Path integrals and propagators

In summary, the SE equation has a propagator given by (i\hbar \partial _{t} - \hbar ^{2} \nabla +V(x,y,z) )K(x,x')=\delta (x-x') with m=1/2 for simplicity, which can be obtained from the evaluation of the path integral. This can be approximated using a WKB approach, denoted by K_{WKB}(x,x'). The notation WKB indicates that the propagator was evaluated in a semiclassical way. The propagator is also known as a Green's function and is the operator inverse of the field equation operator.
  • #1
Klaus_Hoffmann
86
1
we know that for the SE equation we find the propagator

[tex] (i\hbar \partial _{t} - \hbar ^{2} \nabla +V(x,y,z) )K(x,x')=\delta (x-x') [/tex]

with m=1/2 for simplicity

then we know that the propagator K(x,x') may be obtained from the evaluation of the Path integral.

[tex] K(x,x')=C \int \mathcal D[x] e^{iS[x]/\hbar} [/tex] (sum over all path X(t) )

my question is, since we can't know the evaluation of the path integral exactly, but give a WKB approach of this if we name the result of the path integral by [tex] K_{WKB}(x,x') [/tex].

then my question is if at least as an approximation this function satisfies.

[tex] (i\hbar \partial _{t} - \hbar ^{2} \nabla +V(x,y,z) )K_{WKB}(x,x')=\delta (x-x') [/tex]

the notation WKB means that we have evaluated the propagator and so on in a semiclassical way.
 
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  • #2
Klaus_Hoffmann said:
we know that for the SE equation we find the propagator

[tex] (i\hbar \partial _{t} - \hbar ^{2} \nabla +V(x,y,z) )K(x,x')=\delta (x-x') [/tex]

with m=1/2 for simplicity

then we know that the propagator K(x,x') may be obtained from the evaluation of the Path integral.

[tex] K(x,x')=C \int \mathcal D[x] e^{iS[x]/\hbar} [/tex] (sum over all path X(t) )
you seem to have lost your x and x' dependence... at least on the RHS of the above equation
 
  • #3
Klaus_Hoffmann said:
my question is, since we can't know the evaluation of the path integral exactly, but give a WKB approach of this if we name the result of the path integral by [tex] K_{WKB}(x,x') [/tex].

then my question is...

twice you said "my question is" but you never asked a question.
 
  • #4
Klaus_Hoffmann said:
we know that for the SE equation we find the propagator

[tex] (i\hbar \partial _{t} - \hbar ^{2} \nabla +V(x,y,z) )K(x,x')=\delta (x-x') [/tex]

with m=1/2 for simplicity

Isn't that K(x,x') a Green's function? I think the propagator is the 'operator inverse' of the Green's funciton.
 
  • #5
no. "propagator" and "green's function" are synonymous.
 
  • #6
Yeah I remember now, propagator = greens function = operator inverse of the field equation operator
 

Related to Path integrals and propagators

What are path integrals and propagators?

Path integrals and propagators are mathematical tools used in quantum mechanics to describe the evolution of a quantum system over time. They are used to calculate the probability of a particle moving from one position to another in a given time period.

How are path integrals and propagators related?

Path integrals and propagators are closely related, as the propagator is the mathematical object used to calculate the path integral. The propagator describes the probability amplitude for a particle to move from one position to another in a given time, while the path integral is the sum of all possible paths the particle could take between those positions.

What is the significance of path integrals and propagators in quantum mechanics?

Path integrals and propagators are essential tools in quantum mechanics, as they provide a way to calculate the behavior of quantum systems over time. They are used to solve a wide range of problems, from simple particle movement to more complex systems such as quantum field theory.

How do path integrals and propagators differ from other mathematical approaches in quantum mechanics?

Path integrals and propagators offer a different approach to solving problems in quantum mechanics compared to other methods, such as wave equations or matrix mechanics. They provide a more intuitive way to understand the behavior of quantum systems, by considering all possible paths a particle could take rather than focusing on a single trajectory.

What are the applications of path integrals and propagators in other fields of science?

While path integrals and propagators were initially developed for use in quantum mechanics, they have found applications in other areas of science as well. They are used in statistical physics, fluid dynamics, and even in financial modeling. These mathematical tools are also being explored for potential applications in machine learning and artificial intelligence.

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