Time-dependent Schrodinger Equation

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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
• tnt0n ?t xsx0s ?x ?(x,t) gt ?(xs,tn)

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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)

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Algorithm

• Numerical solution of these equations is based on

• The probability density is conserved if we use

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Initial Wavefunction

• Consider a Gaussian wave packet

• The expectation value of the initial velocity is
  ltvgtp0/m ?k0/m
• in the simulation set m ?1

tdse1
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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 (V0) we
  have

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Diffusion Monte Carlo

• Compare with the classical diffusion equation

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

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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

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Algorithm

• The general solution of the Schrodinger equation
  in imaginary time is

• For large ?, the dominant term comes from the
  eigenvalue of lowest energy E0

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

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Algorithm

• We can measure E0 from an arbitrary reference
  energy Vref and we can adjust Vref until a steady
  population of walkers is obtained

Using
It is easy to show
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Random Walkers

• Hence

• ni is the density of walkers at xi

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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)22D ??
• if m ?1, then D1/2
• 4. compute ?V V(x)-Vref and a random number
  r in the interval 0,1
• if ?Vgt0 and r lt ?V ??, then remove the walker
• if ?Vlt0 and r lt -?V ??, then add a walker at x
• 5. Repeat 3. and 4. for all N0 walkers

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Possible Algorithm

• Compute the new number of walkers N
• compute ltVgt
• 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 ltVgt has small fluctuations

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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

N0 50 mcs1000 ds0.1
qmwalk
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Diffusion QuantumMonte Carlo

• Introduce the concept of a Greens function or
  propagator defined by

• G propagates the wave function from time t0 to
  time ?
• similar to electrostatics

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Diffusion QuantumMonte Carlo

• Operate on both sides with ?/?? and then with
  (Hop-Vref)
• hence G satisfies

• With solution

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• But HopTop Vop and Top,Vop? 0
• only for short ?? can we factor the exponential

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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

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Diffusion Quantum Monte Carlo

• 3. Weight the configuration x? by

• For example, if w2, 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 wr where r is a
  random number in the unit interval

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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

qmwalk

• 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