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1.3 The situation around 1900
1.3.1 Black body radiation
On of the key problems at the turn of the last century was black body radiation, explained in more detail below. Two of the most successful physical theories of the time
were statistical mechanics as well as Maxwell’s theory of Electromagnetism. Combinding both, people tried to predict the energy distribution of eletromagnetic waves emitted by a body in thermal equilibrium with its environment at temperature T. When we
have an object that perfectly absorbs EM-radiation and that is in thermal equilibrium
with its environment it also emits EM-radiation at different frequencies n. For instance,
when a body is very hot, it mostly emits bright light. When it’s a bit cooler it the peak
radiation may be in the infrared. We can measure the distribution of EM-radiation frequencies and will find something like depicted in Fig. 1.5. The distribution has a peak
frequency at some value that depends on the temperature. Very low and very high frequencies are not present. One of the goals of physics around 1900 was to explain this
type of distribution using classical theories in combination, like Boltzmann’s statistical
physics approach in combination with Maxwell’s theory as mentioned.
1.3.1.1 Electromagnetic waves
Maxwells equation for the electric and magnetic field in vacuum can be written has
∂2t E(r, t) = c2 r2 E(r, t)
∂2t B(r, t) = c2 r2 B(r, t)
where r = ( x, y, z) is the position in 3d space, c is a constant and E and B are electric
and magnetic fields, respectively and r = (∂ x , ∂z , ∂z ) is the spatial vector derivative.
These equations are wave equations like
∂2t y(r, t) = c2 r2 y(r, t)
that have travelling plane waves as solutions
y(r, t) = A cos(kr
wt)
The wave vector k points into the direction in which the wave travels, its magnitude is
related to the wave length
2p
l=
|k|
and
w = 2pn
is the angular frequency (n is the regular frequency measure in Hertz). Frequency and
wave length are related to c according to
c = nl = w/k
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and in the case of Maxwell equations we have
c ⇡ 3 ⇥ 108 m/s.
The success of Maxwell’s theory clarified that light, e.g. is an electro-magnetic wave in
vacuum. The cos wave is not the only solution that is a plane wave. Sine waves are also
solutions and any combination of sine and cosine wave. For the sake of argument, it’s
ok to just use the cosine here.
1.3.1.2 Statistical mechanics
According to Boltzmann theory, a body in thermal equlibrium with N
1 degrees of
freedom and at temperature T will distribute its thermal energy equally among all its
degrees of freedom, each degree of freedom getting an energy of E = k B T where
k B = 1.3806488 ⇥ 10
23
m2 kg/Ks2
is the Boltzmann constant. A little bit of stastistical physics argument will tell us that
every wave emitted by the black body will aquire an equal amount of energy proportional to E = k B T. If we want to know what the frequency distribution is, i.e. what is
the concentration of EM waves in an interval [w, w + Dw ] this is equivalent to asking
how many k vector fullfill
[|k|, |k| + dk].
Because k is a three-dimensional vector the number of waves in the inverval above
scales as
u(|k|) ⇠ k2 dk
and because |k| µ w we would expect an energy distribution
u(w ) ⇠ E w 2 dw
⇠ k B Tw 2 dw
This is called the Rayleigh-Jeans law. It predicts that more and more energy is emitted at
higher and higher frequencies. This law, however, is inconsistent with observations and
is only true for low frequencies. Sometimes this is called the ultraviolett catastrophy
because it would imply an infinite amount of energy emitted by such a body.
1.3.1.3 Planck
Planck approached the problem heuristically. This is often a good start. He proposed
that energy bits k B T are NOT distributed equally among the possible degrees of freedom. Instead he assumed that the energy is given by
E(n, T ) =
hn
e
12
hn/k B T
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Figure 1.6: Photons as wave pakets with a frequency w and particle like properties, e.g.
localized in space.
