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Transcript

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 ( xb , tb ) dxa G ( xb , tb ; xa , ta ) ( xa , ta ) 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 ( xb , tb ) dxa G ( xb , tb ; xa , ta ) ( xa , ta ) 2 i( xb xa ) 1 G( xb , tb ; xa , ta ) exp 2 i(tb ta ) 2(tb ta ) 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 x ,t S x0 ,0 Ldt 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 G ( x, x0 , t ) A e paths iS / Propagator G ( x, x0 , t ) A e iS / paths • 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 Path Integral ( x, t ) G ( x, x0 , t ) ( x0 , 0)dx0 • We can express G as G( x, x0 , t ) n ( x)n ( x0 )e iEnt / n • Using imaginary time =it/ G( x, x0 , ) n ( x)n ( x0 )e n En t 0 Path Integrals G( x, x0 , ) n ( x)n ( x0 )e En n • Consider the ground state • as G ( x0 , x0 , ) 0 ( x0 ) e 2 E0 • 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 2 1 dx L V ( x) E 2 d • divide the imaginary time interval into N equal steps of size and write E as 1 ( x j 1 x j ) E ( x j , j ) V (x j ) 2 2 ( ) 2 x ,t S x0 ,0 Action Ldt 1 ( x j 1 x j ) E ( x j , j ) V (x j ) 2 2 ( ) 2 • Where j = j and xj is the displacement at time j N 1 S i E ( x j , j ) j 0 N 1 1 ( x j 1 x j )2 i V ( x j ) 2 j 0 2 ( ) Propagator • The propagator can be expressed as G ( x, x0 , t ) A e iS / paths e e iS N 1 1 ( x j 1 x j )2 V ( x ) j j 0 2 ( )2 G ( x, x0 , N ) A dx1...dxN 1e N 1 1 ( x j 1 x j ) 2 V ( x j ) 2 j 0 2 ( ) Path Integrals G ( x, x0 , N ) A dx1...dxN 1e • • • • N 1 1 ( x j 1 x j ) 2 V ( x j ) 2 j 0 2 ( ) 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 x’s and sum j from 1 to N Path Integral G ( x0 , x0 , N ) A dx1...dxN 1e N 1 ( x j x j 1 )2 V ( x j ) 2 j 1 2 ( ) • 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 G ( x0 , x0 , N ) A dx1...dxN 1e N 1 ( x j x j 1 )2 V ( x j ) 2 j 1 2 ( ) • 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 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 ->xj +(2r-1) where r is a random number on [0,1] • 4. Compute the change E in the energy Algorithm 1 x j 1 xtrial 1 xxtrial x j 1 E V ( xtrial ) 2 2 2 2 1 x j 1 x j 1 x j x j 1 V (x j ) 2 2 2 2 • 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=xj) => P(x=xj)+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