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Determining whether an operator is Hermitian

  1. Nov 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the set of functions [itex] {f(x)} [/itex] of the real variable [itex] x [/itex] de fined on the interval [itex] -\infty< x < \infty [/itex] that go
    to zero faster than [itex] 1/x [/itex] for [itex]x\rightarrow ±\infty[/itex] , i.e.,
    [tex]
    \lim_{n\rightarrow ±\infty} {xf(x)}=0
    [/tex]
    For unit weight function, determine which of the following linear operators is Hermitian when acting upon [itex] {f(x)} [/itex]:
    [tex] (a) \frac{d}{dx} + x[/tex] [tex](b) -i \frac{d}{dx}+x^2 [/tex][tex](c) ix \frac{d}{dx} [/tex][tex](d) ix \frac{d^3}{dx^3} .[/tex]


    2. Relevant equations

    [itex] Hf(x)=λf(x) [/itex] has real values of [itex] λ [/itex] where [itex] H [/itex] is a Hermitian operator and [itex] λ [/itex] are it's eigenvalues



    3. The attempt at a solution

    [tex] a) \frac{df(x)}{dx} + xf(x) = λf(x) [/tex][tex] \frac{df(x)}{dx} + (x-λ)f(x) = 0 [/tex] [tex]\text{Integrating factor is }e^{\int (x-λ)dx}=e^{\frac{1}{2} x^2-λx}[/tex]
    [tex]e^{\frac{1}{2} x^2-λx}f(x)=constant[/tex]

    I've done a similar thing for parts a), b), and c) but I'm not sure what to do with this or if it even helps. For d) I've tried to work out the eigenfunctions but get to mess and didn't really want to continue down the route I was going without knowing if this was useful or not.

    [tex]\text{b) leads to } e^{ix(\frac{x^2}{3}-λ)}f(x)=constant[/tex]
    [tex]\text{c) leads to } x^{iλ}f(x)=constant[/tex]
    [tex]\text{d) } ix \frac{d^3f(x)}{dx^3}=λf(x) [/tex]
    [tex] x \frac{d^3f(x)}{dx^3}+iλf(x)=0 [/tex]
    [tex]\text{let }x=f(t)[/tex]
    [tex]\frac{d^3f(x)}{dx^3}=\frac{d^3f(x)}{dt^3}\frac{d^3t}{dx^3} [/tex]
    [tex]\text{want }\frac{d^3t}{dx^3}=\frac{1}{x}[/tex]
    [tex]\text{(After integrating 3 times }t=\frac{1}{2}(x^2(ln(x)-\frac{3}{2})) [/tex]

    This is were I decided not to continue until I knew whether or not I was actually doing anything right.

    Thanks in advance for any help.
     
  2. jcsd
  3. Nov 21, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    No, that's really not the right way to go about it. You've likely defined an inner product ##<f,g>= \int_{-\infty}^\infty f^*(x) g(x) dx ##. In terms of that inner product H is Hermitian if <f,Hg>=<Hf,g> for any two functions f and g. Does that sound familiar? Checking that is the way to check if an operator is Hermitian. Start with the two parts of the first one. Is x Hermitian? That's pretty easy. Now is d/dx Hermitian? That's a little harder. Try looking at an integration by parts.
     
  4. Nov 22, 2012 #3
    Oh, thank you I knew I must've been doing something wrong.
     
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