Can the Uncertainty Principle Prove Electrons Cannot Be Inside the Nucleus?

[teleport]

Homework Statement

It is given the maximum energy of electrons emitted in beta decay. I'm asked to show using that and the uncertainty principle that electrons cannot be inside the nucleus.

Homework Equations

The Attempt at a Solution

I can't show my work because I don't even know how to start!

 

[Dick]

Think about what the uncertainty principle would say about the kinetic energy of an electron confined in a nucleus. KE=p^2/2m.

 

[teleport]

For the uncertainty principle to say something about that, wouldn't it need one of two uncertainty inputs in order to return a value? But where is this input given or derived from?

 

[Dick]

For the uncertainty principle to say something about that, wouldn't it need one of two uncertainty inputs in order to return a value? But where is this input given or derived from?

The delta x corresponds to the size of the nucleus.

 

[teleport]

OK, by using the greatest nucleus (I believe around 15 fm but not sure of the exact number) I get a lower bound on the minimum momentum uncertainty which is greater than the upper bound on the momentum itself derived from the given max energy of electrons by about a factor of 10. Is this what you meant? If so, why is such a (although high)fractional uncertainty the answer to the impossibility of such an event from occurring? Because it's high enough?

 

[Dick]

The physical picture they are suggesting is that the electron is running around in the nucleus confined by a potential barrier which it eventually tunnels through. Which means it's emission energy should be comparable to it's kinetic energy confined in the nucleus. Like anything done with the uncertainty principle, this an estimate. But a large deviation from the estimate suggests that picture is not correct.

 

[teleport]

Not sure I'm following you completely. I assumed like you said that the emission energy was comparable to the kinetic energy inside the nucleus. That way I got a max value for the momentum inside the nucleus from the emission energy alone. My calculation was the estimate, so what is the value that diviates? The huge uncertainty I got? (more than 100% uncertainty relative to the momentum)

 

[Dick]

Maybe you should post what numbers you are getting. I'm getting that the emission energy of a confined electron is implausibly large compared with the confined electron estimate. By more than a factor of 10.

 

[teleport]

I substracted the rest energy, which makes it smaller if you didn't. Good thanks, I'm ok with the answer.