Quantization of angular momentum?

To date all models for the structure of the atom are based on a hypothesis first put forward by the British physicist John Nicholson in 1911.  Nicholson recognized that the units of Planck’s constant were those of angular momentum and so he reasoned that Planck’s constant was the angular momentum of the electron orbiting the hydrogen nucleus in the base energy state.  However Nicholson went further and argued that Planck’s constant was the fundamental unit of angular momentum and that it could only take on values which were integer multiples of this base value.

This hypothesis was sufficient for Niels Bohr to come up with a model for the hydrogen atom in which the differences between the various energy levels matched the spectral lines predicted by the empirically derived Rydberg formula. In the Bohr model each increase in energy state is associated with an increment in angular momentum of one “unit” of angular momentum . The Bohr model contained substantial flaws and other models have since superseded it, however they all incorporate and rely on Nicholson’s basic assumption.

While Nicholson’s assumption yields the correct results for the energy levels of the hydrogen atom there is no other supporting evidence to suggest that angular momentum is quantized in this way.  In particular there is nothing to suggest that there is any mechanism which would cause it to be quantized.

Angular momentum is a compound value, dependent on three variables.  For a point mass m rotating around an axis at radius r with velocity v, such as is the case of the electron orbiting the hydrogen nucleus, the angular momentum is the product of the mass, the radius and the tangential velocity of the object and is given by:

 Equation 1 Equation 1

Nicholson argued that it can only take on values which are integer multiples of Planck’s constant which means that

Equation 2 Equation 2

Where n is a positive integer.

For angular momentum to be quantized in this way each of its three constituent variables must themselves take on a series of discrete values and they must do so in a way which is in lock step with one another.

The concept that mass can take on a discrete value is not too difficult to grasp.  The mass of an object is after all made up of the masses of all of its constituent particles and since any physical object can only contain a finite number of such particles then its mass must be discrete.  In any event when we are dealing with objects such as the hydrogen atom where there is only one electron, then the mass must be a discrete quantity.

However even for the simplest of the three variables there is a complication. It is only the rest mass of an object which has a definite fixed value.  As soon as the object is in motion with respect to an observer then it gains mass due to relativity.  From the point of view of a stationary observer located at the centre of the electron’s orbit the electron is seen to be moving with a constant tangential velocity v.  To such an observer the mass of the electron is seen to increase due to the effects of relativity.  This increase is not discrete in nature; it is continuous, the mass increases by a factor Gamma as the velocity increases.

At the velocities calculated by Bohr for the various energy levels of the hydrogen atom these effects are small.  Small they may be, but they are nevertheless significant.  Perhaps more important is that they occur at all, since even the slightest such variation in the mass of the electron would mean that angular momentum is not quantized.  However the Bohr model correctly predicts the absorption and emission spectra for all so called hydrogenic atoms.  These are atoms which are ionized such that they only have a single orbiting electron.  Here the orbital velocity in the base energy state increases with the atomic number (the number of protons in the atomic nucleus).   So in Helium for example which has two protons in the nucleus the orbital velocity of the electron in the base energy state is twice that of hydrogen.  For Calcium with atomic number 20 it is twenty times and so on.  The effects of relativity on the mass of these orbiting electrons become more and more significant with increasing atomic number.  For Hydrogen with an orbital velocity of 2187691 m/s the value of Gamma is 1.000026627 and so the error in the mass term is almost insignificant.  For calcium, with atomic number 20 value of Gamma is  1.010823491 and so the error in the mass term of Bohr’s equation is around 1%.  While for the heaviest known element Ununoctium Gamma has a value of 1.9667361 and the error in the mass term is around 97%.  Perhaps it is just as well that no heavier elements have been found, since the electron in orbit around an atomic nucleus of atomic number 138 would have to be travelling faster than the speed of light!

The current received wisdom is that velocity is invariant with respect to relativity. Both the moving object and the stationary observer agree on their relative speed.  Radius too is unaffected by relativity, since it is measured normal to the direction of travel.

Setting aside the problems with the mass of the orbiting electron and how it is affected by relativity for the moment.

In Bohr’s model the radius of the orbiting electron increases with the square of the energy level, so in the second energy state the orbital radius is four times that of the base energy state, in the third it is nine times and so on. Meanwhile the orbital velocity varies as the reciprocal of the energy level, so in the second energy state the orbital velocity is half that of the base state, in the third energy state it is one third and so on.

We can therefore rewrite the equation for angular momentum as

 Equation 3 Equation 3

where the suffix n denotes that this is the velocity and radius relating to the nth energy level.

The orbital radius in the base energy state is referred to as the Bohr radius and has a value of 0.529 177 210 92 x 10-10 m.  This is not a particularly special length, except in this particular context.  There are lengths which are smaller than this in the real world and lengths that are longer than this.  So just why this length should be the discrete value which forms the base of the quantization of angular momentum remains a mystery

Similarly the orbital velocity is nothing special, there are objects which travel faster than this and objects which travel slower.  On the scale of velocities which extends from zero to c, this is just an arbitrary point.  The orbital velocity of the electron in the base state does have one peculiar characteristic and that is it is related to the speed of light by a mysterious constant called the Fine Structure Constant.

The real question that needs to be answered is this – What is it that links the successive radii and velocities with one another?  How, in other words, does the electron “know” that in the second energy state it should have four times the radius and one half the velocity that it did in the base energy state. What is it about the electron orbiting at that particular state which tells it to orbit at that particular velocity?  What is the mechanism that drives this process?

What in other words is the relationship between rn and vn?

The answer that comes back is always the same.  The only thing that supposedly connects these variables is the need to constrain angular momentum to be an integer multiple of Planck’s constant.  The argument therefore is an ontological one.  Angular momentum is quantized because it needs to be in order to produce the correct energy levels within the atom.  And the energy levels within the atom derive from angular momentum being quantized.

What is really lacking here is a mechanism – some sort of relationship between the velocity and the orbital radius which constrains them to be bound together in this way.  There is no such mechanism, or at least nobody to date has ever found one and so the question of what underlies the discrete energy levels within the atom remains a question of belief and not one of science.  You must believe that angular momentum is quantized in order that the energy levels of the hydrogen atom are discrete.

Unless that is the orbital velocity is also taken to be affected by relativity in a way that is the inverse of the effect on mass

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