Action for a discrete potential

In summary, the problem at hand is to calculate the action integral S(x_b, t_b; x_a, t_a) for a potential function given numerically at evenly spaced points. Possible approaches include finding an analytical expression for the potential, using piecewise interpolation methods, non-piecewise interpolation methods, or numerical integration techniques.
  • #1
hattori
3
0
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:
Physics news on Phys.org
  • #2





Thank you for sharing your problem with us. I would suggest the following approaches to calculate the action integral S(x_b, t_b; x_a, t_a) for your potential function:

1. Analytical expression for potential: If possible, try to find an analytical expression for your potential function. This will make the calculation of the action integral much easier and more accurate. You can try to fit your given numerical data points to a known potential function or try to derive an expression from the physical principles governing your system.

2. Piecewise interpolation: If finding an analytical expression is not feasible, you can use piecewise interpolation methods such as cubic splines or Hermite interpolation. These methods can give a smooth and continuous function that passes through your given data points. However, the accuracy of the interpolation will depend on the number and distribution of your data points.

3. Non-piecewise interpolation: If you want to avoid using piecewise functions, you can try using non-piecewise interpolation methods such as polynomial interpolation or rational interpolation. These methods can also give a smooth and continuous function, but they may require more data points to accurately represent the potential function.

4. Numerical integration: If you do not want to use any interpolation methods, you can directly calculate the action integral using numerical integration techniques such as the trapezoidal rule or Simpson's rule. This approach may be more time-consuming, but it will give an accurate result.

I hope these suggestions will help you in calculating the action integral for your potential function. If you have any further questions or need more assistance, please do not hesitate to ask. Good luck with your research.


A fellow scientist
 
  • #3


I understand your predicament and the importance of finding a solution for your problem. It seems that your main concern is finding a way to calculate the action integral using a piecewise function or an interpolation function that is quadratic and not piecewise. I would suggest exploring other interpolation methods, such as cubic spline interpolation, which can provide a smoother and more continuous potential function. Additionally, you could also try to approximate the potential function using a series expansion, such as a Taylor series, to obtain an analytical expression that can be used to calculate the action integral. However, keep in mind that any approximation method will introduce some error, so it's important to carefully consider the accuracy and precision of your results. Another option could be to use numerical integration techniques, such as the trapezoidal rule or Simpson's rule, to approximate the action integral without the need for an analytical expression for the potential function. I hope these suggestions are helpful and I wish you success in finding a solution for your problem.
 

1. What is a discrete potential?

A discrete potential refers to a system in which the potential energy is only dependent on the discrete positions of the particles involved, rather than being a continuous function of position.

2. How does action relate to a discrete potential?

Action is a physical quantity that describes the motion of particles in a system. In the case of a discrete potential, the action is given by the sum of the potential energy at each discrete position of the particles, multiplied by the corresponding displacement.

3. How is the action for a discrete potential calculated?

The action for a discrete potential can be calculated using the principle of least action, which states that the actual path taken by a particle between two points is the one that minimizes the action. This involves finding the path that satisfies the equations of motion and minimizes the action.

4. What is the significance of action for a discrete potential?

The action for a discrete potential is significant because it allows us to understand and predict the behavior of particles in a system with discrete potential energy. It provides a mathematical framework for determining the equations of motion and predicting the future state of the system.

5. How does a discrete potential differ from a continuous potential?

A discrete potential differs from a continuous potential in that it only takes into account the potential energy at discrete positions, while a continuous potential considers the potential energy at all points in space. This can lead to different equations of motion and behaviors for particles in the two types of systems.

Similar threads

  • Thermodynamics
Replies
4
Views
621
  • Introductory Physics Homework Help
Replies
25
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
806
  • Classical Physics
Replies
17
Views
2K
  • Classical Physics
Replies
12
Views
4K
  • Quantum Interpretations and Foundations
Replies
6
Views
1K
Replies
4
Views
1K
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
1K
  • Special and General Relativity
Replies
19
Views
3K
Back
Top