Why does the tensor product in QM produce unentangled states?

[meBigGuy]

In QM the tensor product of two independent electron's spin state vectors represents the product state which represents the possible unentangled states of the pair. I don't understand why the tensor product produces that result. |A⟩=|a⟩⊗|b⟩

 

[meBigGuy]

Hmmm. I must be asking a really dumb or off-forum question. Can anyone give me classical examples of tensor products and what the inputs and output represent physically. Should I ask this in a different forum?

 

[Office_Shredder]

You're asking about quantum mechanics which requires the reader to know what a spin state vector is and what an entanglement is, so I would venture you'll have far more success in the quantum physics forum here

https://www.physicsforums.com/forumdisplay.php?f=62

 

[meBigGuy]

I'm trying, but it still doesn't feel right and I'm sure my phrasing is all wrong.

The "broadest" vector state product is the Cartesian product, and it includes the "new product" terms that one might try to associate with classical multiplying. The tensor product somehow eliminates those terms.

But classical multiplying is all about mixing, and its application expresses an interaction between elements. The tensor product provides a type of vector stae multiplication that works such that associative and communicative laws hold, but so does classical multiplication in its domain. The subtleties of what the tensor product operation actually represents and the nature of what is being excluded escapes me.

Saying that it represents all the possible states of two fully independent electron's spin vectors is well and fine, and I see the value of it in subsequent manipulations, but I don't get the why of it.

 

[naima]

One of the benefits of using tensor product is that
(a Phi1,b phi2) and (ph1, phi2) are mapped to states that are collinear viz that have
the same physical content

 

[meBigGuy]

I have no idea wht you just said. Is there no one here that can give me the people's magazine view of tensor product. That is, explain it to a dumb guy.

 

[stevendaryl]

I have no idea wht you just said. Is there no one here that can give me the people's magazine view of tensor product. That is, explain it to a dumb guy.

The idea behind the tensor product, as applied to quantum mechanics, is pretty simple: If you have one particle in state [itex]|\psi\rangle[/itex] and another particle in state [itex]|\varphi\rangle[/itex], then the composite system, made up of those two particles, is in the state [itex]|\psi\rangle \otimes |\varphi\rangle[/itex]. The technical details of what makes it a tensor product are how the operation [itex]\otimes[/itex] works on superpositions.

If [itex]|\psi\rangle = a |\psi_1\rangle + b |\psi_2\rangle[/itex] and [itex]|\varphi\rangle = c |\varphi_1\rangle + d |\varphi_2\rangle[/itex], where [itex]a, b, c, [/itex] and [itex]d[/itex] are complex numbers, then

[itex]|\psi\rangle \otimes |\varphi\rangle
= a c (|\psi_1\rangle \otimes |\varphi_1\rangle)
+ a d (|\psi_1\rangle \otimes |\varphi_2\rangle)
+ b c (|\psi_2\rangle \otimes |\varphi_1\rangle)
+ b d (|\psi_2\rangle \otimes |\varphi_2\rangle[/itex])​

I don't think that there is anything else you really need to know about tensor products. Are you wondering why the tensor product is used for composite systems?

 

[meBigGuy]

Are you wondering why the tensor product is used for composite systems?

Maybe, but not why in the simple sense that it is communicative and associative and therefore usefull, but why at a higher level. Why it turns out to represents the unentangled system states. Is it like "we want an operation that can work in this equation and we will call it a tensor product"

I think I read that the tensor product is the cartesian product with certain terms removed so it becomes communicative and associative, but that is still kind of abstract.

Is there a classical system analogue to tensor product? What are other physical systems where it is useful? I didn't find any in Arfken.