0Introduction
IB Quantum Mechanics
0.1 Light quanta
In quantum mechanics, light (or electromagnetic waves) consists of quanta called
photons. We can think of them as waves that come in discrete “packets” that
behave like particles.
In particular, photons behave like particles with energy
E
=
hν
=
~ω
, where
ν
is the frequency and
ω
= 2
πν
is the angular frequency. However, we usually
don’t care about ν and just call ω the frequency.
Similarly, the momentum is given by
p
=
h/λ
=
~k
, where
λ
is the wavelength
and k = 2π/λ is the wave number.
For electromagnetic waves, the speed is
c
=
ω/k
=
νλ
. This is consistent
with the fact that photons are massless particles, since we have
E = cp,
as entailed by special relativity.
Historically, the concept of quanta was introduced by Planck. At that time,
the classical laws of physics did not accurately depict how the spectrum of black-
body radiation will behave. In particular, it predicted that a black body will emit
an infinite amount of energy through radiation, which is clearly nonsense. Using
the concept of quanta and the energy-frequency relation, Planck was to derive the
spectrum of “black-body” radiation that is consistent with experimental results.
However, they were not yet sure that light indeed come in quanta physically. It
could have just been a mathematical trick to derive the desired result.
The physical reality of photons was clarified by Einstein in explaining the
photo-electric effect.
When we shine some light (or electromagnetic radiation
γ
) of frequency
ω
onto certain metals, this can cause an emission of electrons (
e
). We can measure
the maximum kinetic energy K of these electrons.
γ
e
Experiments show that
(i)
The number of electrons emitted depends on the intensity (brightness) of
the light, but not the frequency.
(ii)
The kinetic energy
K
depends only (linearly) on the frequency but not the
intensity.
(iii) For ω < ω
0
(for some critical value ω
0
), no electrons are emitted at all.
This is hard to understand classically, but is exactly as expected if each electron
emitted is due to the impact with a single photon. If
W
is the energy required
to liberate an electron, then we would expect
K
=
~ω − W
by the conservation
of energy. We will have no emission if ω < ω
0
= W/~.