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Gram Schmidt procedure, trouble finding inner product

  1. Sep 10, 2015 #1
    1. The problem statement, all variables and given/known data
    Given basis |x>,|y>,|z> such that <x|x> = 2,<y|y> = 2,<z|z> = 3,<x|y> = i, <x|z> = i, and <y|z> = 2. Build an orthonormal basis|x'>,|y'>,|z'>. Each of the new basis vectors should be expressed in terms of the old ones multiplied by coefficients.

    2. Relevant equations

    |x'> = |x>/(<x|x>).5 (normalizing the original x ket)

    |y'> = |y> - <x'|y> |x'>



    3. The attempt at a solution
    So far I have worked through that |x'> = (1/2).5|x>
    Using this result I have moved through the calculation of |y'> and found it equals

    |y> - (i/2) |x'>

    I am now trying to normalize this expression, but I am not sure what my inner product on the denominator will look like for this normalization, but my best guess is
    <y| - (i/2)<x'|y> - (i/2)|x'>

    but that does not look legitimate to me, so I am stuck here.

    How do I take the inner product of a bra and and ket, both of which are sums of two kets/bras?
     
    Last edited: Sep 10, 2015
  2. jcsd
  3. Sep 11, 2015 #2

    vela

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    Use parentheses. If you had a = b + c, you wouldn't say ##a\times a = b + c\times b + c##, would you? That's effectively what you're doing right now, so it's not surprising it doesn't look right to you.
     
  4. Sep 11, 2015 #3
    In the "bra"-"ket" formalisim the bra is the complex conjugate of the ket. In your normalizing denominator you must take the inner-product of the vector with its complex conjugate.
     
  5. Sep 11, 2015 #4
    Thanks a million! I always mix up how kets and bras correlate to normal vector notation. I worked it through being mindful of parentheses and I got a much more logical answer :biggrin:
     
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