Using Path Integral to calculate propagator

In summary, there are multiple classical paths connecting (x', t') and (x, t), and the path integral calculation will still hold.
  • #1
cattlecattle
31
0
Hi, It's great to find this forum.
I'm teaching myself QM using Shankar, it's a great book, I've covered 8 chapters so far.
My question is about the notion of using Path Integral method to calculate the propagator. The recipe given by Shankar says the propagator is
[itex]
U(x,t;x')=A\int \exp{\frac{iS[x(t)]}{\hbar}}\mathcal{D}[x(t)]
[/itex]
where the integral sums over all possible paths connecting (x',0) and (x, t)

He went on to prove that if the potential is in the form of
[itex]
V=a+bx+cx^2+d\dot{x}+ex\dot{x}
[/itex]
then the propagator has a simpler form:
[itex]
U(x,t;x')=e^{iS_{cl}/\hbar}\cdot A(t)
[/itex]

In his proof for this, he simply assumed that given x, t and x', the there exists one and only one classical path - at least the action S is well defined. But I found this part confusing, two counter examples:

1. In a harmonic oscillator of angular frequency ω, let x=x'=0, and t=2π/ω, apparently there are infinite classical paths connecting (x', 0) to (x, t) - no matter what your initial velocity, you always come back to the same spot after one full period.

2. Similarly, if x=x'=a > 0, and t=π/ω, there will be no classical paths (there's no way to come back to the same non trivial spot after half period!

In case 1, luckily all such classical paths have the same action S=0, so we can still find the propagator following [itex] U(x,t;x')=e^{iS_{cl}/\hbar}\cdot A(t) [/itex]

But in case 2, there is no classical path at all, what does this imply for the propagator, does the propagator not exist? In QM, the propagator connects one state to the other, so it should always be well-defined.

I'm sure this is treated more rigorously in non-elementary Path Integral texts, but before diving into those, maybe someone here can explain in simple words what this means.
Thanks
 
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  • #2
If there is no classical path connecting the points, then his derivation doesn't obtain. In those cases, you have to use eqn 8.6.1 directly -- you don't have his shortcut. Even though there is no way for the particle to get from x to x' in time t' classically (potential barrier for example), there may be a non-zero probability of finding the particle there.
 
  • #3
RUTA said:
If there is no classical path connecting the points, then his derivation doesn't obtain. In those cases, you have to use eqn 8.6.1 directly -- you don't have his shortcut. Even though there is no way for the particle to get from x to x' in time t' classically (potential barrier for example), there may be a non-zero probability of finding the particle there.

Thanks for the reply, that makes sense. What about the case where there are multiple classical paths connecting (x', t') and (x, t)? I have a hunch that this problem will go away if they all yield the same action S, but I don't know if this is true.
 
  • #4
cattlecattle said:
Thanks for the reply, that makes sense. What about the case where there are multiple classical paths connecting (x', t') and (x, t)? I have a hunch that this problem will go away if they all yield the same action S, but I don't know if this is true.

I think his derivation would still hold, you would just pick a particular classical path and proceed. Multiple classical paths for the same potential require unusual circumstances (e.g., curved spacetime or wormholes). Your oscillator example produces multiple classical paths for the same potential (same frequency), because you supplied boundary conditions that are automatically satisfied per periodicity and therefore you don't have a unique solution to the second-order, classical differential equation of motion.
 
  • #5

FAQ: Using Path Integral to calculate propagator

What is the path integral method used for in physics?

The path integral method is used to calculate the probability amplitude for a particle to travel from one point to another in a given amount of time. It is based on the principle that a particle can take all possible paths to reach its destination, and the probability of each path is calculated and summed together to determine the final probability amplitude.

How does the path integral method work?

The path integral method involves breaking down the path of a particle into infinitesimal steps, and calculating the probability amplitude for each step. These amplitudes are then multiplied together and summed to give the total probability amplitude for the particle to travel from one point to another.

What is the role of the propagator in path integral calculations?

The propagator is a mathematical function that describes the evolution of a particle from one point to another in a given amount of time. It takes into account all possible paths of the particle and their associated probabilities, and is crucial in calculating the final probability amplitude using the path integral method.

How is the path integral related to the Schrödinger equation?

The path integral method can be used to derive the Schrödinger equation, which is a fundamental equation in quantum mechanics. This equation describes the evolution of a quantum system over time and can be used to predict the behavior of particles on a microscopic level.

What are the advantages of using path integral over other methods of calculation?

The path integral method allows for a more intuitive and visual understanding of quantum systems, as it takes into account all possible paths of a particle rather than just the most probable one. It also has applications in various fields such as statistical mechanics, quantum field theory, and condensed matter physics.

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