Simple Harmonic Oscillator - Hamiltonian

[Confundo]

See post two.

 

[Confundo]

Just noticed the latex thing, I'll try and rewrite the equations in that format.

Homework Statement

From

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

$$H = 0.5(p^2 + w^2q^2)$$

Homework Equations

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

The Attempt at a Solution

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

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

I'm not really sure how I'm supposed to integrate over dV here.