This function has the following interesting behavior. For small frequencies hn ⌧ k B T
we have E(n, T ) = k B T consistern with the classical law. However when hn
kB T
then E(n, T ) ! 0. So, physically, it means that frequencies that are larger than k B T/h
are not excited and do not aquired energy and are thus not emitted. As such the law
was a phenomenological law with a fitting parameter h. Planck used the letter h to
symbolize “Hilfskonstante”. When fitting the new law to the observed data one finds
an excellent fit and a numerical value for h:
h ⇡ 6.5 ⇥ 10
34
Js
This fitting parameter had the unit of an action.
Later Planck showed that one could explain this law if one makes the physical assumption that EM radiation is quantized and comes in energy quanta that depend on
the frequency
E0 (n) = hn = h̄w
In other words light with a certain frequency comes in packets of quantized energy
given by the above equation. This revolutionized physics. It seemed that classical,
Maxwell EM theory predicted radiation to be waves and wave properties like frequency
and have length were correct concepts. However, according to classical theory the enery in a wave is related to the amplitude of the wave and quantum theory shows that
the energy of a quantum of EM radiation is related to the frequency. This was something quite different. The image people probably had in mind after this discovery is
that light was a phenomenon that looked a bit like that in Fig. 1.6.
1.3.2 Photo Electric Effect
A few years later, in 1905, Einstein published 4 papers each one of them revolutionized
physics. One paper was about Brownian Motion and a theory that proved the atomistic nature of matter and molecular collission causing the observed random movement
patterns in Brownian motion. A second paper called “zur Elektrodynamik bewegter
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Körper” Einstein introduced the special theory of relativity. In a sequel to this paper
he discussed the consequences of this theory, and his famous formula E = mc2 . This
paper is called “Ist die Trägheit eines Körpers von seinem Energiegehalt anhängig?”. In
a forth paper Einstein explained the photo-electric effect, which is aside from Planck’s
paper the second paper that can be considered the foundation of quantum mechanics.
Ironically Einstein didn’t like the implications of QM that evolved in the following two
decades. In particular the probabilistic aspect of quantum mechanics.
The photo-electric effect is the following. If one shines EM radiation onto a thin metal
plate, electrons will be emitted from the plate. IF one measures this carefully one finds
that the kinetic energy of the emitted electron is related to the frequency of the EM
radition that hits the metal plate
1
me v2e = h̄w
2
W0
h̄w > W0
If the frequency exceeds a minimum value, such that h̄w > W0 the kinetic energy increases with the frequency of the EM radiation. Einstein was able to explain this based
on the assumption that the incoming light came in as energy quanta, photons, whose
energy was h̄w. Then Einstein remembered some of his earlier work. According to the
theory of relativity the energy of a particle is given by
⇣
⌘1/2
E = p2 c2 + m2 c4
In this equation p is the momentum, c is the speed of light and m is the restmass of
a particle. The speed of a moving body can be expressed in terms of its energy and
restmass.
pc2
v=
1/2
( p2 m2 + m2 c4 )
This equations imply that for bodies at rest we have p = 0 and thus the famous
E = mc2 .
But what happens if we let v = c for instance what are these relations for photons that
move at the speed of light? Then it follows that
m=0
which in turn implies
E = pc
Recall that for EM radition we also have
E = wh̄
which also implies that
p=
wh̄
= |k|h̄
c
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Figure 1.7: Photons as wave pakets with a frequency w and particle like properties, e.g.
localized in space.
So what Einstein derived was a relation of classical concepts like energy and momentum and related them to wave properties of classical EM waves
✓
◆
E = h̄w
p = h̄k
Energy is proportional to the frequency and momentum proportional to the wavevector.
