Collision of a free electron and a hydrogen atom - energies

[kapitan90]

Homework Statement

An electron of know KE collides with a hydrogen atom in its ground state. With what possible KE may it rebound?
KE = 11.5 eV

2. The attempt at a solution
I assumed that the electron may either hit an orbiting electron and excite him (maximum layer is n = 2, change in KE = 11.5 - (13.6 - 3.4) = 1.3 eV)
or miss it (KE unaltered = 11.5eV)

Those answers are correct, yet I don't understand them fully.

3. Relevant questions
How do we know that the orbiting electron will take the maximum amount of energy it can take? Is it a simplification or a free electron has to give this amount of energy to the orbiting electron? Why?

Also, can we speak of 'rebounding off the atom' if the electron doesn't hit the free electron? It cannot collide with atom's nucleus as it's energy is unaltered, so how may it rebound?

 

[AlexChandler]

If it collides with the nucleus, since the nucleus is so massive, it will just bounce right back with its energy unchanged. It is like throwing a ball against a brick wall.

The orbiting electron can only exist in quantized states. To see why, you will have to solve the Schrödinger equation. The energy of any state is

$$E_n = - \frac{E_0}{n^2}$$

where n is an integer, and E0 is the ground state energy 13.6eV. So the incident electron can only rebound with quantized energy levels.

$$K_f = K_i - \Delta E_{atom} = K_i - (E_n - E_0)$$

 

[kapitan90]

But why can't an orbiting electron be excited to n = 1, only to n = 2?

 

[AlexChandler]

But why can't an orbiting electron be excited to n = 1, only to n = 2?

For higher transitions, it will take more and more energy. Does the incident electron have enough energy to make the orbiting electron transfer from n=1 to n=3?