Recovering SE from Path integral

[Karlisbad]

I know someone posted it before..but i would like to know if given the factor:

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

and knowing the progpagator satisfies:

$$\Psi (x2,t2)=\int_{-\infty}^{\infty}dxdtK(x2,t2,x1,t1)\Psi(x1,t1)$$
Where S is the action and the propagator is related to (1)
:zzz:

 

[dextercioby]

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.

 

[StatMechGuy]

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 Schrödinger equation or the path integral is both nontrivial in most cases. I'm curious about known solutions for the Schrödinger equation versus the path integral...

For the Schrödinger 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?

 

[Karlisbad]

- "Semiclassical expansion"------> You expand the action $$S[\phi]$$ of the field near its classical solution so $$\delta S[\phi _{classical}]=0$$ 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:

$$\sum_{n=1}^^{\infty}a(n,r,t)(\hbar ^{n})^{n}$$

Unfortunately we can't sum the series for any h except h=0 since it diverges... 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 )

 

[Epicurus]

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.

 

[quetzalcoatl9]

the propogator is:

$$\Psi_f(x,t) = e^{\frac{i}{\hbar}S} \Psi_i(x,t)$$

where we are propagating some initial wavefunction $$\Psi_i$$to some final wavefunction $$\Psi_f$$

take the derivative with respect to the final time $$t_f$$:

$$\frac{\partial \Psi_f}{\partial t_f} = \frac{i}{\hbar} \frac{\partial S}{\partial t_f}e^{\frac{i}{\hbar}S} \Psi_i$$

$$\frac{\partial \Psi_f}{\partial t_f} = \frac{i}{\hbar} \frac{\partial S}{\partial t_f} \Psi_f$$

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 $$\frac{\partial S}{\partial t_f} = -E_f$$ so:

$$\frac{\partial \Psi_f}{\partial t_f} = -\frac{i}{\hbar} E_f \Psi_f$$


$$i \hbar \frac{\partial \Psi_f}{\partial t_f} = E_f \Psi_f$$

which of course is the time-dependent Schrödinger equation