Overview of electrodynamics and quantum mechanics

1

Abstract
In these notes I present an overview of electrodynamics and quantum mechanics
which (together with statistical mechanics) are the foundation of much of today’s
technology: electronics, chemistry, communication, optics, etc.

CONTENTS

1 Introduction: the Unity of Science 4
2 Quantum Mechanics 5
2.1 The puzzles of matter and radiation 6
2.1.1 Planck’s Black-body radiation 8
2.1.2 The photo-electric effect 12
2.1.3 Bohr’s atom 14
2.1.4 Adsorption, stimulated emission and the laser 16
2.2 Quantum Mechanical formalism 19
2.3 Simple QM systems 21
2.3.1 The chiral amonia molecule 21
2.3.2 The amonia molecule in a constant electric field 24
2.3.3 The amonia maser and atomic clocks 26
2.3.4 The energy spectrum of aromatic molecules 28
2.3.5 Conduction bands in solids 29
2.4 Momentum and space operators 31
2.4.1 Heisenberg uncertainty principle 34
2.5 Schroedinger’s equation 35
2.5.1 Diffraction of free particles 35
2.5.2 Quantum interference observed with C60 37
2.5.3 QM tunneling and the Scanning Tunneling Micro-
scope 38
2.6 The correspondance principle 41
2.6.1 Gauge invariance and the Aharonov-Bohm effect 42
2.7 Dirac’s equation: antiparticles and spin 44
2.7.1 Angular momentum and spin 49
2.8 The Hydrogen atom and electronic orbitals 52
2.8.1 Spin-orbit coupling 54
2.8.2 Many electron systems 55
2.8.3 The periodic table 56
2.9 The chemical bond 59
¨
2.9.1 Huckel’s molecular orbital theory 63
2.9.2 Molecular vibrational spectrum 65
2.9.3 Molecular rotational spectrum 67
2.10 Time independent perturbation theory 68
2.10.1 The polarizability of atoms in an electric field 71
2.10.2 Atom in a constant magnetic field: the Zeeman
effect 73
2.10.3 Degenerate eigenstates 76
2.11 Time dependent perturbation theory 77

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1
INTRODUCTION: THE UNITY OF SCIENCE

4

2
QUANTUM MECHANICS

Quantum mechanics (QM) is a theory of matter and its interactions with force fields
(here we will only care about electromagnetic fields). While classical mechanics and
electromagnetism are intuitive (one has a direct experience of gravitation, light, elec-
tricity, magnetism, etc.) quantum mechanics is not. The description of matter that arises
from the QM formalism is totally at odds with our daily experience: particles can pass
through walls, can be at two different places and in different states at the same time,
can behave as waves and interfere with each other. Worse, QM is a non-deterministic
description of reality: it only predicts the probability of observing events. This aspect
deeply disturbed Einstein who could not accept that QM was the correct final descrip-
tion of reality (as he famously quipped: ”God does not play dice”). He and many others
came up with alternative descriptions of QM introducing hidden variables (unknow-
able to the observer) to account for its non-deterministic aspects. But in 1964 John
Bell showed that if hidden variables existed some measurements would satisfy certain
inequalities. The experiments performed by Alain Aspect and his collaborators in the
1970’s showed that the Bell inequalities were violated as predicted by QM, but not by
the hidden variable theories thereby falsifying them.
Yet, for all its technical prowess Aspect’s experiment was only addressing a philo-
sophical issue concerning the foundations and interpretation of QM. The theory itself
had been amply vindicated earlier by its enormous predictive power: QM explains the
stability of atoms, their spectra, the origin of the star’s energy and of the elements and
their properties, the nature of the chemical bond, the origin of magnetism, conductivity,
superconductivity and superfluidity, the behaviour of semi-conductors and lasers, etc.,
etc.. All of today’s micro-electronic industry is derived from applications of QM (tran-
sistors, diodes, integrated circuits, etc.), the development of the chemical industry is a
result of the QM understanding of the chemical bond and the nuclear industry would of
course been impossible without an understanding of the nucleus and the nuclear forces
that QM provided.
So, for all its weirdness Quantum Mechanics is the most successfull explanation of
the World ever proposed by Mankind. It beats Platonicism, the Uppanishads, Kabbalah,
Scholasticism, etc., yet is non-intuitive and cannot be understood except by following
its mathematical formalism to its logical conclusions. ”The great book of Nature is
written in the language of mathematics”, Galileo’s quip is truer for QM more than for
any other scientific theory. More recently one of the founder’s of QM, Eugene Wigner
wrote in an article entitled ”the unreasonable effectiveness of mathematics in the natural
sciences”, that ”the miracle of the appropriateness of the language of mathematics for
the formulation of the laws of physics is a wonderful gift which we neither understand
nor deserve”.

5

6 QUANTUM MECHANICS

It is with this mind set that I would like you to approach the study of QM. Like an
apprentice sorcerer learning the tricks of his master without fully understanding them,
yet always at awe confronting their power. As we have done with electromagnetism,
we will approach QM by following as far as we can the historical narrative. We will
see why the radiation of a Black Body was such a puzzle that it prompted Max Plank to
introduce the idea that energy was quantized; why the stability of atoms and their spectra
prompted Bohr, Sommereld and others to suggest that the energy levels of atoms were
also quantized; how the idea that particles could also have wavelike behaviour was first
suggested by de Broglie and brought to fruition by Schroedinger, Heisenberg and Dirac.
And how from then on, QM revolutionized the understanding of matter, the chemical
bond, magnetism, conduction, etc.

