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Lindsayyyy
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Hi everyone
I have to particles without a potential. The coordinates are r_1 and r_2 (for particle 1 and 2). Both have orthonormal states |↑> and |↓>. I shall show that the expectation value is the following, where as |↑↓> is a product state
[tex] d^2=\langle \uparrow \downarrow \mid (r_1-r_2)^2 \mid \uparrow \downarrow \rangle = \langle \uparrow \mid r^2 \mid \uparrow\rangle +\langle \downarrow \mid r^2 \mid \downarrow \rangle -2 \langle \uparrow \mid \vec r \mid \uparrow \rangle \langle \downarrow \mid \vec r \mid \downarrow \rangle[/tex]
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Well, my attempt so far isn't very good I think because I have many problems understanding this.
I think I can write my r_1 as:
[tex] \vec r_1 = \frac {1}{\sqrt 2} (\mid \uparrow \rangle + \mid \downarrow \rangle) [/tex]
and the 2nd one as
[tex] \vec r_2 = \frac {1}{\sqrt 2} (\mid \uparrow \rangle + \mid \downarrow \rangle) [/tex]
I can now to the tensor product, but that doesn't lead to anywhere ( I tried to use it to get my up down product state, but this term looks so complicated I can't use it to ease up my euqations)
Thanks for your help
Homework Statement
I have to particles without a potential. The coordinates are r_1 and r_2 (for particle 1 and 2). Both have orthonormal states |↑> and |↓>. I shall show that the expectation value is the following, where as |↑↓> is a product state
[tex] d^2=\langle \uparrow \downarrow \mid (r_1-r_2)^2 \mid \uparrow \downarrow \rangle = \langle \uparrow \mid r^2 \mid \uparrow\rangle +\langle \downarrow \mid r^2 \mid \downarrow \rangle -2 \langle \uparrow \mid \vec r \mid \uparrow \rangle \langle \downarrow \mid \vec r \mid \downarrow \rangle[/tex]
Homework Equations
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The Attempt at a Solution
Well, my attempt so far isn't very good I think because I have many problems understanding this.
I think I can write my r_1 as:
[tex] \vec r_1 = \frac {1}{\sqrt 2} (\mid \uparrow \rangle + \mid \downarrow \rangle) [/tex]
and the 2nd one as
[tex] \vec r_2 = \frac {1}{\sqrt 2} (\mid \uparrow \rangle + \mid \downarrow \rangle) [/tex]
I can now to the tensor product, but that doesn't lead to anywhere ( I tried to use it to get my up down product state, but this term looks so complicated I can't use it to ease up my euqations)
Thanks for your help