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Outer product on operators? 
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#1
Nov2812, 10:46 PM

P: 440

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. :\ 


#2
Nov2912, 01:40 AM

P: 623

[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. 


#3
Nov2912, 04:22 AM

P: 123

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'} 


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