**Unformatted text preview: **23. Lecture 23
23.1 De Broglie waves
We have seen that light appears to be a wave but, under careful examination, behaves
as if composed of particles. Nevertheless all the wave properties of interference and
diﬀraction are still valid. In view of this “particle-wave” duality, de Broglie proposed
that particles should also behave as waves. The frequency and wave-length are given
by the same relation as for photons:
Eel. = ω , Pel. = h
λ (23.1) In the case of the photon we were more familiar with ω , λ and computed E, P . In
the case of the electron we are more familiar with E, P and compute the associated
angular frequency ω and wave-length λ. Although these relations are the same as for
the photon, it should be emphasized that the relation between energy and momentum
is diﬀerent:
2
Pph.
Pel.
Eel. =
,
Eph. =
(23.2)
2m
c
This was a very strange proposal but it received spectacular experimental conﬁrmation
when an experiment analogous to the two slit experiment showed an interference pattern
for electrons exactly the same as for light. In fact any experiment of diﬀraction and
interference of electrons can be analyzed in the same way as for light. We only need to
compute the wave-length using λ = h .
p
Now we want to see how this helps us in the hydrogen atom. First think of a
string in a violin or guitar. It is well known that such a string only vibrates at speciﬁc
frequencies which is why is used in a musical instrument. In fact the lowest frequency
is such that the length of the string is half a wave-length. We say that the vibrations
of a string have a discrete spectrum of frequencies. The proposal is that the discrete
spectrum of the hydrogen atom is due precisely to the wave-like behavior of the electron.
More quantitatively, we require the the size of the orbit is an integer multiple of the
wave length of the electron. Namely in each orbit an integer number of wave-lengths
ﬁt:
2π r = nλ, n = 1, 2 , 3 , 4 , . . . (23.3) Now we use the Newtonian relation
v2
e2
¯
m =2
r
r ⇒ – 148 – v= e2
¯
mr (23.4) to compute the velocity of the electron and thus the momentum and wave-length:
me 2
¯
2π
r
p = mv =
, λ=
= 2π
(23.5)
r
p
me 2
¯
Using now eq.(23.3), namely 2π r = nλ we get:
r
2 π r = 2π
n⇒
me 2
¯ r= n 2 2
me 2
¯ (23.6) We indeed ﬁnd a discrete set of orbits!. The energy is computed by replacing the
velocity from eq.(23.4) into
1 2 e2
¯
e2
¯
1 me 4
¯
1
E = mv −
= − = − 2 2 = −13.6eV 2
2
r
r
n
n (23.7) where we replaced the known values of e, m and . Now we have the energy of all
¯
possible orbits ﬁtting precisely eq.(22.4)!. This is a good check that the discreteness of
the hydrogen spectrum is due to the wave-like nature of the electron.
23.2 Other results and applications
We basically have ﬁnished what we wanted to discuss about quantum mechanics. For
the last hundred years we have been improving our understanding of how quantum
mechanics applies to diﬀerent physical phenomena. For example one can understand
the periodic table of the elements, chemistry, material science, particle physics, etc.
as ﬁelds where quantum mechanics is of paramount importance. However it has only
been recently that experimental progress has been enough that one can start thinking
of practical application in which controlling the state of a quantum system can be used
to our advantage. One such possibility would be to use quantum mechanics to create
better computers, a ﬁeld which is still in its infancy. For illustration we will discuss
two devices which are practical applications of quantum mechanics: lasers and atomic
clocks.
A laser creates a beam of coherent light which is well collimated, namely does not
spread much. In quantum mechanical terms we generate a large number of photons
all in the same state. To understand how that is achieved we need another important
property: if an atom is in an excited state it can transition to a lower energy state by
emitting a photon with the corresponding energy. However, if there is already a number
of photons of that energy present, the probability of decay is enhanced for the photon
to go to the same state in which those photons are. A laser utilizes this by having a
material whose atoms are excited by electric discharges, light, electric ﬁelds, etc. In
ﬁg. 125 we see an example, the active zone, where the material is excited, is situated – 149 – Figure 123: The Bohr atom explained through the de Broglie hypothesis. An integer number
of wave-lengths ﬁt into each orbit. between two mirrors. A decay produces a photon. The other atoms, when they decay
prefer to emit the photon in the same state as the one already present. As more and
more photons accumulate on a state, more likely is for others to join. In this manner
light is ampliﬁed which gives its name to the device: Light Ampliﬁcation by Stimulated
Emission of Radiation. The words stimulated emission of radiation makes reference to
the idea that a photon already present stimulates the atoms to emit radiation in the
same state. The mirrors can be thought as determining standing waves similarly as in
a sound waves in a pipe. Alternatively we can think that a beam of light goes back
and forth between them being ampliﬁed all the time. To extract the energy, one of
the mirrors is partially transparent. It is clear that only photons moving along the
cylinder, perpendicular to the mirrors are ampliﬁed. The others are lost since they are
not reﬂected back into the active zone. Furthermore, all the photons are in the same
state, so they are in phase and the light is coherent. This should be contrasted with – 150 – the usual thermal emission in which the same medium is heated and each atom decays
independently of the others. In that case we have a large number of sources all emitting
independently. In the laser they all contribute to the same wave generating a highly
coherent pulse.
The same idea is used with microwaves. A resonant cavity as it is called contains
atoms and an alternating frequency is applied. When the external frequency agrees
with the energy of an atomic transition a resonance occurs that can be easily detected.
Since the frequency of the atomic transition is very precise, we can create an alternating
voltage with a very precise frequency or period. But a periodic signal with very precise
period is exactly what we need to build a clock. This type of clocks are called atomic
clocks and are the most precise clocks available at the moment. They are used in
numerous applications, for example atomic clocks aboard the GPS satellites produce
timig signals that allow us to determine our position by getting timing signals from
diﬀerent satellites.
Finally, another phenomenon that we wanted to discuss is Compton scattering.
When photoelectric eﬀect occurs but the energy of the photon is much larger than the
binding energy of the electron, the electron can be considered as a free particle. In such
situation, the photon cannot be absorbed and is deﬂected. If the deﬂection angle is θ,
see ﬁg.124 then, conservation of energy and momentum give the relation between the
wave-length λ of the incoming photon and λ the wave-length of the outgoing photon:
λ − λ = h
(1 − cos θ)
mc (23.8) Here, m is the mass of the electron. In fact to obtain this result you need to use the
relativistic relation between momentum an energy E = m2 c4 + p2 c2 . We leave this
as an exercise for anyone who is interested. – 151 – Figure 124: Compton scattering is analogous to the photoelectric eﬀect but the electron
either is not bound or the binding energy is small compared to the energy of the photon. The
photon deﬂected by an angle θ and changes its wave-length. Figure 125: Schematic of a laser. A beam of light goes back and forth in an active medium
which ampliﬁes light. One of the mirrors is partially transparent and lets the laser light go
out. – 152 – ...

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