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Math Help - Help with a problem from Atkins Molecular Quantum Chemistry

  1. #1
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    Question Help with a problem from Atkins Molecular Quantum Chemistry

    Hi all,

    Problem
    ------------------------------------------------------------------------------------------------------

    Show that if   ( /Omega f )* = - /Omega f* , then < /Omega > = 0 for any real function  f

    ------------------------------------------------------------------------------------------------------

    This is probably disgustingly simple, but it has me stumped

    Sorry for the presentation, can anyone show me how to use TeX equations in my post? I am new to the forums.
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  2. #2
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    Quote Originally Posted by MasterAir View Post
    Hi all,

    Problem
    ------------------------------------------------------------------------------------------------------

    Show that if   ( /Omega f )* = - /Omega f* , then < /Omega > = 0 for any real function  f

    ------------------------------------------------------------------------------------------------------

    This is probably disgustingly simple, but it has me stumped

    Sorry for the presentation, can anyone show me how to use TeX equations in my post? I am new to the forums.
    Could you define your terms? I am assuming the < > are the standard expectation value brackets, but what are \Omega and f?

    See there first few posts in this thread for LaTeX.

    -Dan
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    Could you define your terms? I am assuming the < > are the standard expectation value brackets, but what are and f?

    See there first few posts in this thread for LaTeX.

    -Dan
    Omega is an operator, (I think it is Hermitian, but I am unsure.) f is any real function, <\Omega> Is the expectation value for the operator Omega.

    In words - if I have understood correctly - if the complex conjugate of the result of an (hermitian) operator acting on a function is equal to minus the effect of the same (hermitian) operator acting on the complex conjugate of the same real function, then the expectation value of the (hermitian) operator is zero for any real function f.
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by MasterAir View Post
    Omega is an operator, (I think it is Hermitian, but I am unsure.) f is any real function, <\Omega> Is the expectation value for the operator Omega.

    In words - if I have understood correctly - if the complex conjugate of the result of an (hermitian) operator acting on a function is equal to minus the effect of the same (hermitian) operator acting on the complex conjugate of the same real function, then the expectation value of the (hermitian) operator is zero for any real function f.
    Okay, so the complex conjugate of \Omega f is equal to -\Omega(f^*)?

    I'm still confused on one point. According to the definitions I know (Quantum Mechanics in general, not specifically Molecular Quantum Chemistry)
    \left < \Omega \right > = \int_V \psi ^* \Omega \psi \ d^3x
    where \psi (x, y, z) is your wavefunction.

    Where is f coming into this?

    -Dan
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  5. #5
    Forum Admin topsquark's Avatar
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    Okay, let me try this and you can correct me if I'm wrong on any point. (This would be easier if we were to use "bra-ket" notation. If you'd like me to rewrite it, just let me know.)

    \Omega is a Hermitian operator, so
    \left < \Omega \right > = \int_V \psi ^* \Omega \psi \ d^3x is a real number, where \psi is a wavefunction.

    So we know that
    \left < \Omega \right >^* = \left < \Omega \right >

    From here on I am taking \psi = f as a real valued function. And we know that ( \Omega f)^* = - \Omega (f ^*) = - \Omega f. (Given.)

    Now
    \left < \Omega \right > ^* = \left ( \int_V f ^* \Omega f \ d^3x \right ) ^* = \left ( \int_V f \Omega f \ d^3x \right ) ^*<-- Since f is a real valued function
    = \int_V f^*  ( \Omega f )^* \ d^3x = \int_V f^*  (- \Omega f ) \ d^3x = - \int_V f^* \Omega f \ d^3x  = - \left < \Omega \right >

    But \left < \Omega \right >^* = \left < \Omega \right > since the expectation value is real.

    Thus
    \left < \Omega \right > = - \left < \Omega \right >

    Thus
    \left < \Omega \right > = 0

    -Dan
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  6. #6
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    Thumbs up

    Yes, I think it would be easier in bra-ket notation. However, thanks very much, I follow your answer.

    I think the main reason I was stumped is that I failed to realise that  f=f* for a real function. Some, sort of mental block, I don't quite know.

    Thanks again for your help.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by MasterAir View Post
    Yes, I think it would be easier in bra-ket notation. However, thanks very much, I follow your answer.

    I think the main reason I was stumped is that I failed to realise that  f=f* for a real function. Some, sort of mental block, I don't quite know.

    Thanks again for your help.
    No problem. It was odd for me as well. I don't recall if I have ever done this assuming a real wavefunction. (Which may mean that I did the proof in a longer way than necessary. I tend to do that. )

    -Dan
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