref and we can adjust V ref until a steady population of walkers is obtained Using It is easy to show

10. Random Walkers Hence n i is the density of walkers at x i

11. Possible Algorithm 1. Place N 0 walkers at the initial set of positions x i 2. compute the reference energy V ref =  V i /N 0 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)-V ref ] 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 N 0 walkers

12. Possible Algorithm Compute the new number of walkers N compute The new reference potential is The constant a is adjusted so that N remains approximately constant 6. Repeat steps 3-5 until the ground state energy estimate has small fluctuations

13. Program Input parameters are: number of initial walkers N 0, number of Monte Carlo steps mcs, and step size ds consider a harmonic oscillator potential V(x)= (1/2)kx 2 qmwalk N 0 =50 mcs=1000 ds=0.1

14. Diffusion Quantum Monte Carlo Introduce the concept of a Green’s function or propagator defined by G propagates the wave function from time t=0 to time similar to electrostatics:

15. Diffusion Quantum Monte Carlo Operate on both sides with  /  and then with (H op -V ref ) hence G satisfies With solution

16. But H op =T op + V op and [T op,V op ]  0 only for short  can we factor the exponential

18. 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

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

20. 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 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 N 0 qmwalk