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Operator in non-orthogonal basis

  1. Oct 4, 2007 #1
    Hi, is possible make up a operator in a non-orthogonal basis, if is possible how I can contruct the operator.

  2. jcsd
  3. Oct 4, 2007 #2
    why not form your operators as |b><a|
  4. Oct 5, 2007 #3
    which are the consequence of choice a basis non-orthogonal?
  5. Oct 5, 2007 #4


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    Why do you want to form an operator in a non-orthogonal basis in the first place?
  6. Oct 5, 2007 #5


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    Of course.

    All you need to know is the effect of the operator on all the basis states. So if you know all the values of [itex] <a_i|A|a_j>[/itex] then you know everything about the operator.

    Alternatively, as quetzalcoatl9 pointed out, an arbitrary operators can be written as

    [itex] A = \sum c_{ij} |a_i><a_j| [/itex]

    One consequence of having a non orthonogonal basis is that you can't read off directly from the above expression what is the effect of applying the operator to a basis state gives.

    If the basis is orthogonal, then applying A to, say, [itex] |a_3> [/itex] will simply give [itex] c_{13} |a_1> + c_{23} |a_2> + \ldots [/itex] (I am assuming that the labels of the states are discrete and start at 1). If the basis is not orthogonal, the expression is of course more complicated.
  7. Oct 5, 2007 #6
    I can construct a basis depent of basis non-orthogonal, how might make up? and what happen with the eigenvalues and elements of the operator.

    someone know if the situation present in some quantum system.
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