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Feynman ammonia molecule states

  1. Mar 4, 2012 #1
    I'm trying to follow Feynman's explanation on page 9-3 of Volume 3 of The Feynman Lectures on Physics. I've attached a copy of the section in question.

    To normalize CII he notes that
    < II | II > = < II | 1 >< 1 | II > + < II | 2 >< 2 | II > = 1
    I am not clear how he derives the conclusion
    CII = 1/√2 (C1 + C2)

    I tried to solve the first equation unsuccessfully like this-
    < II | 1 >< 1 | II > + < II | 2 >< 2 | II > = 1
    CII C1 + CII C2 = 1
    [1/√2 (C1 + C2)] C1 + {1/√2 (C1 + C2)} C2 = 1
    [1/√2 (C1 + C2)] (C1 + C2) = 1
    1/√2 (C1 + C2) (C1 + C2) = 1
    which isn’t working for me.

    Thank you for clarifying.
     

    Attached Files:

  2. jcsd
  3. Mar 4, 2012 #2
    The step from the first to the second line of your attempt is wrong. You've lost half of the matrix elements.
     
  4. Mar 5, 2012 #3
    Thank you for your response, M Quack, which is clear and helpful.

    If CII = < II | Φ >, I’m assuming I can substitute it for both < II | 1 > and < II | 2 >. You can see I’ve taken the same approach with C1 and C2. I suspect to go further I may have to use C1 = C2 and possibly C1*, which I am frankly weak on, and < 2 | 1 > = < 1 | 2 > = 1 / 2.

    The source of my confusion may be a basic misunderstanding of state transitions and the meaning of the coefficients, C1, C2, and CII. I am really struggling to move forward. Thank you.
     
  5. Mar 5, 2012 #4
    <ii | 1> <1 | ii> = (cii^* c1) (c1^* cii)

    = (1/√2 (c1+c2)^* c1) (c1^* 1/√2 (c1+c2)
    = 1/2 (c1^* c1 + c2^*c1) (c1^*c1+c1^*c2)
    = 1/2 (1 + 0) (1 + 0)

    Do you see what I mean when I say you dropped half the matrix elements?

    (Please don't ask me why preview insists on making everything lowercase)
     
  6. Mar 6, 2012 #5
    That’s really helpful. The fog begins to lift. Thanks again.

    I now realize the difference between < II | Φ >, which is CII, and < II | 1 >, which is the product of | 1 > and < II |.
    Also, < II | = | II >* = CII*.

    No, I can’t say I see what you mean about the matrix elements (although implying I was able to identify half of them seems generous).
     
  7. Mar 6, 2012 #6
    <II|i> is nothing else than a matrix element between the old and new wave functions. Maybe that is speaking a bit loosely as usually a matrix element is <x|M|x> where |x> is the wave function or state, and M some operator or matrix, depending on the notation.
     
  8. Mar 6, 2012 #7
    I appreciate your help.
     
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