Editing Old quantum theory (section)

=== One-dimensional potential: ''U'' = 0 ===
One-dimensional problems are easy to solve. At any energy ''E'', the value of the momentum ''p'' is found from the conservation equation:
: <math>
\sqrt{2m(E - U(q))}=\sqrt{2mE} = p = \text{const.}
</math>
which is integrated over all values of ''q'' between the classical ''turning points'', the places where the momentum vanishes. The integral is easiest for a ''particle in a box'' of length ''L'', where the quantum condition is:
: <math>
2\int_0^L p \, dq = nh
</math>
which gives the allowed momenta:
: <math>
p= {nh \over 2L}
</math>
and the energy levels
: <math>
E_n= {p^2 \over 2m} = {n^2 h^2 \over 8mL^2}
</math>