Electron in box considering effects of gravity

In summary, the conversation discusses estimating the thickness of the layer occupied by an electron at the bottom of a large rectangular box, considering the effects of gravity. The uncertainly principle and Schrodinger's equation are suggested as possible tools to use for estimation. The conversation also includes a discussion on the energy levels and potential energy of the electron in the box, as well as the uncertainty in momentum. Ultimately, the conversation ends with a formula for estimating the thickness of the layer, but questions remain about the values of energy and the size of the box.
  • #1
mps
27
0
An electron is enclosed in a large rectangular box. Considering gravity, estimate the order of magnitude of the thickness of the layer that is occupied by the electron at the bottom of the box.

I thought about this for a long time and I really have no idea how to proceed. Any thoughts?
 
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  • #2
Hi mps, welcome to PF. In the future, please do not shirk the problem template: it's there for a reason!

What tools do you have for estimating the spatial extent of a particle? For example, could the uncertainty principle help you out here? Alternatively, could you try and solve for the vertical wave function for the particle under the (approximately linear) potential of gravity?
 
  • #3
Thanks Steely Dan! I don't know how the uncertainly principle can help here...could you please explain? As for the wave function, does this mean I have to use Schrodinger's equation?
Do I have to use both the uncertainly principle and Schrodinger's or just one or the other?
 
  • #4
mps said:
Thanks Steely Dan! I don't know how the uncertainly principle can help here...could you please explain? As for the wave function, does this mean I have to use Schrodinger's equation?
Do I have to use both the uncertainly principle and Schrodinger's or just one or the other?

Well, since you're just being asked for a rough estimate, using the uncertainty principle sounds like the way to go if you can do it; if there's no way to proceed, then one would appeal to the Schrodinger equation (there is a solution to the equation for the gravitational potential, the Airy function). Clearly the electron is more likely to be near the bottom than the top; in particular, all other things being equal, it would prefer to be exactly at the bottom, but the uncertainty principle forbids this. So the "layer" it exists in can be estimated if you can estimate the uncertainty in the momentum. How can this be done? Well, try using the relationship between momentum and energy ([itex]p = \sqrt{2m(E-U)}[/itex]).
 
  • #5
Thanks so much Steely Dan!
I feel really stupid but how do you know what E and U are?
 
  • #6
I mean, i know its total energy and potential energy but what are the values?
 
  • #7
Steely Dan's given you a starting point. You need to make an effort at solving the problem yourself.
 
  • #8
Sorry but I have thought about it really hard and I really can't come up with anything.
If the electron is happy at the bottom, why would it have any momentum?
If the electron is at the bottom, it has no potential energy... How can we know the kinetic energy K (it's not given)?

p = sqrt (2mK) so to calculate p we need K, but I can't think of a way to get K.

I don't think rest energy helps because that's not mechanical.

Could you please help me some more? thanks!
 
  • #9
There are two competing effects here: considering the classical potential, the electron would simply sit at the bottom, but considering the uncertainty principle, it cannot be exactly at the bottom, for that would imply zero uncertainty in position (which we know cannot be right). So the electron exists in some layer of width [itex]\Delta x[/itex] where the minimum potential is zero and the maximum potential is [itex]mg\Delta x[/itex].
 
  • #10
Oh! Well then
[itex]Δp = \sqrt{2m(E-0)} - \sqrt{2m(E-mgΔx)}[/itex]
and we substitute this into ΔpΔx≥h/2∏ to find Δx.

What is E though?
Many thanks! :)
 
  • #11
Well, what are the energy levels of a well with infinite potential boundaries?
 
  • #12
Okay En=n22ħ2/(2mL2)

Problems
1. Do we just set n = 1 since when know the particle is near the ground?
2. The equation is derived for when potential energy is zero (when kinetic and potential energy are equal). Is En going to be E in our case? So we get the equation below?*
3. We don't know L. Since the box is "large", L -->∞ so E --> 0?

*
[itex]Δp = \sqrt{2m(∏2ħ2/(2mL2))} - \sqrt{2m(∏2ħ2/(2mL2) - mgΔx)}[/itex]
 

1. What is an "Electron in box considering effects of gravity"?

An "Electron in box considering effects of gravity" is a theoretical model used in quantum mechanics to study the behavior of an electron confined in a box-like potential while also taking into account the effects of gravity. In this model, the electron is treated as a particle in a one-dimensional box with infinite potential walls, with the additional consideration of the gravitational force acting on the electron.

2. How does the electron behave in this model?

The behavior of the electron in this model is described by the Schrödinger equation, which takes into account both the confinement of the electron in the box and the gravitational potential. The electron's behavior can be characterized by its energy levels and wavefunction, which determine the probabilities of finding the electron at different positions in the box.

3. What are the implications of considering gravity in this model?

The inclusion of gravity in this model has important implications for understanding the behavior of quantum particles in strong gravitational fields, such as those near black holes. It also allows for the study of quantum phenomena, such as tunneling and interference, in the presence of gravity.

4. How does the size of the box affect the electron's behavior?

The size of the box, or the length of the potential well, plays a crucial role in determining the energy levels and wavefunction of the electron. As the size of the box decreases, the energy levels become more closely spaced and the electron's wavefunction becomes more confined, leading to different behaviors and probabilities for the electron's position.

5. Can this model be applied to other particles besides electrons?

Yes, this model can be extended to study the behavior of other particles, such as protons and neutrons, in a similar potential well with the effects of gravity. It can also be extended to study the behavior of multiple particles in the same potential well, allowing for a better understanding of complex systems and interactions.

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