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Homework Help: Find a ket orthogonal to a given ket

  1. Sep 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Given a state [tex]\mid \psi \rangle=\frac{1}{\sqrt{3}}[(i+1)\mid 1 \rangle + \mid 2 \rangle][/tex], find the normalized state [tex]\mid \psi^{'} \rangle[/tex] orthogonal to to it.

    2. Relevant equations

    [tex]\langle \psi^{'} \mid \psi \rangle = 0[/tex]

    [tex]\langle \psi^{'} \mid \psi^{'} \rangle = 1[/tex]

    3. The attempt at a solution

    I seek a state such that [tex]\langle \psi^{'} \mid \psi \rangle = 0[/tex] so I am looking for a vector such that [tex]A\frac{1+i}{\sqrt{3}} + B\frac{1}{\sqrt{3}}=0[/tex]

    Solving for B...


    Desiring a normalized vector, I let [tex]A=\frac{1}{\sqrt{2}}[/tex], and therefore [tex]B=\frac{-1-i}{\sqrt{2}}[/tex].

    Since this is the vector that is dotted with [tex]\mid \psi \rangle[/tex], I find the ket as the transposed conjugate of this vector, hence: [tex]\mid \psi^{'} \rangle = 1 \mid 1 \rangle + (-1 + i) \mid 2 \rangle[/tex].

    Now, [tex]\langle \psi^{'} \mid \psi \rangle = \frac{1}{\sqrt{3}}(1+i-1-i) =0[/tex] OK

    But, [tex]\langle \psi^{'} \mid \psi^{'} \rangle = 1+(-1-i)(-1+i)=1-i^2=2[/tex] NO

    So I found a vector that is orthogonal to [tex]\mid \psi \rangle[/tex] but it is not normalized.

    I think one of the following is happening:

    1) I need to re-normalized the vector I found.

    2) I am approaching this problem in the wrong way entirely. I did try to apply a rotation operator to the given vector (rotating by [tex]\frac{\pi}{2}[/tex] but this gave me nothing better.

    How should I proceed?
    Last edited: Sep 11, 2009
  2. jcsd
  3. Sep 11, 2009 #2


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    What makes you think that [itex]A=\frac{1}{\sqrt{2}}[/itex] produces a normalized vector?

    You have found [itex]\langle\psi'|=A\langle 1|+B\langle2|=A\langle 1|-(1+i)A\langle2|[/itex]...To normalize the state, take the norm of that Bra, set it equal to one and solve for [itex]A[/itex].
  4. Sep 11, 2009 #3


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    This is easier than most problems of this kind because the dimensionality is 2. Prove first then use the following shortcut

    Given a normalized state

    [tex]|\psi> = a|1> + b|2>[/tex]

    show that

    ** Edit **

    [tex]<\psi '| = -b<1| + a<2|[/tex]

    is othonormal to it.
    Last edited: Sep 11, 2009
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