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Propagator/transition amplitude through intermediate integrations

  1. Jun 14, 2008 #1
    Hi all,

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

    [tex](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)[/tex].

    the situation is this:
    i already have an approximation of [tex](a,T_{i-1}|-a,T_i)[/tex] which is qualitively different from [tex](a,T|-a,-T)[/tex], 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)
  2. jcsd
  3. Jun 14, 2008 #2
    The infinitesmal amplitudes can be exponentiated, since [tex]1 + \epsilon \approx e^{\epsilon}[/tex]. 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 [Broken]
    Last edited by a moderator: May 3, 2017
  4. Jun 14, 2008 #3
    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?
    Last edited by a moderator: May 3, 2017
  5. Jun 14, 2008 #4
    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 [itex]G(q_{j+1}, t_{j+1}; q_j, t_j)[/itex] that casts it in the form:
    [tex]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]}[/tex].
  6. Jun 14, 2008 #5
    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


    because i already know much about [tex](a,T_{n-1}|-a,T_n)[/tex].
  7. Jun 14, 2008 #6
    What approximation do you already have?
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