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The dot or cross product of two operators acting on a state

  1. May 21, 2014 #1
    If a system is made up by two subsystems, for example, the atom and the photon. and let's assume the state of the atoms is described by [itex]|\phi\rangle[/itex], while the state of the photons can be described by [itex]|n\rangle[/itex], The Kronecker product of the [itex]|\phi\rangle[/itex] and [itex]|n\rangle[/itex] can be used to describe the whole system. and that would be:

    [itex]|\Psi\rangle=|\phi, n\rangle=|\phi\rangle\otimes|n\rangle[/itex]

    I always treat the [itex]|\phi\rangle[/itex] and [itex]|n\rangle[/itex] as vectors, so the operation of [itex]\otimes[/itex] means the elements of the first vector (here [itex]|\phi\rangle[/itex]) times the "whole" following vector which is [itex]|n\rangle[/itex] here; that will gives us a vector which is [itex]|\phi, n\rangle[/itex]. so if the numbers of the elements of [itex]|\phi\rangle[/itex] and [itex]|n\rangle[/itex] is [itex]m[/itex] and [itex]n[/itex] respectively, the vector [itex]|\phi, n\rangle[/itex] has [itex]m\times n[/itex] elements.

    now for example, we have two operators, [itex]\hat{\mathbf{A}}[/itex] and [itex]\hat{\mathbf{N}}[/itex], and they satisfy the following equations:
    [itex]\hat{\mathbf{A}}|\phi\rangle=\mathbf{a}|\phi'\rangle[/itex]
    [itex]\hat{\mathbf{N}}|n\rangle=\mathbf{n}|n'\rangle[/itex].

    Of course, [itex]\hat{\mathbf{A}}[/itex] can only act on the atomic states, and [itex]\hat{\mathbf{N}}[/itex] can only act on the photons states.

    Now, my question, what is [itex]\hat{\mathbf{A}}\cdot \hat{\mathbf{N}}|\phi, n\rangle[/itex], and what is [itex]\hat{\mathbf{A}}\times \hat{\mathbf{N}}|\phi, n\rangle[/itex]? The idea just not clear to me, if the operation [itex]\otimes[/itex] is involved.
    if [itex]\hat{U}=\hat{\mathbf{A}}\cdot \hat{\mathbf{N}}[/itex], for example, how to write [itex]\langle \phi, n|\;|U|^2\; |\phi', n'\rangle[/itex] on the base of [itex]|\phi\rangle[/itex] and [itex]|n\rangle[/itex]?
     
  2. jcsd
  3. May 21, 2014 #2

    strangerep

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    Science Advisor

    You need the tensor product of operators. (See the section "Tensor product of linear maps".)
     
  4. May 21, 2014 #3
    are you talking about a section of a book?
     
  5. May 22, 2014 #4
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