Path Integral Quantum Monte Carlo

1
Path Integral QuantumMonte Carlo

• Consider a harmonic oscillator potential
• a classical particle moves back and forth
  periodically in such a potential
• x(t) A cos(?t)
• the quantum wave function can be thought of as a
  fluctuation about the classical trajectory

2
Feynman Path Integral

• The motion of a quantum wave function is
  determined by the Schrodinger equation
• we can formulate a Huygens wavelet principle for
  the wave function of a free particle as follows
• each point on the wavefront emits a spherical
  wavelet that propagates forward in space and time

3
(No Transcript)
4
Feynman Paths

• The probability amplitude for the particle to be
  at xb is the sum over all paths through spacetime
  originating at xa at time ta

5
Principal Of Least Action

• Classical mechanics can be formulated using
  Newtons equations of motion or in terms of the
  principal of least action
• given two points in space-time, a classical
  particle chooses the path that minimizes the
  action

Fermat
6
Path Integral

• L is the Lagrangian LT-V
• similarly, quantum mechanics can be formulated in
  terms of the Schrodinger equation or in terms of
  the action
• the real time propagator can be expresssed as

7
Propagator

• The sum is over all paths between (x0,0) and
  (x,t) and not just the path that minimizes the
  classical action
• the presence of the factor i leads to
  interference effects
• the propagator G(x,x0,t) is interpreted as the
  probability amplitude for a particle to be at x
  at time t given it was at x0 at time zero

8
Path Integral

• We can express G as

• Using imaginary time ?it/?

9
Path Integrals

• Consider the ground state
• as ???

• hence we need to compute G and hence S to obtain
  properties of the ground state

10
Lagrangian

• Using imaginary time ?it the Lagrangian for a
  particle of unit mass is

• divide the imaginary time interval into N equal
  steps of size ?? and write E as

11
Action

• Where ?j j?? and xj is the displacement at time
  ?j

12
Propagator

• The propagator can be expressed as

13
Path Integrals

• This is a multidimensional integral
• the sequence x0,x1,,xN is a possible path
• the integral is a sum over all paths
• for the ground state, we want G(x0,x0,N?? ) and
  so we choose xN x0
• we can relabel the xs and sum j from 1 to N

14
Path Integral

• We have converted a quantum mechanical problem
  for a single particle into a statistical
  mechanical problem for N atoms on a ring
  connected by nearest neighbour springs with
  spring constant 1/(?? )2

15
Thermodynamics

• This expression is similar to a partition
  function Z in statistical mechanics
• the probability factor e- ?E in statistical
  mechanics is the analogue of e- ?E in quantum
  mechanics
• ? N ?? plays the role of inverse temperature
  ?1/kT

16
Simulation

• We can use the Metropolis algorithm to simulate
  the motion of N atoms on a ring
• these are not real particles but are effective
  particles in our analysis
• possible algorithm
• 1. Choose N and ?? such that N ?? gtgt1 ( low T)
  also choose ?( the maximum trial change in the
  displacement of an atom) and mcs (the number of
  steps)

17
Algorithm

• 2. Choose an initial configuration for the
  displacements xj which is close to the
  approximate shape of the ground state
  probability amplitude
• 3. Choose an atom j at random and a trial
  displacement xtrial -gtxj (2r-1) ? where r
  is a random number on 0,1
• 4. Compute the change ?E in the energy

18
Algorithm

• If ?E lt0, accept the change
• otherwise compute pe- ?? ?E and a random number
  r in 0,1
• if r lt p then accept the move
• if r gt p reject the move

19
Algorithm

• 4. Update the probability density P(x). This
  probability density records how often a
  particular value of x is visited Let
  P(xxj) gt P(xxj)1 where x was position
  chosen in step 3 (either old or new)
• 5. Repeat steps 3 and 4 until a sufficient
  number of Monte Carlo steps have been performed

qmc1
20
(No Transcript)
21
(No Transcript)
22
Excited States

• To get the ground state we took the limit ? ??
• this corresponds to T0 in the analogous
  statistical mechanics problem
• for finite T, excited states also contribute to
  the path integrals
• the paths through spacetime fluctuate about the
  classical trajectory
• this is a consequence of the Metropolis algorithm
  occasionally going up hill in its search for a
  new path