Download Path Integral Quantum Monte Carlo

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Elementary particle wikipedia, lookup

Quantum computing wikipedia, lookup

Measurement in quantum mechanics wikipedia, lookup

Hydrogen atom wikipedia, lookup

Noether's theorem wikipedia, lookup

Max Born wikipedia, lookup

Quantum entanglement wikipedia, lookup

Coherent states wikipedia, lookup

Quantum key distribution wikipedia, lookup

Ensemble interpretation wikipedia, lookup

Density matrix wikipedia, lookup

Bell's theorem wikipedia, lookup

Aharonov–Bohm effect wikipedia, lookup

History of quantum field theory wikipedia, lookup

Atomic theory wikipedia, lookup

Bohr–Einstein debates wikipedia, lookup

Wheeler's delayed choice experiment wikipedia, lookup

Symmetry in quantum mechanics wikipedia, lookup

Scalar field theory wikipedia, lookup

Quantum teleportation wikipedia, lookup

Renormalization group wikipedia, lookup

Renormalization wikipedia, lookup

EPR paradox wikipedia, lookup

Particle in a box wikipedia, lookup

Wave function wikipedia, lookup

Copenhagen interpretation wikipedia, lookup

Instanton wikipedia, lookup

Quantum state wikipedia, lookup

Interpretations of quantum mechanics wikipedia, lookup

Identical particles wikipedia, lookup

Wave–particle duality wikipedia, lookup

Relativistic quantum mechanics wikipedia, lookup

T-symmetry wikipedia, lookup

Hidden variable theory wikipedia, lookup

Canonical quantization wikipedia, lookup

Feynman diagram wikipedia, lookup

Matter wave wikipedia, lookup

Quantum electrodynamics wikipedia, lookup

Theoretical and experimental justification for the Schrödinger equation wikipedia, lookup

Probability amplitude wikipedia, lookup

Double-slit experiment wikipedia, lookup

Propagator wikipedia, lookup

Path integral formulation wikipedia, lookup

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