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

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

( , ) ( , ; , ) ( , )b b a b b a a a ax t dx G x t x t x t

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

2( )1( , ; , ) exp

2 ( ) 2( )b a

b b a ab a b a

i x xG x t x t

i t t t t

( , ) ( , ; , ) ( , )b b a b b a a a ax t dx G x t x t x t

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

0

,

,0

x t

x

S 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

/0( , , ) iS

paths

G x x t A e

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

/0( , , ) iS

paths

G x x t A e

Path Integral

• We can express G as

0 0 0( , ) ( , , ) ( ,0) 0x t G x x t x dx t

/0 0( , , ) ( ) ( ) niE t

n nn

G x x t x x e

0 0( , , ) ( ) ( ) nEn n

n

G x x x x e

• Using imaginary time =it/

Path Integrals

• Consider the ground state

• as

0200 0 0( , , ) ( ) EG x x ex

0 0( , , ) ( ) ( ) nEn n

n

G x x x x e

• 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 is2

1( )

2

dxL V x E

d

21

2

( )1( , ) ( )

2 ( )j j

j j j

x xE x V x

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

Action

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

1

0

211

20

( , )

( )1( )

2 ( )

N

j jj

Nj j

jj

S i E x

x xi V x

21

2

( )1( , ) ( )

2 ( )j j

j j j

x xE x V x

0

,

,0

x t

x

S Ldt

Propagator• The propagator can be expressed as

211

20

( )1( )

2 ( )

Nj j

jj

x xV x

iSe e

21

12

0

( )1( )

2 ( )

0 1 1( , , ) ...

Nj j

jj

x xV x

NG x x N A dx dx e

/0( , , ) iS

paths

G x x t A e

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 x’s and sum j from 1 to N

211

20

( )1( )

2 ( )

0 1 1( , , ) ...

Nj j

jj

x xV x

NG x x N A dx dx e

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

21

21

( )1( )

2 ( )

0 0 1 1( , , ) ...

Nj j

jj

x xV x

NG x x N A dx dx e

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

21

21

( )1( )

2 ( )

0 0 1 1( , , ) ...

Nj j

jj

x xV x

NG x x N A dx dx e

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

• 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

2 2

1 1

2 2

1 1

1 1( )

2 2

1 1( )

2 2

j trial xtrial jtrial

j j j jj

x x x xE V x

x x x xV x

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