Hello,(adsbygoogle = window.adsbygoogle || []).push({});

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?

Anyhelp 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].

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Action for a discrete potential

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**