Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Action for a discrete potential

  1. Apr 16, 2008 #1
    Hello,

    I have a potential function given numerically at points evenly spaced. That is to say, I have the numerical values of [tex]V(0), V(\delta), V(2\delta), V(3\delta), ...[/tex], in some interval. I need to calculate the action integral in terms of initial and end-points: [tex]S(x_b, t_b; x_a, t_a)[/tex].

    I think I first need an analytical expression for potential to start with. Since I only know potential for some few points, I tried to write down an interpolation function. However, there's one restriction: the potential function can be quadratic in [tex]x[/tex] at most. Quadratic spline interpolation gives just that sort of function, but the problem with it is, the method gives a piecewise function for potential. One function defined within [tex]x \textrm{ within } [0, \delta][/tex], another for [tex]x \textrm{ within } [\delta, 2\delta][/tex], and so on. The potential (and it's derivative) generated with quadratic spline is continuous, but well, it's piecewise.

    So, I wonder:
    • Is there a way for calculating [tex]S(x_b, t_b; x_a, t_a)[/tex], using a piecewise function?
    • Or is there a way to write down an interpolation function that is quadratic,and not piecewise?
    • Or any way around without ever writing down an interpolation?

    Any help or advice will be appreciated.
    Thanks.


    Note: The problem is actually quantum-mechanical, and the restriction to quadratic functions stems from the fact that, when the potential does not contain 3rd or higher orders, one can use [tex]K(x_b, t_b; x_a, t_a) = F(t_b,t_a) exp{(i/\hslash)S(x_b, t_b; x_a, t_a)}[/tex].
     
    Last edited: Apr 16, 2008
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Action for a discrete potential
  1. Capillary Action (Replies: 6)

  2. Capillary action? (Replies: 5)

  3. Least action (Replies: 1)

  4. What is action ? (Replies: 19)

Loading...