Semiclassical propagator:

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In summary, the conversation discusses the possibility of defining a "semiclassical" operator that would approximately solve a given differential equation in WKB representation. There is also a question about whether this operator satisfies a Hamilton-Jacobi equation. Additionally, there is a clarification about the variables used in the equation. The overall question is about how to obtain a propagator when the Schrödinger equation cannot be solved directly.
  • #1
Karlisbad
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If we have SE (or other differential equation) defining the propagator by:

[tex] (\frac{\partial}{\partial t}-H(q,p))G(x,s)=\delta (x-s) [/tex]

then my question is..can you define a "semiclassical" operator?..i mean in the sense that it would solve approximately the equation (1) above but in WKB representation, ... does it satisfy some kind of Hamilton-Jacobi equation?..:grumpy: :grumpy:
 
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  • #2
Can you define "t,x,s,q,p" in your eq.? By the looks of it, it doesn't make too much sense to me.

Daniel.
 
  • #3
Sorry "dextercioby" ^_^ perhaps..is it clearer if i put [tex] G(x,x') [/tex] where here x means [tex] x=(x,y,z,t) [/tex] :redface:

My question is that if you can't solve SE (in most cases) how you can expect to obtain G?..Path integral formulation allows you to calculate G(x,x') but only in the "Semiclassical" approach, by expanding the functional near its classical solution...my question is..DOes The semiclassical-propàgator satisfy a Hamilton-Jacobi equation? :confused:
 

1. What is a semiclassical propagator?

A semiclassical propagator is a mathematical tool used in quantum mechanics to calculate the probability of a particle moving from one point to another in a given time. It takes into account both classical and quantum factors, making it a more accurate representation of particle movement than classical mechanics alone.

2. How is the semiclassical propagator calculated?

The semiclassical propagator is calculated by combining the classical propagation of a particle with the quantum mechanical phase associated with the particle's wavefunction. This is done using the Feynman path integral, which sums up all possible paths a particle can take between two points.

3. What is the difference between a semiclassical propagator and a classical propagator?

The main difference between a semiclassical propagator and a classical propagator is that the semiclassical version takes into account the wave-like nature of particles, while the classical propagator treats particles as classical point objects. This allows the semiclassical propagator to provide more accurate predictions for particle movement.

4. When is a semiclassical propagator used?

A semiclassical propagator is typically used in situations where both classical and quantum effects are relevant, such as in atomic and molecular systems. It is also commonly used in the study of wave phenomena, such as in optics and acoustics.

5. What are the limitations of the semiclassical propagator?

One limitation of the semiclassical propagator is that it is not applicable in systems where quantum effects dominate, such as in subatomic particles. It also does not take into account relativistic effects, so it is not suitable for high-speed or high-energy systems. Additionally, the semiclassical propagator is an approximation and may not always provide accurate results.

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