Path Integral Quantum Monte 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

Feynman Path Integral The motion of a quantum wave function is determined by the Schrodinger equation we can formulate a Huygen’s 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

Feynman Paths The probability amplitude for the particle to be at x b is the sum over all paths through spacetime originating at x a at time t a

Principal Of Least Action Classical mechanics can be formulated using Newton’s 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

Path Integral L is the Lagrangian L=T-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

Propagator The sum is over all paths between (x 0,0) 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,x 0,t) is interpreted as the probability amplitude for a particle to be at x at time t given it was at x 0 at time zero

Path Integral We can express G as Using imaginary time =it/ 

Path Integrals Consider the ground state as  hence we need to compute G and hence S to obtain properties of the ground state

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

Action Where j = j  and x j is the displacement at time j

Propagator The propagator can be expressed as

Path Integrals This is a multidimensional integral the sequence x 0,x 1,…,x N is a possible path the integral is a sum over all paths for the ground state, we want G(x 0,x 0,N  ) and so we choose x N = x 0 we can relabel the x’s and sum j from 1 to N

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

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

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  >>1 ( low T) also choose  ( the maximum trial change in the displacement of an atom) and mcs (the number of steps)

Algorithm 2. Choose an initial configuration for the displacements x j which is close to the approximate shape of the ground state probability amplitude 3. Choose an atom j at random and a trial displacement x trial ->x j +(2r-1)  where r is a random number on [0,1] 4. Compute the change  E in the energy

Algorithm If  E <0, accept the change otherwise compute p=e -   E and a random number r in [0,1] if r < p then accept the move if r > p reject the move

Algorithm 4. Update the probability density P(x). This probability density records how often a particular value of x is visited Let P(x=x j ) => P(x=x j )+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

Excited States To get the ground state we took the limit  this corresponds to T=0 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