# Wilson Sommerfeld Method to solve for Energy

I have an example in my notes to find the quantum energy levels when the Hamiltonian is $$H(p,q)={p^2}/{2m}+(mw^2q^2)/2$$. However when given the Hamiltonian $$H(p,q)={p^2}/{2m}$$, I'm having difficulties as there is no q dependence and therefore cannot figure out what to do once I have the equations of motion. Any help would be greatly appreciated. It states that q is in the range [0,L] and p is a real number.

There's no need for the equations of motion. The key is that $$q$$ is constrained in the interval $$[0, L]$$, so the Wilson-Sommerfeld condition will be an integral from $$0$$ to $$L$$ to $$0$$: $$\oint p \, dq = 2 \int_0^L p \, dq = nh,$$ (the "orbit" in phase space will be a rectangle with the top at $$+|p|$$, the bottom at $$-|p|$$ and sides at $$q=0$$ and $$q=L$$, whose area is $$2|p|L$$). Fixing an energy $$E = p^2/2m$$, we may write $$p$$ in terms of $$E$$ and take it outside the integral. After solving for $$E$$ the result is identical to the one dimensional well: $$E_n = \frac{n^2 h^2}{8mL^2}.$$
$$H(p,q)={p^2}/{2m}$$ is the equation for a free particle.
Look at how the Hamiltonian is constructed from Lagrange's equations. Then solve the differential equation for $$d{q}/d{t}$$.
I'm ignoring the last part of the question regarding the limits because $$p$$ and $$L$$ - and I'm assuming $$L$$ denotes the Lagrangian - are functions of time.