Previous:  Elliptic Orbits

Summary of Elliptic Orbit

The results of the previous section are very significant in terms of what the full theory of quantum mechanics tells us about the energy levels and angular momenta of electrons in the hydrogen atom.  In the previous section we saw that, if we assume an elliptic electron orbit around the nucleus:

The allowed motion is determined by two integers, n and k;

n can take any value from 1 upwards.  k can only take values such that it is less than or equal to n.

The energy of the electron depends only on n;

The angular momentum of the electron depends only on k;

A third integer, m, defines the orientation of the plane of the orbit.

Quantum Theory Results

Why are these results significant?  Because the full quantum mechanical treatment of the hydrogen atom gives almost identical results to our simple little model.  The only difference is that the integer k in our simple theory is replaced by an integer l such that

and the angular momentum of the electron is given by

The integer m can be shown to take the values 0, ±1, ±2, etc, all the way up (and down!) to ±l.  These allowed values are derived from the assumptions and requirements of the wavefunction that describes the motion of the electron in a quantum-mechanical framework.

In quantum mechanics, the integers n, k and m are referred to as “quantum numbers”.  To summarise, they have the following properties.

The quantum number n is referred to as the total quantum number, and it determines the allowed energies of an electron in the hydrogen atom.  It can take values

            n=1,2,3 …

The quantum number l is called the orbital quantum number, and determines the angular momentum of the electron.  It can take values

            l = 0,1,2 … n-1

The quantum number m is called the azimuthal quantum number, and it determines the orientation of the electron motion.  It can take values

            m = 0, ±1, ±2 … ±l

It should be noted that these three quantum numbers emerge naturally from the solution of the Schrodinger equation for the hydrogen atom, whereas in our simple Bohr model, they were in effect introduced as postulates (via the Wilson-Sommerfeld relationship).

Quantum Energies for Hydrogen

Although rather more involved mathematically than in the simple Bohr model, quantum mechanics can be used to show that the allowed energies of the electron in the hydrogen atom are given by

That is, the energies predicted by quantum mechanics are identical to those predicted by the simple Bohr theory!  The radius of the electron orbit is a rather more tricky concept in quantum mechanics, compared with the Bohr theory, because quantum mechanics predicts a probability of finding the electron at a certain location, rather than specifying exactly where the electron is to be found.

However, it can be shown that for a given total quantum number n and an orbital quantum number l = n-1, the orbital radius with the greatest probability of occurrence is the one equal to that predicted in the simple Bohr theory.  This is given by

             metres.

This is a very interesting result.  This radius of orbit was derived from the simple Bohr model on the basis of a circular orbit.  In the previous section, we saw that in the simple Bohr model with elliptic orbits, when the integer k (= l+1) equals n, the orbit is circular.  This is the equivalent case to the one we are discussing here, where l = n-1.

It therefore comes as no surprise to find that in the quantum mechanical case with l = n-1, the most likely radius to find the electron is equal to that given by the simple Bohr model with a circular orbit.  Personally I find this a fascinating result.

Next:  The Correspondence Principle