Estimate energy of infinite well (ground state)

So, the argument gives an estimate for the ground state energy ofE = px2/(2m) \approx \frac{\hbar^{2}}{2mL^2}In summary, we can estimate the ground state energy of an infinite potential well using the Heisenberg uncertainty principle. By taking the uncertainty in position to be the length of the well, we can estimate the uncertainty in momentum and use that to estimate the ground state energy. The argument may be shaky, but it gives an estimate of E = \hbar^{2}/(2mL^2). We can then compare this with the exact value from the eigenvalue equation to see how accurate our estimate is.
  • #1
thatguy14
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Homework Statement


We have to estimate the ground state energy of an infinite potential well (1d) using an argument based on the Heisenberg uncertainty principal. We then are supposed to compare it with the exact value from the eigenvalue equation.


Homework Equations



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The Attempt at a Solution



I am not really sure where to start on this one but this is what I tried:

ΔxΔp = h and since there is no potential where the particle is all the energy is kinetic so classically Ke = E = p^2/2m. then to look at change i just did

Δp = [itex]\sqrt{ΔE2m}[/itex]

and subbed it into the above unvertainty principle. This gives

Δx[itex]\sqrt{ΔE2m}[/itex] = h
[itex]Δx^{2}[/itex]ΔE2m = [itex]h^{2}[/itex]
ΔE = [itex]\frac{h^{2}}{2mΔx^2}[/itex]

The problem is that even if this is somewhat right I am not sure what I really did. Can anyone give me some hints?
 
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  • #2
Usually Δ represents "change" in a quantity. But in the uncertainty principle, it indicates the uncertainty in a quantity. The uncertainty is loosely the "spread" in possible results of measuring that quantity.

You know the particle is somewhere in the box of length L. That's all the information you have to go on when trying to estimate the uncertainty in position. So, as a crude estimate, you can take the uncertainty in x to be L: Δx [itex]\approx[/itex] L. Using this estimate for Δx, the uncertainty principle gives an estimate for the uncertainty in momentum Δpx.

To estimate the ground state energy, you need an estimate of the momentum px since E = px2/(2m). You have an estimate for Δpx, but you need an estimate of px itself. Here's where the argument gets "shaky" in my opinion. It seems reasonable that px should be at least of the order of magnitude of Δpx. The usual assumption is to just estimate px by its uncertainty Δpx.

Now, it is easy to think of examples where the value of some quantity is orders of magnitude greater than the uncertainty in the quantity. So, you could worry that estimating px by Δpx might be a bad estimate. But that is what is done in this case.

You can sort of support this by considering the classical particle in a box. The classical particle bounces back and forth in the box with a fixed magnitude of momentum but with a changing direction. So, the momentum has two possible values [itex]\pm[/itex] px and each is equally likely. So, the "spread" in momentum is of the same order of magnitude as the momentum itself.
 
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FAQ: Estimate energy of infinite well (ground state)

1. What is an infinite well in physics?

An infinite well in physics is a theoretical model used to represent a confined particle, such as an electron, in a potential well. It assumes that the particle is confined to a specific region and cannot escape its boundaries.

2. How do you estimate the energy of an infinite well in its ground state?

To estimate the energy of an infinite well in its ground state, you can use the Schrödinger equation, which is a mathematical equation used to describe the behavior of quantum systems. By solving this equation for the infinite well potential, you can obtain the energy of the particle in its ground state.

3. What factors affect the energy of an infinite well?

The energy of an infinite well is affected by the size of the well, the mass of the particle, and the properties of the potential barrier. The energy also depends on the quantum state of the particle, such as its position and momentum.

4. How does the energy of an infinite well change with different quantum states?

The energy of an infinite well increases with each quantum state, also known as the energy level. This is because as the quantum state increases, the particle has more energy and is able to occupy higher energy levels within the well.

5. How is the energy of an infinite well related to the uncertainty principle?

The energy of an infinite well is related to the uncertainty principle, which states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position and momentum, can be known simultaneously. In the case of an infinite well, the energy and position of the particle are related, and as the energy increases, the position becomes more uncertain.

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