Lecture 4
25
Oct 16, 2002
4
Review of Quantum Mechanics
Quantum Mechanics is truly a 20
th
century development, and signified the start of “Modern Physics”.
Quantum mechanics (QM) is the basis for understanding diverse phenomena as electric conduction,
photo-electrics, radiation of light and heat, transistors, lasers, etc.
. Without QM we would have no mi-
croelectronics, no TL lighting, no lasers or IR data transmissions, no microwave cookers, and so on.
Without QM we would not understand the Sun or the age of the earth and the universe. The most im-
portant tenet of Quantum Mechanics is that particles should be ascribed a wave character, and we will
therefore start this discussion with a review of classical waves.
A wave is the variation and propagation of a physical quantity through vacuum of a physical medium.
In particular, waves
transport energy
. Any wave, even when it is non-repetitive, can be represented as
a sum of sine and cosine term of different frequencies and amplitudes (Fourier theorem). In general,
that sum has an infinite number of terms, but often can be truncated after a finite number of terms.
Waves have a few properties that set them apart from particles:
1.
Superposition principle
: waves add up linearly in amplitude wherever they overlap in space-
time. E.g. density fluctuations from sound waves add up when they meet (principle of kidney
stone fragmentation by focused ultra sound). The superposition principle is responsible for the
prime characteristic of waves:
interference and diffraction
. The superposition principle follows
naturally from the property from addition of the vector and scalar quantities that are propagated
by the waves.
2. The simplest wave is characterized by a single
propagation speed
v
, a repetition length (
wave-
length
λ
), and a repetition
period
T
or
frequency
f
= 1/
T
. The relationship
=
/
T
=
f
holds.
The Fourier theorem tells us that any complicated waveform may be composed from a (infinite)
set of different frequency waves – which may propagate with different velocities in a particular
medium (dispersion).
3. The energy transported in a wave is proportional to the
square of the amplitude
. Thus, the in-
tensity of a light wave is proportional to the electric field strength squared, the sound energy in
a sound wave to the density amplitude squared, etc.
.
Clearly, waves do have momentum and energy: just watch a storm on the ocean or a tornado do its de-
structive work. The momentum of light waves is easy to find: because the light has zero mass (after all
it goes at the absolute maximum speed c in vacuum!), it follows that
E
=
pc
, or
p
=
E
/
c
for massless
objects, see equation (I.33). This “momentum of light” is the driving agent for the solar “wind”, and
the principle behind sailing space probes.