0Introduction
IB Quantum Mechanics
0.3 Matter waves
The relations
E = hν = ~ω
p =
h
λ
= ~k
are used to associate particle properties (energy and momentum) to waves. They
can also be used the other way round — to associate wave properties (frequency
and wave number) to particles. Moreover, these apply to non-relativistic particles
such as electrons (as well as relativistic photons). This
λ
is known as the de
Broglie wavelength.
Of course, nothing prevents us from assigning arbitrary numbers to our
particles. So an immediate question to ask is — is there any physical significance
to the “frequency” and “wavenumber” of particles? Or maybe particles in fact
are waves?
Recall that the quantization of the Bohr model requires that
L = rp = n~.
Using the relations above, this is equivalent to requiring that
nλ = 2πr.
This is exactly the statement that the circumference of the orbit is an integer
multiple of the wavelength. This is the condition we need for a standing wave
to form on the circumference. This looks promising as an explanation for the
quantization relation.
But in reality, do electrons actually behave like waves? If electrons really are
waves, then they should exhibit the usual behaviour of waves, such as diffraction
and interference.
We can repeat our favorite double-slit experiment on electrons. We have a
sinusoidal wave incident on some barrier with narrow openings as shown:
λ
wave
density of
electrons
δ
At different points, depending on the difference
δ
in path length, we may
have constructive interference (large amplitude) or destructive interference (no
amplitude). In particular, constructive interference occurs if
δ
=
nλ
, and
destructive if δ = (n +
1
2
)λ.
Not only does this experiment allow us to verify if something is a wave. We
can also figure out its wavelength λ by experiment.
Practically, the actual experiment for electrons is slightly more complicated.
Since the wavelength of an electron is rather small, to obtain the diffraction
pattern, we cannot just poke holes in sheets. Instead, we need to use crystals as
our diffraction grating. Nevertheless, this shows that electrons do diffract, and
the wavelength is the de Broglie wavelength.
This also has a conceptual importance. For regular waves, diffraction is
something we can make sense of. However, here we are talking about electrons.
We know that if we fire many many electrons, the distribution will follow the
pattern described above. But what if we just fire a single electron? On average,
it should still follow the distribution. However, for this individual electron, we
cannot know where it will actually land. We can only provide a probability
distribution of where it will end up. In quantum mechanics, everything is
inherently probabilistic.
As we have seen, quantum mechanics is vastly different from classical mechan-
ics. This is unlike special relativity, where we are just making adjustments to
Newtonian mechanics. In fact, in IA Dynamics and Relativity, we just “derived”
special relativity by assuming the principle of relativity and that the speed
of light is independent of the observer. This is not something we can do for
quantum mechanics — what we are going to do is just come up with some theory
and then show (or claim) that they agree with experiment.