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.
a. 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, determine the value of the energy E_{q} where quantum effects become important. Write your answer in terms of some or all of g, m, and Planck’s constant h.
b. A second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height H, we can write
where p is the momentum as a function of height x above the ground, n is a non-negative integer, and h is Planck’s constant.
i. Determine the allowed energies E_{n} as a function of the integer n, and some or all of g, m, and Planck’s constant h.
ii. Numerically determine the minimum energy of a bouncing neutron. The mass of a neutron is m_{n}=1.675 \times 10^{-27} \mathrm{~kg}=940 \mathrm{MeV} / \mathrm{c}^{2}; you may express your answer in either Joules or \mathrm{eV}.
iii. Determine the bounce height of one of these minimum energy neutrons.
c. Let E_{0} be the minimum energy of the bouncing neutron and f be the frequency of the bounce. Determine an order of magnitude estimate for the ratio E / f. It only needs to be accurate to within an order of magnitude or so, but you do need to show work!