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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.

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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: $$ E_n = \frac{n^2 h^2}{8mL^2}. $$

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$H(p,q)={p^2}/{2m}$ 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.

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