Solution for differential equation

In summary, the conversation discusses a problem where a particle moves towards a bounded spherical potential with a given energy. The correct trajectory is discussed, and the problem is approached analytically by deriving a differential equation. The potential and energy conservation equations are used to solve the problem, and the conservation of angular momentum is assumed. It is stated that the particle cannot penetrate the potential if its energy is lower than the potential energy. The problem is compared to scattering off a hard sphere and the sudden jump in potential is treated as a limiting case. The question of whether all nonlinear models produce the same limiting solution is raised.
  • #1
PhysicsRock
114
18
Greetings,
in one of the exercise sheets we were given by our Prof, we were supposed to draw the trajectory of a patricle that moves toward a bounded spherical potential that satisfies

##
V(\vec{r}) = \begin{cases}
V_0 & | \vec{r} | \leq a \\
0 & else \\
\end{cases}
##

for ##E_\text{particle} < V_0##. I drew a circular path within the potential boundary, but apparently that is wrong. A friend of mine brought to my attention that the particle couldn't enter the potential at all, if it's Energy is lower than the potential energy. I wanted to take a look at this analytically and derived the differential equation

##
m \ddot{\vec{r}} = V_0 \left[ \delta(r - a) - \delta(r + a) \right] \frac{\vec{r}}{r}
##

with ##r = | \vec{r} |##. Intuitively, it makes sense to me as there is no Force on the particle unless it's right at the boundaries where the potential changes, thus satisfying ##\vec{F} = - \nabla V##.

To come to my question, has anyone got any idea how this could be solved, if it is even possible to solve an equation of that form?
 
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  • #2
You have written a three-dimensional equation for what is essentially a one-dimensional problem because the potential depends on ##r## only. For that reason, I think it is safe to assume that angular momentum is conserved, write the standard energy conservation equations, one for each region, used in central force problems and see where it takes you. You will have to figure out how to quantify the "kick" that the particle receives at the boundary. I have not solved this problem, but that would be my approach.

Your friend is correct. If the particle has energy less than ##V_0##, it will not penetrate (classically) the potential, i.e. it will bounce off with no energy loss. This sounds like scattering off a hard sphere to me.
 
  • #3
The trajectory is straightforward everywhere except at the boundary.
Focusing in on the boundary, you can treat it as a flat surface.
The sudden jump in potential should be thought of as a limiting case. E.g., start with a simple linear rise over some short distance, solve, then take the limit as the distance tends to zero.
The awkward question is whether all nonlinear models produce the same limiting solution. If not, the question is not well posed.

Fwiw, it smells to me like a light ray striking a boundary where the refractive index drops .
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives. It describes the relationship between a function and its rate of change.

2. Why are differential equations important in science?

Differential equations are important in science because they are used to model and describe many natural phenomena, such as the growth of populations, the motion of objects, and the flow of fluids. They help us understand and predict the behavior of complex systems.

3. What is the difference between an ordinary and a partial differential equation?

An ordinary differential equation involves only one independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations are used to describe phenomena in one dimension, while partial differential equations are used to describe phenomena in multiple dimensions.

4. How do you solve a differential equation?

The method for solving a differential equation depends on the type of equation and its complexity. Some common methods include separation of variables, substitution, and using an integrating factor. More complex equations may require numerical methods or computer simulations.

5. What are some real-world applications of differential equations?

Differential equations are used in many fields of science and engineering, including physics, chemistry, biology, economics, and engineering. They are used to model and predict the behavior of systems such as population growth, chemical reactions, heat transfer, and electrical circuits.

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