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Homework Help: Simple Harmonic Oscillator - Hamiltonian

  1. Oct 16, 2008 #1
    See post two.
     
    Last edited: Oct 16, 2008
  2. jcsd
  3. Oct 16, 2008 #2
    Just noticed the latex thing, I'll try and rewrite the equations in that format.

    1. The problem statement, all variables and given/known data

    From

    [tex] 0.5 \int dV[e_0 E^2_x (x,t) + (\frac{1}{u_0} B^2_y (z,t)] (1) [/tex]
    Show that

    [tex] H = 0.5(p^2 + w^2q^2) [/tex]

    2. Relevant equations

    [tex] B_y (z,t) = (u_0e_0 / k)(2w^2 / Ve_0)^{0.5}p(t)cos(kz) (2) [/tex]
    [tex] E_x (z,t) = (2w^2 / Ve_0)^{0.5}q(t)sin(kz) (3) [/tex]

    3. The attempt at a solution

    Substituting (2) and (3) into (1)

    [tex] H = 0.5 \int dV [(2w^2 / V)(q(t))^2(sin(kz))^2 + (u_0e_0 / k^2)(2w^2 / V)(p(t))^2(cos(kz))^2] [/tex]

    I'm not really sure how I'm supposed to integrate over dV here.
     
    Last edited: Oct 16, 2008
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