- #1

PhysicsRock

- 38

- 4

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?