1.3.3 Bohr-Sommerfeld Atom
Here’s on last example of the quantization of action or energy introduced in the BohrSommerfeld theory of the hydrogen atom. It was known at the time that atoms had
internal structure, e.g. that the hydrogen atom consisted of a positively charged heavy
proton and a negatively charged electron that circulated the central proton. However,
the classical idea that the electron moved on circular paths around the atom’s core implied that the electron, performing an accelerated motion, would have to emit electromagnetic radiation and energy which would slow the electron down which would
imply that the electron would spiral towards the positive proton and collapse. Bohr
and Sommerfeld proposed that the electron’s orbits must be quantized at certain energy levels. When the electron is energetically excited it will move to a higher energy
level and when it collapses to a lower level it would emit a quantized bit of energy in
form of a photon. Once in the lowest energy level it cannot move any lower and is stuck
to the minimal energy.
1.3.4 Double Slit experiments
A bunch of experiments were designed to test the idea whether for example light is a
wave or a particle. A class of problems are the so called double slit experiments. Where
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Figure 1.8: The classical light experiment that showed that light behaves like waves because of the observed interference patterns when two circular waves that
emerge from closely spaced slits emerge. When one slit is blocked, the interference pattern disappears.
a source emits light, the light hits a wall with two slits, and at each slit a circular wave
emerges that propages forward. The two waves interact and generate an interference
pattern on a screen that measures the intensity. The interference can be calculated, and
we will do that later, but essentially what happens ist that whem the wave maxima and
minima of each wave meet the generate a larger maximum or minimum. On the other
hand in locations where the maximum of one wave meets the minimum of another,
they annihilate and cancel. On the screen this type of interference generates a sequenct
of peaks and minima that are spaced at a certain distance, see Fig. 1.8. The interference
pattern disappears, when one slit is closed and only one circular wave produces an
intensity distribution on the screen. If we call this intensity I1 and the intensity that we
would get if the other slit is closed by I2 and the interference pattern intensity by I12 we
see that
I12 6= I1 + I2 .
This is what we would get if we were to shoot particles through the holes (in a thought
experiment). A particle would either go through one hole, maybe bounce off it, and
then hit the screen. Many particles would generate a particle intensity which would be
the sum of I1 and I2 , see. Fig.1.9.
Because this the original light experiments clearly proved the wave nature of light
and electro-magnetic radiation.
1.3.4.1 The problem....
But what happens if we turn down the intensity of the lamp that produces the light.
The interference pattern will also go down in intensity but the pattern will not change.
Now let’s assume we can have a screen with photo-detectors that can measure very
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Figure 1.9: If particles are shot at the board with two slits and either got through one or
the other whole, no interference is observed, as expected.
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tiny energy bits. What we see in such an experiment is that light actually hits the screen
in particle fashion for instance every photon leaves a little mark, clear evidence that
photons are particles. If we now wait long enough such that very many photons leave
marks on the screen eventually an intensity pattern will emerge and this pattern is an
interference pattern!!! This is cleary very strange.
If photons are particles, they should go thtrough only one slit. But if that’s the case,
there should not be an interference pattern! Something is very strange. It seems like the
photon is a particle that behaves as a wave as it goes though BOTH slits and the suddenly behaves like a particle when it hits the screen. This is actually the case. Noone
knows why. If you actually try to figure out which slit the photon went through you’ll
find that as soon as you try to measure this, the interference patterns disappears. You
cannot measure this without disturbing the system such that the effect is gone. Therefore the question: “Which slit did the photon go through” is meaningless in this experiment.
1.3.5 Matter waves
Ok, you might think, photons, light, electromagnetic waves, all very strange stuff anyway. Let’s look at stuff we can connect to more easily. Say electrons. Those are nice
and massive point particles that are always somewhere, e.g. they have a position r in
space. Let’s shoot these at a wall with two slits. So, if you prepare the slits right, you
will actually also see an interference pattern. Which means that even massive particles
behave like wave in the sense that one observes interference of the electron with itself as
“it goes through both slits”. You can, in principle do this with any body. However, the
more massive the body, the more closely spaced the intensity peaks in the interference
pattern to such a degree that you cannot resolve them.
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