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Introduction

Thus far, we have considered the Bohr theory on the assumption that the electron orbits around the nucleus are circular.  However, the orbits need not be circular – we know, for example, that the orbits of the planets around the sun are elliptic in shape.  These elliptic orbits stem from the inverse-square nature of the gravitational attraction between the planets and the sun.  In the simple Bohr model, we also have an inverse-square attractive force between the electron and the nucleus.  Thus, the electron orbits around the nucleus could also be elliptic in shape.

Wilson-Sommerfeld Rule

The major difficulty in considering elliptic orbits is how to extend the quantisation rule that we used for the circular orbit in the previous sections.  For the circular orbit, the quantisation rule is

Now, this cannot be applied directly to the elliptic orbit because r is not constant throughout the orbit.  Consider the following figure:

For the circular orbit, because r is constant, the motion of the electron can be specified simply in terms of the angle θ.  However, the for the elliptic orbit (with the nucleus located at one focus of the ellipse), we also need to specify the value of r, since this changes value at different points of the orbit.

The quantisation rules for an elliptic orbit can be derived from the so-called Wilson-Sommerfeld quantisation rule.  This states that for any physical system whose variables are periodic functions of time (e.g. as they are for an orbiting system), the following must hold:

where P_i is the i^th generalised momentum associated with the generalised coordinate q_i.  This all sounds rather complicated, but in fact if the variable q is a physical coordinate, then P is just the linear momentum associated with that coordinate.  For example, q could be the variable representing the x-direction, in which case P is the x-component of momentum.  When q is an angle, then P is the component of angular momentum associated with that angle.

Also in this expression, the integral is taken around one complete cycle of the motion.  For the circular orbit, there is only one generalised coordinate, the angle θ.  The generalised momentum associated with this is just the angular momentum, mvr.  Thus, the Wilson-Sommerfeld quantisation condition is

which can be seen to be identical to the quantisation rule stated above.

Quantisation of the Elliptic Orbit

For the elliptic orbit, we have two generalised coordinates, the distance r and the angle θ.  We therefore have two quantisation conditions:

where k and n_r are both integers.  If these integrals are evaluated (a rather messy but straightforward task), it can be shown that the eccentricity ε of the elliptic orbit is related to k and n_r through the relationship

where n obviously is also an integer.  The eccentricity of an ellipse is a measure of how “elliptical” the ellipse is.  A circle is an ellipse with an eccentricity of zero.  An eccentricity of one would imply a straight line.  With this in mind, and noting that an eccentricity of one would not be permitted as it doesn’t correspond to an orbit, we must have

Thus, one could choose a value for n, and for that value of n there would a series of “allowed” values of k.  For example, if we take n=4, then k could take the values 1,2,3 or 4.  Now, the eccentricity of the orbit is given by

Therefore, for a given value of n, the greater the value of k, the more circular the orbit.  This is illustrated in the following figure, for a value n=4.

Energy and Angular Momentum

The energy of the elliptic orbit is given by the same expression as that for the circular orbit, namely

Thus the energy of the electron in its orbit is independent of the integer k.  Note that the “size” of the orbit is also determined by the integer n.

From the second of the quantisation conditions above,

and noting that the angular momentum of the electron in its orbit is constant, it is easy to show that

where we have defined

Thus, whereas the energy of the electron orbit is independent of k, so the angular momentum is independent of n.  All of the orbits in the figure shown above correspond to the same electron energy, but the angular momentum of the electron differs between the orbits.

The observant reader will note that we could extend our simple model further.  Up to now we have considered only the two-dimensional motion of the electron around an elliptic orbit in a fixed plane.  We could relax this assumption and allow the plane of the ellipse to vary as well.  It is not too hard to guess that the introduction of a third variable (describing the orientation of the plane of the ellipse) gives rise to a third integer, m, that governs the orientation of the plane of the elliptical orbit.  We will consider this further in the next section, as the properties of m are not easily obtained from our simple model.

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