Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:

    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


    User Avatar
    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook