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Thread: Hermiticity Question

  1. #1
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    Hermiticity Question

    Please help, I have no idea!

    Consider the set of functions {f(x)} of the real variable x defined on the interval −inf < x < inf
    that go to zero faster than 1/x for x --> ħinf, i.e.
    lim(x-->ħinf) xf(x) = 0
    For unit weight function, determine which of the following linear operators is Hermitian when acting upton {f(x)}:
    (a) d/dx + x
    (b) −i d/dx + x^2
    (c) ix d/dx
    (d) i d3/dx3


    I tried using definition of a hermitian operator as <u, Hv> = (<v, Hu>)* but I don't get what u and v are, unless they are just general members of the set {f(x)} in which case I'm still confused!

    Thanks!
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  2. #2
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    Quote Originally Posted by joker_900 View Post
    Please help, I have no idea!

    Consider the set of functions {f(x)} of the real variable x defined on the interval −inf < x < inf
    that go to zero faster than 1/x for x --> ħinf, i.e.
    lim(x-->ħinf) xf(x) = 0
    For unit weight function, determine which of the following linear operators is Hermitian when acting upon {f(x)}:
    (a) d/dx + x
    (b) −i d/dx + x^2
    (c) ix d/dx
    (d) i d3/dx3


    I tried using definition of a hermitian operator as <u, Hv> = (<v, Hu>)* but I don't get what u and v are, unless they are just general members of the set {f(x)} in which case I'm still confused!
    In these example, the space V on which H acts is the set of functions f(x) on the real line for which $\displaystyle \textstyle\lim_{|x|\to\infty} xf(x) = 0$. (They had better be differentiable functions for the question to make sense, though this is not specified.) The definition of H being hermitian is: $\displaystyle \langle Hf,g\rangle = \langle f,Hg\rangle$, for all f, g in V. The inner product $\displaystyle \langle f,g\rangle$ means the integral (with respect to the unit weight function) of the product of f(x) and the complex conjugate of g(x).

    In part (a), for example, the operator H is differentiation plus multiplication by x, so that $\displaystyle Hf(x) = f'(x) + xf(x)$. You have to check whether the integrals of $\displaystyle \bigl(f'(x) + xf(x)\bigr)g(x)^*$ and $\displaystyle f(x)\bigl(g'(x) + xg(x)\bigr)^*$ (where the asterisk denotes the complex conjugate) are equal. It looks as though integration by parts will figure somewhere in the calculation.
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