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In quantum mechanics, there exist some systems where the potential energy of some particle is a Dirac delta function of position: ##V(x) = A\delta (x-x_0 )##, where ##A## is a constant with proper dimensions.

Is there any classical mechanics application of this? It would seem that if I approximate the delta with a Gaussian of nonzero width

##V(x) = Ae^{-k(x-x_0 )^2}##,

then a particle coming from the left with velocity ##v## could either

1. Have enough kinetic energy to get over the barrier and continue to right with same velocity ##v##

2. Have exactly the right amount of kinetic energy to get on the top of the barrier and stay there in unstable equilibrium

3. Have less kinetic energy than needed to get over the barrier and bounce back, returning to the left direction with velocity ##-v##.

Here I'm assuming that ##A>0##. Is there any reason why this wouldn't also hold when ##A\rightarrow\infty## and ##k\rightarrow\infty## ?

Is there any classical mechanics application of this? It would seem that if I approximate the delta with a Gaussian of nonzero width

##V(x) = Ae^{-k(x-x_0 )^2}##,

then a particle coming from the left with velocity ##v## could either

1. Have enough kinetic energy to get over the barrier and continue to right with same velocity ##v##

2. Have exactly the right amount of kinetic energy to get on the top of the barrier and stay there in unstable equilibrium

3. Have less kinetic energy than needed to get over the barrier and bounce back, returning to the left direction with velocity ##-v##.

Here I'm assuming that ##A>0##. Is there any reason why this wouldn't also hold when ##A\rightarrow\infty## and ##k\rightarrow\infty## ?

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