# Simple Harmonic Oscillator - Hamiltonian

1. Oct 16, 2008

### Confundo

See post two.

Last edited: Oct 16, 2008
2. Oct 16, 2008

### Confundo

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

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

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

2. Relevant equations

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

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.

Last edited: Oct 16, 2008