Loss of Coherence and Bohr’s Correspondence Principle

E=m_{o}c^{2} + m_{o}v^{2}/2 vs. E=m_{o}v^{2}/2

G.R.Dixon, 2/12/2007

Experiments with non-relativistic electrons indicate that the deBroglie relation

(1)

is essentially correct. The total energy for a particle with kinetic energy m_{o}v^{2}/2 is

. (2)

And according to deBroglie, the electron’s energy is proportional to the "electron wave’s" frequency:

. (3)

Thus

. (4)

Even when m_{o} is very small (e.g. in the case of an electron), Eq. 2 results in very large values of n.

In the case of an infinite square well in which monoenergetic electrons are trapped, a standing wave ideally exists. This wave has nodes at each wall, and one or more antinodes between the walls (depending on how many half waves are present).

Of course in any real potential there are no perfectly sharp angles. If the walls nominally exist at x=0 and x=L, then between x=Dx_{1} and x=Dx_{1}+Dx_{2} (where Dx_{i} << L) the potential bends very sharply (but not perfectly sharply) to a near vertical line and then, at some U >> m_{o}v^{2}/2, bends sharply back again and levels out at x~0. If Y_{L} is the wave traveling in the negative x direction, then the reflection process is analogous to a plane light wave being reflected at a mirror. There is partial reflection right at the "surface," a bit more a very small distance below the surface, etc. Add to this the microscopic irregularities in any real potential (or "surface") and it is not difficult to appreciate that Y_{R} and Y_{L} will not be precise mirror images between the two walls.

Indeed given the very high rotation rate (i.e. n) of the
spinning spirals for Y_{R} and Y_{L}, it is not unreasonable to expect less than sharply defined values of |Y_{R} + Y_{L}|^{2} even when only a few half waves are accommodated between the walls. As the walls are widened out (and l remains unchanged) the fuzziness can be expected to increase. When L=100 or more half wavelengths, we can expect |Y_{R} + Y_{L}|^{2} to actually (i.e., not ideally) approach a single value across the whole breadth of L. In other words, the situation would approach the single-valued classical probability quite as Bohr’s Correspondence Principle requires. It is noteworthy that if one uses simply E=m_{o}v^{2}/2 in Eq. 2 then n is much smaller, and the loss of coherence between Y_{R} and Y_{L} may be less pronounced as L grows.

In effect (and in theory), the loss of coherence as L is increased in half wavelength increments could be experimentally monitored in order to decide whether Eq. 2 should be used for E, or whether the simpler E= m_{o}v^{2}/2 is more appropriate.