Chapter 7: Wave Mechanics and Wave Particle Duality

Schrödinger's theory of the quantum world is called *wave
mechanics*. He worked out the exact solutions of the wave
equation for the hydrogen atom, and the results perfectly agreed
with the known energy levels of these atoms, seemingly without any
of the complications and metaphysical speculations associated with
the uncertainty principle. Moreover, the equation could also
be applied to more complicated atoms, and even to particles not
bound in atoms at all. It was soon found that in *every*
case, Schrödinger's equation gave a correct description of a
particle's behaviour, provided it was not moving at a speed near
that of light. [1]

In spite of this success, the very meaning of the waves remained
unclear. Schrödinger believed that the intensity of the
wave at a point in space represented the 'amount' of the electron
that was present at that point. In other words, the electron
was spread out, rather than concentrated at a point. However,
it was soon found that this interpretation was untenable, because
observations revealed that particles never spread out. For
example, it follows from the wave equation that when a wave,
representing an electron, strikes a target, it spreads out in
*all directions*. Experimentally, on the other hand, the
electron scatters in some *specific* direction but *never
breaks up*.

Max Born, another German physicist and a good friend of Einstein, interpreted this result as follows: the wave associated with the electron is not a tangible 'matter wave', but one that determines the

This indicated that the simple interpretation of the wave equation as a description of the physical matter waves in ordinary space, as was originally assumed by Schrödinger himself, is incorrect. Born noted the following conclusions:

- Matter waves represent
*probability amplitudes*associated with the occurrence of an event. - If there are multiple alternatives for the event to happen,
the total amplitude is the sum of the
*alternative*amplitudes. - Finally, the absolute square of the overall amplitude, the
*intensity*of the wave, must be interpreted as the*probability that the event will happen*.

This difference between the world of the very small and the
everyday world we experience cannot be understated. When you
are walking across a street and an automobile is approaching you,
it is at a definite place at any given time, and it is possible to
describe with a great deal of *precision* where exactly the
automobile is, and where it will be at another moment in time if it
continues on the same path. For objects in the realm of the
very small, this is simply *not* possible.

These discoveries are at the heart of quantum physics. The
importance of the introduction of statistical probabilities into
physical law cannot be understated. In earlier physics, that
is to say the classical mechanics of Isaac Newton, probabilities
had no place. In the Newtonian world-view, an object of any
kind could at least in theory be described as having a definite
position at any given time. This was not the case at all in
the new quantum theory: *the probabilities describing material
particles could never be replaced by old-style definite
positions*.

**Fig. 7-1: Electron probability clouds for the s-, p- and
d-orbitals ^{*}**

With de Broglie's principle, we can also extend this to
electrons and all other matter particles, and hence account for
their wave nature (and hence the double-slit experiment).
They have been experimentally found to be correct; in 1994,
interference fringes, a typical characteristic of waves, were
generated with beams of iodine *molecules*, which are about
500,000 times more massive than electrons.

By 1930s, this indeterministic interpretation of quantum
physics, mainly put forward by Niels Bohr with support from Born's
probability interpretation and Heisenberg's uncertainty principle,
came to be known as the *Copenhagen interpretation*, mainly
because Bohr ran an influential physics institute there during this
period. What happened next is dealt with in the next chapter.

2. It is impossible to give an everyday world analogy for this 'cloud of probabilities'. Nevertheless, this is the best description of how these particles exist and behave. Back

** Fig. 7-1 Courtesy: Creation
Explanation*