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Problem with product states

  1. Dec 7, 2012 #1
    Hi everyone

    1. The problem statement, all variables and given/known data

    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]



    2. Relevant equations

    -

    3. 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
     
  2. jcsd
  3. Dec 7, 2012 #2
    Basically, the tensor product is simply a product of two separate subspaces. So,
    [itex]\mid \uparrow \downarrow \rangle[/itex] can be more explicitly written as [itex]\mid \uparrow \rangle_{1}\mid\downarrow \rangle_{2}[/itex]

    Furthermore, [itex]r_{1}[/itex] and [itex]r_{2}[/itex] are really [itex]r \otimes I[/itex] and [itex]I \otimes r[/itex] respectively.

    You just have to expand [itex](r_1 - r_2)^2[/itex] and then "act them" on the appropriate subspaces.
     
  4. Dec 7, 2012 #3
    thanks for your help so far.

    yeah I know that (r1-r2)^2 is on of the binomial theorems. But I don't know actually how, let's say (r_1)^2 acts on ∣↑↓⟩. That's where I'm stuck.

    edit: actually, I don't know what r even is (without the index). I thought it might have been a typing mistake by the task given, but you posted it aswell. Or did they just leave out the indices?
     
  5. Dec 8, 2012 #4
    My best guess is what I posted earlier: [itex]r_{1} = r \otimes I[/itex] and [itex]r_{2} = I \otimes r[/itex] where I is identity. The subscripts 1 and 2 refer to the particle number.

    Let me work out the trickier part explicitly: [itex]r_{1}r_{2} = r \otimes r[/itex]
    Lets act it on the state:
    [tex](\langle \uparrow \mid \otimes \langle \downarrow \mid)(r \otimes r)(\mid \uparrow \rangle \otimes \mid \downarrow \rangle)[/tex]
    Now, operations on each subspace are independent of each other ie.
    [tex](A \otimes B) (C \otimes D) = AC \otimes BD[/tex]
    So, the previous expression simplifies to
    [tex](\langle \uparrow \mid r \mid \uparrow \rangle) \otimes (\langle \downarrow \mid r \mid \downarrow \rangle)[/tex]
    But these are just c-numbers. So the tensor product becomes a normal product and we arrive at
    [tex]\langle \uparrow \mid r \mid \uparrow \rangle\langle \downarrow \mid r \mid \downarrow \rangle[/tex]
     
  6. Dec 9, 2012 #5
    thanks for your help. I'm back home tomorrow then I will try to understand it a bit better. If I have problems again I will post here.
     
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