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Homework Help: Show Lx is Hermitian

  1. Nov 19, 2007 #1
    1. The problem statement, all variables and given/known data

    I have to show that in 3-d, Lx (angular momentum) is Hermitian.

    2. Relevant equations

    In order to be Hermitian: Integral (f Lx g) = Integral (g Lx* f)
    Where Lx=(hbar)/i (y d/dz - z d/dy)
    and f and g are both well behaved functions: f(x,y,z) and g(x,y,z)

    3. The attempt at a solution

    I know to do this I have to do integration by parts. I got to the point where I had to figure out, using integration by parts,: Integral [f(x,y,z) y (dg(x,y,z)/dz) dx]

    And I cannot figure this out :(

    I set:
    u=f(x,y,z) y
    dv=(dg(x,y,z)/dz) dx

    So then I get: du=[df(x,y,z)/dx]y + f(x,y,z)
    But what is v then?? Unless I'm completely off-track already, in which case, help would be great!
  2. jcsd
  3. Nov 19, 2007 #2


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    You need to multiply by dx dy dz, and integrate over all three, not just dx. This should make the integration by parts much easier.
  4. Nov 19, 2007 #3
    As in...

    dv=(dg(x,y,z)/dz) dxdydz

    So that...

    du = (df/dx)y + (df/dx)y + f + (df/dz)y

    Still not sure :(
  5. Nov 19, 2007 #4


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    Focus on the z integral (because the derivative is with respect to z). So du = (df/dz)y.
  6. Nov 19, 2007 #5
    oh, and then v is just g(x,y,z)...
  7. Nov 19, 2007 #6


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  8. Nov 19, 2007 #7
    great, thanks!!
  9. Nov 21, 2007 #8


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    Since L_{x} (or rather its closure in the strong topology of L^{2}(R^3)) generates a uniparametric subgroup of the group of unitary operators which represent a rotation (around an arbitrary axis) in a Hilbert space, then, by Stone's theorem, L_{x} is e.s.a. and its closure is s.a. But all e.s.a. operators are hermitian/symmetric. QED
  10. Nov 21, 2007 #9


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    Yeah, that's what I *meant* to say ...
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