Propagator/transition amplitude through intermediate integrations

[omg!]

Hi all,

suppose i want to calculate the propagator/transition amplitude which i can write as follows:

$$(a,T|-a,-T)=\int_{-\infty}^\infty dx_1\int_{-\infty}^\infty dx_2(a,T|x_1,T_1)(x_1,T_1|x_2,T_2)(x_2,T_2|-a,-T)$$.

the situation is this:
i already have an approximation of $$(a,T_{i-1}|-a,T_i)$$ which is qualitively different from $$(a,T|-a,-T)$$, so that i can't simply calculate the latter by changing the time variable in the first expression. but i think it should be possible to use this information somehow to obatin the complete propagator. the problem is that i don't know how to deal with the intermediate integrations.

any help would be greatly appreciated.

(i'm using the path integral formalism for the calculation)

 

[lbrits]

The infinitesmal amplitudes can be exponentiated, since $$1 + \epsilon \approx e^{\epsilon}$$. Once inside the exponential, you have a sum instead of a product. An analogous calculation is done here: http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/path_integral

 

[omg!]

The infinitesmal amplitudes can be exponentiated, since $$1 + \epsilon \approx e^{\epsilon}$$. Once inside the exponential, you have a sum instead of a product. An analogous calculation is done here: http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/path_integral

hi, thanks again for your response!

sorry for my ignorance, but i couldn't find anything that would help in the link you provided. could you tell me where to look, or elaborate further?

 

[lbrits]

Well, it isn't really clear what you're doing, but I hazard a guess that it has to do with path integrals :)

Specifically, the derivation of $G(q_{j+1}, t_{j+1}; q_j, t_j)$ that casts it in the form:
$$G(q_{j+1}, t_{j+1}; q_j, t_j) \approx \int\!\frac{dp_j}{2\pi}e^{\frac{i}{\hbar} \left[ p_j (q_{j+1}-q_j) - H(p_j, \bar{q}_j)\delta t \right]}$$.

 

[omg!]

hmm, that looks like some discretized version of the path integral.

what I'm trying to do is to write the expression in my first post into something like

$$(a,T|-a,-T)=C\prod_{n=1}^N(a,T_{n-1}|-a,T_n)$$

because i already know much about $$(a,T_{n-1}|-a,T_n)$$.

 

[lbrits]

What approximation do you already have?