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Adjoint operator in bra-ket notation

  1. Nov 24, 2015 #1

    First of all I want apologize for my bad english!
    Second, I'm doing a physical chemystry course about the main concepts of quantum mechanics !!

    The Professor has given to me this definition of "the adjoint operator":

    <φ|Aψ> = <Aφ|ψ>

    My purpose is to verificate this equivalence so i gave some numeric values at <φ|, ψ> and at the matrix A (rappresentative of an operator).
    Then i calculate the expression <φ|Aψ> multiplicating,at first ,the product |Aψ> = A|ψ> and then doing the scalar product <φ|Aψ>....The bra and ket are 1x3 and 3x1 matrix respectively ,while A is 3x3.

    Now my problem is to calculate the "other" expression: <Aφ|ψ>

    Because this expression says to calculate first <Aφ|.
    BUT i dont' know how to calculate this because according to the linear algebra i can't do the product between A and <φ| (i.e. <Aφ|=A<φ|. In fact it would be a product between a 3x3 matrix and a 1x3 vector...I'm not able to do this but only the product 1x3 | 3X3 at most....

    So how can i calculate the expression <Aφ|ψ> using the linear algebra?? I have to shift the matrix in order to do that product (where?)or what??

    Thanks very much!! :)
  2. jcsd
  3. Nov 24, 2015 #2


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    A bra notation ##\langle \ldots |## is just the instruction to calculate the conjugate transpose of a vector it confines. To express something like ##\langle A^\dagger \phi |## in matrix notation, just first compute the matrix multiplication of the quantity inside the bra, and then take its conjugate transpose.
  4. Nov 24, 2015 #3

    I don't understand why i have to take its conjugate transpose. after having calculated Aφ|
    To calculate <Aφ| ,assumed that i know the numerical values of the BRA <φ| and the matrix A what do i have to do?:

    1) A<φ|


    2) <φ|A

    Because i think that the relative order is important (the matric product usually is not commutative)...and the first expression according to linear algebra is not allowed ( matrix 3x3 * bra 1x3)

    Thanks :)
  5. Nov 24, 2015 #4
    ##\langle \psi | A## means in traditional matrix notation ##\psi ^\dagger A##. Similarly, if you try to put the A inside the bra like ##\langle A^\dagger \psi |##, you'll have ##\left({A^\dagger}\psi \right)^\dagger##, which is the same thing.
  6. Nov 25, 2015 #5


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    Lorran has explained the answer for me.
  7. Nov 26, 2015 #6
    Thanks :)

    Only one last thing please:

    The components of the vector ψ in the traditional matrix expression (Aψ) (corresponding to the bra <Aψ| ) are the same components of the ket vector |ψ> ??

  8. Nov 26, 2015 #7


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    Yes, they are the same.
  9. Nov 26, 2015 #8
    Thanks :) !!!
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