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A Help with proof of eq. 2.64 of Intro. to Quantum Mechanics

  1. May 16, 2017 #1
    I am self studying the Book- Introduction to Quantum Mechanics , 2e. Griffith. Page 47.
    While the book has given a proof for eq. 2.64 but its not very ellaborate
    Integral(infinity,-infinity) [f*(a±g(x)).dx] = Integral(infinity,-infinity) [(a±f)* g(x).dx] . It would be great help if somebody could provide me a more step by step proof of the same.
    Where a+ and a- are roots of Hamiltonian of harmonic oscillator problem.
  2. jcsd
  3. May 16, 2017 #2
    Are a+ and a- real? If so it would be justified.
  4. May 17, 2017 #3
    The operator

    $${d \over dx}$$ picks up a minus sign under hermitian conjugation when the hilbert space is that of functions that vanish fast enough at infinity. The reason why that's true is because:

    $$\int f {d g \over dx} = 0 - \int g {df \over dx} dx$$

    This implies that when you take the hermitian conjugate of the $\hat{a}_+$ operator, just replace ${d \over dx}$ by $-{d \over dx}$ which gives the $\hat{a}_-$ operator
  5. May 17, 2017 #4


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    Be careful there: they are not! There is a constant ##\hbar\omega## difference !
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