Recovering SE from Path integral

In summary, SE's can be derived from path integrals, but solving them is nontrivial. There are known solutions for the Schrodinger equation, but more complicated methods are also possible.
  • #1
Karlisbad
131
0
I know someone posted it before..but i would like to know if given the factor:

[tex] e^{(i/\hbar)S[\phi]} [/tex] (1)

and knowing the progpagator satisfies:

[tex] \Psi (x2,t2)=\int_{-\infty}^{\infty}dxdtK(x2,t2,x1,t1)\Psi(x1,t1) [/tex]
Where S is the action and the propagator is related to (1)
:zzz:
 
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  • #2
A derivation of SE' from path integral approach to QM is to be found in any decent text on quantum mechanics. It's not a big deal.

However, solving problems with path integral rather than solving ODE's (required by writing SE's in the position representation) is (a big deal), since it's usually easier.

Daniel.
 
  • #3
dextercioby said:
A derivation of SE' from path integral approach to QM is to be found in any decent text on quantum mechanics. It's not a big deal.

However, solving problems with path integral rather than solving ODE's (required by writing SE's in the position representation) is (a big deal), since it's usually easier.

Daniel.

I would definitely argue that solving the Schrodinger equation or the path integral is both nontrivial in most cases. I'm curious about known solutions for the Schrodinger equation versus the path integral...

For the Schrodinger equation, I know there are solutions to the free particle, harmonic oscillator, V~r potential, V~1/r potential, and that the hydrogen atom has been solved exactly in a constant magnetic field (although I don't remember where I saw this...).

For the path integral formulation, free particle and harmonic oscillator are solved (this is in any book that deals extensively with path integrals), as well as the hydrogen atom (this is not).

Does anybody know of other existing solutions?
 
  • #4
- "Semiclassical expansion"------> You expand the action [tex] S[\phi] [/tex] of the field near its classical solution so [tex] \delta S[\phi _{classical}]=0 [/tex] i don't know who proved it, i heard about that "Hawking effect" or other Pseudo-Quantum Gravity models were based on this expansion..the corrections to the WKB approach can be taken:

[tex] \sum_{n=1}^^{\infty}a(n,r,t)(\hbar ^{n})^{n} [/tex]

Unfortunately we can't sum the series for any h except h=0 since it diverges...:frown: :frown: in any case if we could take the sum then we could find an EXACT method to evaluate path integrals.

- More complicate methods?..well they told in a seminar at my University that under "Wick rotation" and other methods perturbation theory with path integral could be evaluate using infinite-dimensional Montecarlo integration with a Infinite-dimensional Gaussian Meassure (don't ask :redface: )
 
  • #5
More complicate methods?..well they told in a seminar at my University that under "Wick rotation" and other methods perturbation theory with path integral could be evaluate using infinite-dimensional Montecarlo integration with a Infinite-dimensional Gaussian Meassure (don't ask)

Yes, Wick rotation makes the integrals amenable to numerical solution but I do not think they give any further analytical solutions.

Try look up anything by Hagen Kleinhert. There are a class of transformations which give a path integral with certain potential a gaussian measure, which is then solvable.

One extra for the SE is the Hookian atom, although this is just a 3d H0.
 
  • #6
the propogator is:

[tex]\Psi_f(x,t) = e^{\frac{i}{\hbar}S} \Psi_i(x,t)[/tex]

where we are propogating some initial wavefunction [tex]\Psi_i[/tex]to some final wavefunction [tex]\Psi_f[/tex]

take the derivative with respect to the final time [tex]t_f[/tex]:

[tex]\frac{\partial \Psi_f}{\partial t_f} = \frac{i}{\hbar} \frac{\partial S}{\partial t_f}e^{\frac{i}{\hbar}S} \Psi_i[/tex]

[tex]\frac{\partial \Psi_f}{\partial t_f} = \frac{i}{\hbar} \frac{\partial S}{\partial t_f} \Psi_f[/tex]

there is a result from classical lagrangian mechanics (i won't derive it here, you can look it up in any classical mech. book) that says that [tex]\frac{\partial S}{\partial t_f} = -E_f[/tex] so:

[tex]\frac{\partial \Psi_f}{\partial t_f} = -\frac{i}{\hbar} E_f \Psi_f[/tex]


[tex]i \hbar \frac{\partial \Psi_f}{\partial t_f} = E_f \Psi_f[/tex]

which of course is the time-dependent Schrodinger equation
 

Related to Recovering SE from Path integral

1. What is the purpose of recovering SE from path integral?

The purpose of recovering SE (Schrödinger equation) from path integral is to understand the quantum mechanical behavior of a system. The path integral formulation of quantum mechanics provides a powerful mathematical tool for calculating the probability of various outcomes in a quantum system. By recovering SE from path integral, we can derive the fundamental equation that governs the time evolution of a quantum system.

2. How is SE recovered from path integral?

SE can be recovered from path integral by converting the path integral into an integral equation called the "Feynman-Kac formula". This formula relates the path integral to the solution of SE. By taking the limit of this formula as the time step approaches zero, we can derive the original form of SE.

3. Why is recovering SE from path integral important?

Recovering SE from path integral is important because it provides a rigorous mathematical foundation for quantum mechanics. It allows us to understand the dynamics of a quantum system and make predictions about its behavior. Additionally, the path integral approach is often more intuitive and easier to work with than traditional methods of solving SE.

4. What are the limitations of recovering SE from path integral?

One limitation of recovering SE from path integral is that it is not always possible to do so analytically. In some cases, numerical methods must be used to approximate the solution. Additionally, the path integral approach may not be suitable for all types of quantum systems, particularly those with complex interactions or boundary conditions.

5. How does recovering SE from path integral relate to other formulations of quantum mechanics?

The path integral formulation of quantum mechanics is equivalent to the traditional formulation using SE. This means that any predictions or calculations made using one approach can also be made using the other. However, the path integral approach offers a different perspective and may be more useful in certain situations, such as when dealing with multiple particles or non-linear interactions.

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