Editing Old quantum theory (section)

=== One-dimensional potential: ''U'' = ''Fx'' ===

Another easy case to solve with the old quantum theory is a linear potential on the positive halfline, the constant confining force ''F'' binding a particle to an impenetrable wall. This case is much more difficult in the full quantum mechanical treatment, and unlike the other examples, the semiclassical answer here is not exact but approximate, becoming more accurate at large quantum numbers.
: <math>
2 \int_0^{\frac{E}{F}} \sqrt{2m(E - Fx)}\ dx= n h
</math>
so that the quantum condition is
: <math>
{4\over 3} \sqrt{2m}{ E^{3/2}\over F } = n h 
</math>
which determines the energy levels,
: <math>
E_n = \left({3nhF\over 4\sqrt{2m}} \right)^{2/3}
</math>

In the specific case F=mg, the particle is confined by the gravitational potential of the earth and the "wall" here is the surface of the earth.