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State space and its subspace : finding a basis

  1. Sep 12, 2011 #1
    Hello.

    I really need help with this one:

    1. The problem statement, all variables and given/known data

    I have a 3 dimensional state space H and its subspace H1 which is spanned with

    |Psi> = a x1 + b x2 + c x3
    and
    |Psi'> = d x1 + e x2 + f x3

    Those two "rays" are linearly independent and x1, x2, and x3 is an (orthonormal) basis for H.

    Now I need to find coefficients g, h and i so that
    |psi_othogonal> = g x1 + h x2 + i x3
    is not a trivial element of the subspace H1_orthogonal.

    2. Relevant equations

    not sure

    3. The attempt at a solution

    I think that I need to find another element in H which is orthogonal both to |Psi> and |Psi'>

    In R3 I would normally use cross product to find the third base vector but how does this transforms to the complex valued coefficients?

    I think it may be:

    g = (bf~ - ce~)
    h = (cd~ - af~)
    i = (ae~ - bd~)

    where f~ means complex conjugate of f.

    but is this right?

    Also how would I go about prooving this? What pops in my mind is to build scalar products and see if they give 0...

    Thanks alot

    /Nathan
     
  2. jcsd
  3. Sep 12, 2011 #2

    vela

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    Try calculating the scalar products and see what you get. If they're 0, you have your answer.
     
  4. Sep 14, 2011 #3
    Thanks for the replay.

    I tried to calculate the scalar product but it doesn't work; it all boils down to where I put conjugate. if I put conjugate on

    g = (~bf - ~ce)
    h = (~cd - ~af)
    i = (~ae - ~bd)

    then the scalar product of this vector and |psi> is zero BUT
    it is not zero with |psi'>...

    Or am I missing something?

    ~z w is generally not equal to z ~w ?

    any help is appreciated...
     
  5. Sep 14, 2011 #4

    vela

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    Try going the other way. Set the inner products to 0 and to get two equations involving g, h, and i, which you can solve up to a multiplicative constant.
     
  6. Sep 14, 2011 #5
    I did that before and I got expressions for g and h with i as a parameter. (1 equations and 3 unknowns - no surprise there) But I have no idea what i might be. If I were dealing with real numbers I would just set i = 1 and normalize the (g h i) vector, but these are complex numbers and I don't think I can do the same thing.

    anyway I got:

    g = i (b~ f~ - c~ e~) / (a~ e~ - b~ d~)
    and
    h = (-d~ g - f~ i) / e~

    what am I missing? either my complex algebra is too rusty or there is some fundamental physical relations I didn't think of.

    Please help me; this has been bugging me for a week now.
     
  7. Sep 14, 2011 #6

    vela

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    Why not? You don't actually have to set i to 1. You could just normalize the vector, which will allow you to solve for |i|. You still have the freedom to set an arbitrary complex phase, which you can take to be 0.
     
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