- #1
Suske
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Homework Statement
L.S.,
I'm breaking my head over this problem! To anyone who can help me out: thanks a lot!
Consider a two-point grid (point a en point b) and two electrons occupying those points, (they can both occupy one point at the same time), both wit spin-up or spin-down. Now, they state that you write the state of the spin-electrons as a product of the state of spin and the state of place: X * Psi (I don't understand why you'd do this and what this means, for starters..)
Now the problem is to show that we talk about 6-dimensional Hilbert space with basis:
[itex]\frac{1}{\sqrt{2}}[/itex] Xup, up*(Psia, b - Psib, a)
[itex]\frac{1}{2}[/itex](Xup, down + Xdown, up)*(Psia, b - Psib, a)
[itex]\frac{1}{\sqrt{2}}[/itex](Xdown, down)*(Psia, b - Psib, a)
(spin-triplet)
and
[itex]\frac{1}{2\sqrt{2}}[/itex](Xup, down - Xdown, up)*(Psia, a)
[itex]\frac{1}{2}[/itex](Xup, down - Xdown, up)*(Psia, b + Psib, a)
[itex]\frac{1}{2\sqrt{2}}[/itex](Xup, down - Xdown, up)*(Psib,b)
The Attempt at a Solution
I understand it is 6-dimensional, (23 - the two combinations that cannot exist because of the Pauli principal)
Further, I have no clue how to derive this basis!
Would someone please help me with this problem? I thought I understood this chapter well, but now my brains are vaporizing..
Thanks!
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