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 schrodinger 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?

 

[rwisz]

I would like to clarify that the collision between a free electron and a hydrogen atom is a complex process and there are many factors that can affect the outcome. The 11.5 eV of kinetic energy is just one possible scenario and it is important to consider other possibilities as well.

To answer your first question, the maximum energy that the orbiting electron can absorb is determined by the energy levels of the hydrogen atom. In this case, the maximum energy that can be absorbed is 13.6 eV, which corresponds to the transition from the ground state (n=1) to the first excited state (n=2). This is a fundamental property of the hydrogen atom and is not a simplification.

Regarding the concept of "rebounding off the atom", it is important to note that the electron is not actually colliding with the atom in a traditional sense. Instead, it is interacting with the electric field of the atom, which can either result in an excitation of the orbiting electron or a change in the direction of the free electron. Therefore, the term "rebound" is used to describe the change in direction of the free electron after the interaction.

In conclusion, the collision between a free electron and a hydrogen atom is a complex process and the outcome depends on various factors. The 11.5 eV of kinetic energy is just one possible scenario and it is important to consider other possibilities as well. It is also important to note that the terms used to describe this process, such as "rebound", are simplifications and may not accurately represent the actual physics involved.