How Does Electron Momentum Change Across Hydrogen Atom Energy Levels?

[khorsani]

Consider the hydrogen atom (proton and electron)...
1. the radius of the lowest energy state is about 5x10^(-11) m. How
well can you know the momentum of the electron? In your solution, show
that you get units of momentum.

P=mv

2. If energy is added, so that the electron moves up to the fifth
energy level, will the electron have moreor less momentum? Explain
your reasoning.

3. Consider two transitions:
(a) from level 5 to level 2
(b) from level 3 to level 2
both transitions produce photons in the visible range, one in the red
and the other in the blue. Which transition goes with which photon??
Justify your reasoning.

So does this make any sense, am I on the right path?


f= c/lambda

E upper - E lower = hf

thus it follows that:

1/hc(E upper - E lower) = 1/lambda = R (1/2squared - 1/n squared)

and then I need Balmers formula to find the energy level in terms of the kinetic and potential energy? Am I on the right path?


Here is a response from the forum:

Ok, here what we know so far:

1. the electron has angular momentum.
2. But only certain values of angular momentum which are multiples of Plank's constant.
3. the combination of quantized energy and quantized angular momentum picks out only certain allowed orbits
4. so: the wavefronts are "quantized", only certain orbits are possible, only certain energies are possible, only certain angular momenta are possible and the light is emitted in transitions between orbits.
5. (The electron isn't following orbital paths in hydrogen, it is confined to regions of space)
6. Only two electrons end up in every energy-and-angular momentum combination

here I'm a little lost in understanding all this, but I'll keep trying

anyway:
p = h/lambda kg x m/s
p = 6.63 x 10^(-34) m^2 x kg/s

 

[Dick]

The first one is an uncertainty principle problem. Neither of the other two require an exact calculation. For the second one, just apply the same logic as in the first one. For the third you just need to know whether the energy difference between 5 and 2 is greater than or less between than 3 and 2.

 

[ogb p]

/ (5 x 10^(-11) m)
p = 1.33 x 10^(-23) kg x m/s

So the momentum of the electron in the lowest energy state is 1.33 x 10^(-23) kg x m/s. However, since the electron is confined to a small region of space, its momentum cannot be known with certainty. This is due to the Heisenberg uncertainty principle, which states that the more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa. Therefore, we can only know the momentum of the electron in the lowest energy state within a certain range of values.

If energy is added to the electron and it moves to the fifth energy level, it will have more momentum. This is because the energy level is directly proportional to the momentum of the electron. As the electron moves to a higher energy level, its momentum also increases.

Now, for the transitions between energy levels. The energy of a photon is given by the equation E = hf, where h is Plank's constant and f is the frequency of the photon. The frequency of a photon is related to its wavelength by the equation f = c/lambda, where c is the speed of light. So, we can say that the energy of a photon is inversely proportional to its wavelength.

In the hydrogen atom, the energy levels are given by the equation E = -R(1/n^2), where R is the Rydberg constant and n is the principal quantum number. This means that as n increases, the energy level decreases. Therefore, the transition from level 5 to level 2 corresponds to a higher energy photon (blue) than the transition from level 3 to level 2 (red). This is because the transition from level 5 to level 2 involves a larger change in energy than the transition from level 3 to level 2. So, the transition from level 5 to level 2 goes with the blue photon and the transition from level 3 to level 2 goes with the red photon.

Your understanding of the concept seems to be on the right track. Keep exploring and learning more about quantum mechanics and the hydrogen atom to deepen your understanding. Great job!