Electrons, the nucleus and the uncertainty principle.

[uranium_235]

I read somewhere that one of the explanations for an electron not spiraling into the nucleus is due to the uncertainty principle. If an electron falls into the nucleus both its position and velocity will be certain. How is that possible? Does the nucleus have both certainty in position and velocity? Then would not this explanation contradict its self?

 

[ZapperZ]

I read somewhere that one of the explanations for an electron not spiraling into the nucleus is due to the uncertainty principle. If an electron falls into the nucleus both its position and velocity will be certain. How is that possible? Does the nucleus have both certainty in position and velocity? Then would not this explanation contradict its self?

1. In a sketch, draw a horizontal axis as the r (radial) axis, and the vertical axis as the potential energy (U) axis.

2. Sketch the coulomb potential U=-kQq/r, where Q is the charge of the nucleus, and q is the charge of another charged particle. This is the potential relevant in a simple, hydrogenic-type atom.

3. For a bound charge particle q, it can have a substantial probability to exist confined within the potential well bounded by the vertical axis, and the U potential profile.

4.. Now look at what happens when a charge q gets closer and closer to the nucleus, i.e. as r -> 0. The particle cannot have a substantial probability anywhere else other than within the potential well. And the width of the well is getting smaller and smaller as r approaches zero, meaning we are confining the charge to smaller and smaller region of space. Consequently, we are knowing more and more about where q is radially, thus reducing the uncertainty in its position.

5. If there is no uncertainty principle, this will cause no problem. However, because it is there, there will be an increase in the range of momentum values the charge can have. This will act as a counter effect to oppose being confined to a smaller volume. Thus, there is a minimum ground state that does not allow it to be any "closer".

Zz.

 

[Janitor]

... Does the nucleus have both certainty in position and velocity? Then would not this explanation contradict its self?

The mass of a particle enters into calculations of uncertainty. If the electron were replaced by a muon, which is similar but has a couple hundred times the mass of an electron, the muon would be confined pretty tightly near the nucleus. The nucleus itself is more massive yet than a muon, and so it is effectively confined to a miniscule region near the middle of an atom.

John Baez gives a nice 'back of the envelope' type of calculation here:

http://math.ucr.edu/home/baez/lengths.html

 

[uranium_235]

Ah, I get it. I was reading bits and pieces from different sources, now they seem to come together. Thank you.