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Linearity issue

  1. Nov 25, 2006 #1
    Previously i learnt from maths that the complex inner product satisfied skew symmetry and linearity in the first component, ie - <aA+bB,C> = a<A,C> +b<B,C>

    But after studying Shankar in quantum mechanics, he claims the linearity is in the ket vector, ie - <A,bB+cC> = b<A,B> +c<A,C> which would mean that the complex conjugate of the constants b,c appear in the wrong spots???
     
  2. jcsd
  3. Nov 25, 2006 #2
    This is a matter of convention. Mathematicians do it one way, but physicists do it another. It doesn't actually change anything, as long as you remain consistent.
     
  4. Nov 26, 2006 #3
    But with this alternate complex inner product you actually get the complex conjugate, so both definitions are not equivalent as with one you get the conjugate of the other.
     
  5. Nov 26, 2006 #4
    The definition of "bar" is
    integrating{(complex conjugate of[f(x)]) *g(x)} over x,
    when <f(x)| hit |g(x)>.

    So <aA| equal the complex conjugate of a multiplying <A|.

    The definition is used in J.Griffiths's 《Introduction to Quantum Mechanics》.
     
  6. Nov 26, 2006 #5

    George Jones

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    Yes, it's just convention, and amounts to a change in order of the vectors, i.e., <A,B> is for a physicist what <B,A> is for a mathematician.
     
  7. Nov 27, 2006 #6
    Griffiths's book presumes that we're working in a function space in the position representation. I could be working in a discrete space working in the coherent state representation. This is one of the many problems I have with that book: it tries to make general statements using very specific examples, and that leads to confusion.
     
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