Question

Consider a particle of mass m that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is E and the acceleration of free fall is g. Treat the particle as a point mass and assume the motion is non-relativistic.

An estimate for the regime where quantum effects become important can be found by simply considering when the deBroglie wavelength of the particle is on the same order as the height of a bounce. Assuming that the deBroglie wavelength is defined by the maximum momentum of the bouncing particle,

Answer 1

determine the value of the Energy Eq where quantum effects become important. Write your answer in terms of some or all of g, m, and Plank's constant h.

Answer 2

@JFraser

Answer 3

Let me know if any of this makes sense.
Total mechanical energy is PE + KE which is mgh + 1/2mv^2. And momentum is mv, so if I factor out an mv, I get PE + KE = mv(gh/v + v/2)

Answer 4

I'm going to call PE + KE, E for simplicity
E = p(gh/v + v/2)
p = E/(gh/v + v/2)

Answer 5

And \[\lambda = h/p\]

Answer 6

But I don't have any info about my v. Can I ignore the PE part of the total mechanical energy? It sure would simplify my calcs.

Answer 7

I'll have to think about this one for a while

Answer 8

No worries, thanks!

Answer 9

particle has the maximum momentum when the all energy is in KE, that is just after bouncing from the surface
\[E=\frac{mv^2}{2}\]
from here
\[v=\sqrt{2E/m}\]
\[P=mv=\sqrt{2mE}=\frac{h}{\lambda}\]
at its maximum height same energy is converted to PE
\[E=mgH\]
quantum effects starts at \[H=\lambda\]

Answer 10

\[E=(\frac{mg^2h^2}{2})^{1/3}\]

Answer 11

does it make sense?

Answer 12

Yes, thanks!