Independent of the discussions on wave-particle duality there was another important question to clear. De Broglie waves describe only free electrons . How does the wave behave under the influence of the electromagnetic field, specially how does the electric field of the positive nucleus effect the wave function of the electrons inside the atom ?

The discovery of this equation is one of the major fundamental achievements in physics. Schrodinger’s equation gave correctly the de Broglie waves as solutions in vacuum as expected. But was it valid when there were external fields ? How could this be tested?

The hydrogen atom. (a mystery cleared)

In 1913 to Bohr suggested an atom model with mysteriously non-radiating (Mystery 1 in chapter 2.1) stable electron orbits in an atom. According to this model light is emitted only when an electron “falls” from an higher energy orbit (the one with greater radius) to a lower energy orbit. The energy difference is carried away by the photon. The photon has the energy hv where v is the frequency. Once the energy levels of the stable orbits are known this model could predict which frequencies should occur in the line spectra by considering possible transitions between these orbits. In reality calculations were made first in reverse direction. From a subset of the occurring frequencies one first calculated the energy differences between the orbits. One could then verify if the remaining subset of the occurring frequencies were consistent with the calculated energy levels of the orbits. The calculations were in complete agreement with the observed line spectra. The model allowed to calculate the frequencies of other possible transitions that were not observed until that time. Radiation at these predicted frequencies could then be later actually observed experimentally verifying Bohr’s model.

Bohr’s model helped to gain a limited insight into what happens in an atom but it didn’t solve the really important mystery, namely:

What property distinguished these orbits from other ones so that they were stable? Why were intermediate orbits forbidden ? Starting with the information available from line spectrum, Heisenberg developed successful methods for describing the transitions between these atomic states by using arrays of numbers (known in mathematics as matrices), but these calculations didn’t help to gain a deeper insight in what actually happens in atoms.

Solving the Schrodinger’s equation for a single electron in an electrostatic coulomb potential of a single proton (a combination that is nothing but the hydrogen atom) led to the following interesting finding:

There were bound and stationary solutions for the wave function . Bound means the intensity was very high in a region close around the nucleus (as close as 10^-8 cm approximately as the calculations showed ) and was rapidly decreasing towards 0 when the distance from nucleus is slightly larger. Stationary means that the magnitude of the wave function doesn’t change in time¹ . The surprising finding was that the energy levels of these solutions perfectly fitted the energies of the Bohr orbits.

So, mysteries 1 and 4 in chapter 1 were resolved.

The wave concept therefore gained a new evidence apart from interference experiments. It was verified indirectly by the agreement of calculated and observed energy levels of the atomic states that the Schrodinger equation describes not only the statistical behavior of an ensemble of electrons as the distribution of bright dots in double slit experiment may suggest but the states of even a single electron in the atom. This is because the observed frequency in the line spectrum is also the frequency of even a single photon that is emitted by a single atom.

Now we are here at an important point. Remember what was the reason that made us believe that the picture of an interfering wave was not sufficient and that we needed the concept of pointlike entities to explain the double slit interference experiment discussed in the previous chapter: The reason was the observation of randomly distributed bright dots on the scintillating screen .

These bright dots have however a macroscopical size (i.e. we can see them ) and there is NO EVIDENCE that we have something smaller than the extent of an electron orbital in an atom (10^-8 cm) at the point where the electron seems to hit the scintillating screen. Thus the dots can well be interpreted as indicating only that the electron’s wave function changes its form in interaction with the scintillating crystal and forms a peak around a nucleus like a cloud becoming denser and condensing to a droplet. So it seems we may have a wave changing its form (although very drastically during position measurement) but there seems to be no definite indication that there is a pointlike entity beside the wave function.

This was Schrodinger’s viewpoint initially. Unfortunately things were not so easy. There were some problems with this wave-only viewpoint. The main problem was the following:

The Schrodinger equation was a deterministic equation according to which the same initial conditions lead to same final conditions. Therefore it was not able to account for the random behavior concerning the localization point in the collapse process during position measurement. There were other problems too as we will see in the next chapter.

1.The wave function consists of a magnitude and a phase. The phase is a cyclic property similar to an angle where 360 is equivalent to 0 See appendix-1