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Outer product on operators?

  1. Nov 28, 2012 #1
    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. jcsd
  3. Nov 29, 2012 #2
    [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.
     
  4. Nov 29, 2012 #3
    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 HA and HB), then you can create a new Hilbert space by the direct product of the two of them, H = HA[itex]\otimes[/itex]HB (the vectors of that new space are defined in this way:say ΨΑ[itex]\in[/itex]HA and ΨΒ[itex]\in[/itex]HΒ, then the vectors Ψ=ΨA[itex]\otimes[/itex]ΨB for all ΨA and ΨB are the vectors of H. ΨA[itex]\otimes[/itex]ΨB is a new item that has two independent parts, ΨA and ΨB , 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) (ΨA[itex]\otimes[/itex]ΨB) = (AΨA)[itex]\otimes([/itex]BΨB).
    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' = Aaa'Bbb'
     
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