Editing Prior probability (section)

===Example===
The following example illustrates the a priori probability (or a priori weighting) in (a) classical and (b) quantal contexts.
{{ordered list
| list-style-type = lower-alpha
| 1 = '''Classical a priori probability'''

Consider the rotational  energy E of a diatomic molecule with moment of inertia I in spherical polar coordinates <math> \theta, \phi</math> (this means <math>q</math> above is here <math>\theta, \phi</math>), i.e.
<math display="block"> E = \frac{1}{2I}\left(p^2_{\theta} + \frac{p^2_{\phi}}{\sin^2\theta}\right).</math>
The <math>(p_{\theta}, p_{\phi})</math>-curve for constant E and <math>\theta </math> is an ellipse of area 
<math display="block">\oint dp_{\theta}dp_{\phi} = \pi \sqrt{2IE}\sqrt{2IE}\sin\theta = 2\pi IE\sin\theta .</math>
By integrating over <math>\theta </math> and <math>\phi </math> the total volume of phase space  covered for constant energy E is 
<math display="block">\int^{\phi=2\pi}_{0}\int^{\theta=\pi}_0 2I\pi E\sin\theta d\theta d\phi = 8\pi^2 IE = \oint dp_{\theta}dp_{\phi}d\theta d\phi,</math>
and hence the classical a priori weighting in the energy range <math>dE </math> is
:<math>\Omega \propto</math>  (phase space volume at <math>E+dE</math>) minus  (phase space volume at <math>E</math>) is given by <math> 8{\pi}^2 I dE.</math>
| 2 = '''Quantum a priori probability'''

Assuming that the number of quantum states in a range <math>\Delta q \Delta p </math> for each direction of motion is given, per element, by a factor <math>\Delta q\Delta p/h</math>, the number of states in the energy range dE is, as seen under (a) <math>8\pi^2I dE/h^2 </math> for the rotating diatomic molecule. From wave mechanics it is known that the energy levels of a
rotating diatomic molecule are given by
<math display="block"> E_n = \frac{n(n+1)h^2}{8\pi^2 I}, </math>
each such level being (2n+1)-fold degenerate. By evaluating <math>  dn/dE_n = 1/(dE_n/dn)</math>
one obtains
<math display="block">\frac{dn}{dE_n} = \frac{8\pi^2 I}{(2n+1)h^2}, \;\;\; (2n+1) dn = \frac{8\pi^2 I}{h^2} dE_n.</math>
Thus by comparison with <math>\Omega </math> above, one finds that the approximate number of states in the range dE is given by the degeneracy, i.e.
<math display="block"> \Sigma \propto (2n+1)dn.</math>
Thus the a priori weighting in the classical context (a) corresponds to the a priori weighting here in the quantal context (b).
In the case of the one-dimensional simple harmonic oscillator of natural frequency <math> \nu </math> one finds correspondingly: (a) <math> \Omega \propto dE/\nu </math>, and (b) <math> \Sigma \propto dn </math> (no degeneracy).
Thus in quantum mechanics the a priori probability is effectively a measure of the [[Degeneracy (mathematics)|degeneracy]], i.e. the number of states having the same energy. 

In the case of the hydrogen atom or Coulomb potential (where the evaluation of the phase space volume for constant energy is more complicated) one knows that the quantum mechanical degeneracy is <math> n^2 </math> with <math> E\propto 1/n^2 </math>. Thus in this case <math> \Sigma \propto n^2 dn </math>.
}}