Before embarking upon a survey of the dramatic
consequences of Quantum Theory, it is insightful to examine the prevailing
conditions in 19th century physics.
Such was a time rife with “End of History” sentiments; noted theorists
were proclaiming the End of Physics, a subject that in their eyes had nearly
solved most of nature’s secrets.
Indeed, the bright, young German student Max Planck was dissuaded from
entering physics, the “dead science”.
There were, however, cracks in the façade. Certain problems, which seemed entirely
minute and non-essential, occupied the leading theorists at the time. One of these problems was in the details of
a process that is built upon thermal radiation, the radiation emitted
from a body as a result of its temperature.
From studies of thermal radiation, Max Planck discovered at the turn of
the last century that energy can exist in discrete packets, or quanta. Planck’s Theory remains one of the hallmarks
of modern science, as it laid the foundation for the formulation of quantum
mechanics twenty years later. The key
conclusion to draw from Planck’s work is that the energy of a sinusoidal
oscillator is quantized. The energy
of such an oscillator can only assume values such that
E = nhv
(2.1)
where each allowed value of E is termed a quantum
state and is specified by n, a quantum number. It is important to note here that Planck
only considered the oscillations of the electrons in the walls of the cavity,
and that the quantized energy of oscillation translated into quantized energy
of the emitted radiation. That this energy is not continuous but rather
consists of discrete chunks is such a revolutionary conclusion that its
importance cannot be over emphasized.
The role of Planck’s constant, h, in the above equation is
important as well; it is due to the extreme smallness of the constant (h =
6.626 x 10-34 Js) that the discreteness of energy is not perceptible
in everyday situations and only in the realm of similarly small magnitudes.
The extraordinary implications of Planck’s Theory
were not immediately recognized in the scientific community. Planck himself regarded his theory as a
mathematical sleight of hand. In fact,
it took an Einstein to realize the tremendous importance of quantized energy as
an actual manifestation of physical reality and not merely a calculational
trick. Albert Einstein employed
Planck’s theory in his investigation of the photoelectric effect and in turn
further expanded the concept, as we shall see later.
Common sense indicates that we should expect a
particle to have any possible energy within permissible ranges. For example, imagine a pendulum that swings
from side to side. Recall that when the
pendulum bob is in the extreme right or extreme left positions, the potential
energy of the system is the greatest, while the kinetic energy of
the system is the greatest as the bob moves at maximum speed through the
origin. The higher we hold the bob
initially, the greater the potential energy and the faster it will move once we
let go. Therefore, the initial
potential energy of the bob is a function of the initial position from which it
is started into motion. As we drop the
bob, it follows a semi-circular path as its position decreases towards the
lowest point. Obviously, from bob’s
initial point to its low point, the system exhibits a continuous range of
energies as it travels through a continuous range of positions. Planck’s astounding discovery was that in
reality the energy of such an oscillating system is not continuous, or
in other words, such a system can only exist at discrete values of energy. This would be as if we had dropped the bob
from its high initial position and it disappeared from one point and
immediately appeared at a lower point, without traveling in between! Now from Planck’s work, he derived the
expression in (2.1) that describes such behavior. As mentioned above, the constant h is so small that for
the macroscopic system that I am describing such ‘jumping’ between positions is
completely imperceptible; indeed a detailed analysis shows that the motion of
the macroscopic pendulum is indeed continuous. However, in the tiny world of molecules, whose study is central
to chemistry and biology, the dimensions of analysis are comparable in size to
Planck’s constant h. Therefore
the discreteness of energy fundamentally affects the behavior of such
particles. For instance, a specific
example is one of nature’s own oscillators, the vibrating particle. A microscopic vibrating particle is similar
to a pendulum, or a spring, in many respects, and indeed the energy of
vibration of such a particle exists only in discrete levels (is quantized).