How Does the Outer Product Operate on Quantum Mechanical Operators?

[dipole]

In my QM textbook, there's an equation written as:

[itex] \vec{J} = \vec{L}\otimes\vec{1} + \vec{S}\otimes\vec{1} [/itex]

referring to angular momentum operators (where [itex]\vec{1} [/itex] is the identity operator). I don't really understand what the outer product (which I'm assuming is what the symbol [itex]\otimes[/itex] means here) means when dealing with operators (which can be represented as matrices).

What happens when you outerproduct one operator with another? Unfortunately there is no explanation in the text, I guess it's assumed this is obvious or that the reader knows about this kind of math. :\

 

[Fightfish]

[tex]\otimes[/tex] is not outer product. It is a tensor product.
Could you provide the context?
I am guessing that this means that you act the angular momentum operator only on the first particle but leave the second particle untouched.

 

[cosmic dust]

First of all, I think that the formula should be J = L[itex]\otimes[/itex]1 + 1[itex]\otimes[/itex]S . About it's meaning, when you have two operators (say A and B) which operate on two, in general different, Hilbert spaces (say H_A and H_B), then you can create a new Hilbert space by the direct product of the two of them, H = H_A[itex]\otimes[/itex]H_B (the vectors of that new space are defined in this way:say Ψ_Α[itex]\in[/itex]H_A and Ψ_Β[itex]\in[/itex]H_Β, then the vectors Ψ=Ψ_A[itex]\otimes[/itex]Ψ_B for all Ψ_A and Ψ_B are the vectors of H. Ψ_A[itex]\otimes[/itex]Ψ_B is a new item that has two independent parts, Ψ_A and Ψ_B , pretty much like when you have two reals a and b, you can create a new item (a,b) which represents a point in a plane) . The operators on this new Hilbert space are then created by the direct product of the operators that operate in the two initial spaces, i.e. O = A[itex]\otimes[/itex]B , where this new operator is defined by:
O Ψ [itex]\equiv[/itex](A[itex]\otimes[/itex]B) (Ψ_A[itex]\otimes[/itex]Ψ_B) = (AΨ_A)[itex]\otimes([/itex]BΨ_B).
When the operators are represented by matrices, then the matrix A[itex]\otimes[/itex]B is defined as:
[A[itex]\otimes[/itex]B]_aa',bb' = A_aa'B_bb'