This must be one of the most common misunderstandings about quantum mechanics, that quantum mechanics
is about making things discrete. But is an understandable misunderstanding because the
word “quantum” suggests that quantum mechanics is about small amounts of something. Indeed,
if you ask Google for the meaning of quantum, it offers the definition “a discrete quantity
of energy proportional in magnitude to the frequency of the radiation it represents.”
Problem is that just because energy is proportional to frequency does not mean it is discrete.
In fact, in general it is not.
The reason that quantum mechanics has become associated with discretization is entirely
historical. The first signs that something was not quite right with the fundamental theories
of the 19th century came from atomic spectral lines. Atoms can absorb and emit light only
at certain frequencies. If you think that atoms are basically blobs of particles stuck
together, which was what people thought at the time, then this makes absolutely no sense.
According to quantum mechanics now, the negatively charged electrons occupy shells around the
positively charged nucleus. These shells cannot have any radius, but only certain values of
the radius are allowed. Just what shape the shells have and how large they are can be
calculated with quantum mechanics. And this explains why atoms can only absorb and emit
light of certain frequencies. Because the energy of the light must fit to the energy
that moves an electron from one shell to another. So, yes, the energies of electrons which are
bound to atoms are discrete. But the energies of electrons, or of any particle really, are
not always discrete, and neither are other measureable quantities. The energy of a photon
traveling through empty space, for example, can have any value according to quantum mechanics.
The energy is not discrete. Or, if you look at an electron in the conducting band of a
metal, it can be at any position. The position is not discrete.
What, then, does it mean to have a quantum theory as opposed to a non-quantum theory?
A quantum theory is one in which you have observable quantities that obey Heisenberg’s
uncertainty principle. Mathematically, this is not entirely the correct definition. More
precisely a quantum theory has operators for observables which do not commute. But for
what the physical consequences are concerned, the uncertainty principle is what tells quantum
from non-quantum theories. The other important property of quantum theories is that you can
have entanglement. We will talk about what this means another time.
For today, the lesson to take away is that quantizing a theory does not mean you make
it discrete. This is important also when it comes to the quantization of gravity. Quantizing
gravity does not necessarily mean that space and time have to be discrete. Thanks for watching,
see you next week.