 |
CORRESPONDENCE PRINCIPLE |
See also ...
( Home → Science → Bohr Theory → Correspondence )
|
A very important principle in quantum mechanics is the correspondence principle. This states that, when considering phenomena in the regime of classical physics, the results of quantum mechanics must give rise to the appropriate classical physics results. In this article we will show how the Bohr theory satisfies the correspondence principle. |
Previous: Quantum theory of hydrogen
Large Quantum Numbers
Thus far in our analyses, we have been using the simple Bohr theory to predict the properties of a very small system, namely a hydrogen atom. However, nowhere is it written that we must use the Bohr theory only for very small systems. We have found that the radii (in metres) of the allowed electron orbits in the hydrogen atom are given by the following expression
metres.
Suppose that we take a quantum number n = 100,000. The radius of the electron orbit for this quantum number is 0.53 metres. With such dimensions we are firmly in the regime of classical physics, and yet there is no reason why the Bohr theory can’t also be applied in this situation.
Instinctively, we might expect the Bohr theory to give a result that is consistent with our expectations in classical physics about an electron orbiting a nucleus with a radius of 0.53 m.
So let us consider an electronic transition between two adjacent orbits corresponding to very large quantum numbers, which would correspond to a system that sits in the classical physics regime. The energy emitted during such a transition is found easily from the Bohr theory to be
Now if n is much greater than unity,
and so the energy released is
The frequency of the radiation released is
where again we have used
Now, the radius of the orbit for quantum number n is given by
and the basic equation of motion in the Bohr model is
.
The symbols in
these equations are as follows:
is
the mass of the electron, and r,
ω,
e and E are its orbital radius, angular velocity,
charge and energy, respectively. In addition, h is Planck’s
constant, and 
is
the permeability of free space.
The Classical Result
Let us evaluate ω2. From the equation of motion and the expression for the Bohr radii, we have
Simplifying and comparing with the expression above for f, we see that
That is, the frequency of emitted radiation in the classical limit is simply equal to the frequency of rotation of the electron around the nucleus. This is the classical limit that we expect from the correspondence principle.
It is interesting to note that if we had considered transitions to energy levels separated by more than one level, then the result would become
where p is the numerical "separation" of the energy levels. This corresponds to the higher harmonics of the rotation frequency, and is still consistent with the correspondence principle.