Estimate energy of infinite well (ground state)

[thatguy14]

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

Below

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 = $\sqrt{ΔE2m}$

and subbed it into the above unvertainty principle. This gives

Δx$\sqrt{ΔE2m}$ = h
$Δx^{2}$ΔE2m = $h^{2}$
ΔE = $\frac{h^{2}}{2mΔx^2}$

The problem is that even if this is somewhat right I am not sure what I really did. Can anyone give me some hints?

 

[TSny]

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 $\approx$ L. Using this estimate for Δx, the uncertainty principle gives an estimate for the uncertainty in momentum Δp_x.

To estimate the ground state energy, you need an estimate of the momentum p_x since E = p_x²/(2m). You have an estimate for Δp_x, but you need an estimate of p_x itself. Here's where the argument gets "shaky" in my opinion. It seems reasonable that p_x should be at least of the order of magnitude of Δp_x. The usual assumption is to just estimate p_x by its uncertainty Δp_x.

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 p_x by Δp_x 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 $\pm$ p_x and each is equally likely. So, the "spread" in momentum is of the same order of magnitude as the momentum itself.