How to Calculate Expectation Value of Product State in a Potential-Free System?

[Lindsayyyy]

Hi everyone

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

$$d^2=\langle \uparrow \downarrow \mid (r_1-r_2)^2 \mid \uparrow \downarrow \rangle = \langle \uparrow \mid r^2 \mid \uparrow\rangle +\langle \downarrow \mid r^2 \mid \downarrow \rangle -2 \langle \uparrow \mid \vec r \mid \uparrow \rangle \langle \downarrow \mid \vec r \mid \downarrow \rangle$$

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:

$$\vec r_1 = \frac {1}{\sqrt 2} (\mid \uparrow \rangle + \mid \downarrow \rangle)$$
and the 2nd one as

$$\vec r_2 = \frac {1}{\sqrt 2} (\mid \uparrow \rangle + \mid \downarrow \rangle)$$

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

 

[Fightfish]

Basically, the tensor product is simply a product of two separate subspaces. So,
$\mid \uparrow \downarrow \rangle$ can be more explicitly written as $\mid \uparrow \rangle_{1}\mid\downarrow \rangle_{2}$

Furthermore, $r_{1}$ and $r_{2}$ are really $r \otimes I$ and $I \otimes r$ respectively.

You just have to expand $(r_1 - r_2)^2$ and then "act them" on the appropriate subspaces.

 

[Lindsayyyy]

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?

 

[Fightfish]

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?

My best guess is what I posted earlier: $r_{1} = r \otimes I$ and $r_{2} = I \otimes r$ where I is identity. The subscripts 1 and 2 refer to the particle number.

Let me work out the trickier part explicitly: $r_{1}r_{2} = r \otimes r$
Lets act it on the state:
$$(\langle \uparrow \mid \otimes \langle \downarrow \mid)(r \otimes r)(\mid \uparrow \rangle \otimes \mid \downarrow \rangle)$$
Now, operations on each subspace are independent of each other ie.
$$(A \otimes B) (C \otimes D) = AC \otimes BD$$
So, the previous expression simplifies to
$$(\langle \uparrow \mid r \mid \uparrow \rangle) \otimes (\langle \downarrow \mid r \mid \downarrow \rangle)$$
But these are just c-numbers. So the tensor product becomes a normal product and we arrive at
$$\langle \uparrow \mid r \mid \uparrow \rangle\langle \downarrow \mid r \mid \downarrow \rangle$$

 

[Lindsayyyy]

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.