# Action for a discrete potential

1. Apr 16, 2008

### hattori

Hello,

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

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 $$x$$ 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 $$x \textrm{ within } [0, \delta]$$, another for $$x \textrm{ within } [\delta, 2\delta]$$, 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 $$S(x_b, t_b; x_a, t_a)$$, 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 $$K(x_b, t_b; x_a, t_a) = F(t_b,t_a) exp{(i/\hslash)S(x_b, t_b; x_a, t_a)}$$.

Last edited: Apr 16, 2008