Please post comments and questions here.
Andrew Hudson's question:
"You said in the video that delta is proportional to the inverse of the Coulomb force, and it makes sense to have some sort of correction factor for the wave spilling over into the other well for the Psi plus state, but I'm still kind of unclear as to what delta is representing here."
What delta represents is, as you say, the ability of the electron wave to spill over into the other well. To get a perspective on delta, what it means and represents here, let's go back to the case where there is no coulomb force (between electrons). In that case delta is equal to 1. That will lead to the familiar \(\frac{1}{\sqrt{2}}\) factor and a state that has equal weight in either well. Does that make sense?
In the context of this video, we are in a very different regime, where coulomb repulsion is fairly strong and delta is about .3 to .001, roughly speaking. When delta is .3, then about 10% of the probability density is associated the secondary well, and about 90% with the primary well, so mostly the state has the electron in a particular well, but it allows some freedom to extend into the secondary well (for that state). What delta represents is the freedom for the electron to not be completely constrained to be only in one well. It is a relaxing of constraint toward the more general state, e.g., \(\psi_{A'} = \frac{1}{\sqrt{1+\delta^2}}\psi_A + \frac{\delta}{\sqrt{1+\delta^2}}\psi_B \). Does that make sense?
You said in the video that delta is proportional to the inverse of the Coulomb force, and it makes sense to have some sort of correction factor for the wave spilling over into the other well for the Psi plus state, but I'm still kind of unclear as to what delta is representing here.
ReplyDeleteYes, that is a really good question. I will edit my thoughts on that into the post above.
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ReplyDeleteOn the left corner part of the video when you write out the even spatial state with the odd spin state or vice versa, is there technically also a 1/root(2) factor for each spatial state as well, so you could also just write out each total state multiplied by a (1/2)?
ReplyDeleteYes. Exactly. If you were to write out both states, then those factors could combine.
DeleteI could not quite understand why the up up and down down spin state could pair with the odd state? Also why can you have 2 up state into one I thought pauli exclusion principle only permits one of each spin state?
ReplyDeleteAt about 30 minutes in, I'm confused with the distribution of terms for the psi minus state. Since the (\delta^2)(\psi_A)(\psi_B) in the second term is being subtracted from the (\psi_A)(\psi_B) in the first term, wouldn't that group as (1-\delta^2)(\psi_A \psi_B) instead of (1+\delta^2)(\psi_A \psi_B)? Thanks in advance.
ReplyDeleteyes, you are correct. That is a mistake on my part. It should have been 1-delta^2 there. (and in the normalization). Excellent point. Who are you?
DeletePatrick Nast. Does this mean that the psi plus and their constituent single electron states' normalizations are also 1-delta^2?
Delete@ Patrick. yes.
DeleteThis video was very helpful! Thanks.
ReplyDeleteyou're welcome
Deleteso if you had a spin state for the psi-minus state that was, say, up-up or down-down instead of the up-down plus down-up, would you still have the normalization factor of 1 over root 2 in front? basically what i'm asking is if the normalization factor of 1 over root 2 correspond to there being two separate spin states.
ReplyDeleteyes. exactly. For example, there is no factor for the up-up state.
DeleteA follow up to that question is perhaps a more fundamental question... why is it necessary to have a psi-plus and psi-minus state? Besides the mathematical justification of linear combinations, is it to satisfy the symmetry that is necessary for the spatial states to exist in the first place?
ReplyDeletea Fermion state has to be anti-symmetric... with respect to electron exchange. It has to change sign. Pauli's rule.
DeleteIf I'm not mistaken {(UP UP), (DOWN DOWN), (UP DOWN + DOWN UP), (UP DOWN - DOWN U)} is a basis for the set of possible spins. Am I right?
ReplyDeleteWe could have, for instance, defined a basis (UP UP) (DOWN DOWN) (UP DOWN) (DOWN UP)? Except no! Because (UP DOWN) = - (DOWN UP) doesn't it? Or does it? Is what I'm writing just a bunch of hoopla?
Thanks for the film, professor, and thanks in advance to anyone who answers these questions!
1st. True. That is a basis.
Delete2nd paragraph: true, that is a basis,
but then: false. (up down) does not equal -(down up). That's crazy talk.
and if that were true then those 2 would not be linearly independent, which they are.
DeleteI had a feeling something was strange! Thanks for clearing that up.
DeleteThe video cleared up the questions I had regarding spin states. Thanks for posting
ReplyDelete