2.1 The puzzles of matter and radiation
At the end of the 19th century, scientists disposed of a very succesfull theoretical frame-
work that could explain many of the problems known at that time and which was tech-
nologically revolutionary. Newtonian mechanics was amazingly successful in predict-
ing the motion of celestial bodies. Its most striking success was the prediction by Le
Verrier in 1846 of the existence of the planet Neptune. Analysing some anomalies in
the motion of Uranus, he predicted Neptune’s precise location in the sky, a prediction
which was immediately confirmed by German astronomers. In 1861, Maxwell unified
electricity, magnetism and optics opening the area of electrical appliances and wireless
communication: Edison invented the light bulb in 1879 and founded ”General Electric”
in 1892 while Marconi established the ”Marconi Wireless Telegraph Company” in 1897.
Finally, thermodynamics was sustaining the advance of the industrial revolution as ther-
mal engines were driving industry and railways. In spite of terrible social inequalities
(as described by C.Dickens, E.Zola and others) this was a time of peace, prosperity and
optimism, illustrated by the nascent Impressionist movement.
Yet, many fondamental scientific questions remained unsolved and paradoxical. The
chemical properties of the various elements were not understood. The periodicity of
these properties as a function of the mass of the elements as determined in 1869 by
Mendeleev in his famous Periodic Table of the Elements was a mystery. Nonetheless
on the basis of his ad-hoc classification Mendeleev predicted the existence of two new
elements, Gallium and Germanium, which were duly discovered in 1875 and 1886 and
are essential in today’s semiconductor industry! The existence of atoms (indivisible
particles of matter characteristic of each element) postulated by Dalton to explain the
properties of molecules was not generally accepted. Because of the successful applica-
tions of continuum mechanics (in the design of bridges, buildings (e.g. the Eiffel tower),
etc.) and fluid dynamics (in explaining the tides, water waves, etc.), matter was gener-
ally believed to be some sort of continuum akin to a gel not a swarm of particles. It was
Einstein who in 1905 finally managed to convince the scientific world of the existence
of atoms and molecules by showing that the erratic motion exhibited by dust particles
on the surface of water (first observed by the botanist R.Brown in 1827) was due to the
shocks of the water molecules. The continuum pre-conception also sustained the inter-
pretation of electromagnetic waves. Since all known waves at the times were observed

THE PUZZLES OF MATTER AND RADIATION 7

F. 2.1. The emission spectrum in the visible range for a few elements. Notice the fine
spectral lines and the different spectral characteristics for the different elements.
This was one of the puzzles that QM solved.

to propagate in a continuum medium (such as water, air, etc.) at a velocity v = κ/ρ
(where κ is the compressibility and ρ the density of the medium), the electromagnetic
waves predicted by Maxwell and discovered by Hertz were assumed to propagate in
some continuum medium: the ether which properties determined their velocity. How-
ever all attempts to detect the ”wind” of ether resulting from the motion of the Earth in
that medium proved negative. This prompted Einstein to formulate his theory of rela-
tivity which postulated the constancy of the speed of light and got rid of any notion of
ether, see Appendix.
Then there were questions related to the emission and absorption spectra of elements
that exhibited discrete lines rather than an undifferentiated continuum of absorption or
emission, as was the case for sound and water waves and as we have seen for scattered
and refracted light. Not only did the elements exhibit specific adsorption lines but those
differed from element to element. These observations did not fit with the then prevailing
conception of matter as a continuum.
Finally there was the problem of the radiation from a Black Body, a material (such as
to a good approximation graphite) which adsorbs radiation uniformly at all frequencies
and which can therefore also emit radiation uniformly at all frequencies. Notice how-
ever that many bodies (e.g. the elements just mentioned) are not black-bodies as they
adsorb/emit only at certain frequencies. At a given temperature, the radiation inside a
black body cavity is at thermal equilibrium with the walls of the cavity that absorb and
re-emit it. When computing the electromagnetic radiation energy emitted by a black-
body at a given temperature, one found its energy to diverge because the number of
modes at high frequencies diverged. This was not only absurd but also in contradiction

8 QUANTUM MECHANICS

with the experiments which studied the energy distribution inside the cavity by measur-
ing the energy leaking out of the cavity (for example through a small hole) as a function
of frequency.
To see how this comes about, consider a square cavity (an oven) of size a at tempera-
ture T . As we have seen, a body at a given temperature emits electromagnetic radiation,
see Fig.??. Imagine that the walls of the cavity are made of small oscillators emitting ra-
diation at frequency ω (like the oscillators we considered when studying the frequency
dependence of the refraction index). Stationary waves of the form sin k · r cos ωt will
be present in the cavity if its walls are reflecting (though for the energy to equilibrate
between oscillators the walls cannot be 100% reflective). To satisfy the boundary con-
ditions on the walls we shall require that: k x = πl/a, ky = πm/a, kz = πn/a (n, m, l ≥ 0).
Hence we have: k2 = (l2 + m2 + n2 )(π/a)2 ≡ (πρ/a)2 . The number of modes dNlnm with
such wavelength is:

4πρ2 ka adk k dk
dNlnm = 2 dρ= π( )2 = 8πa3 ( )2
8 π π 2π 2π
The factor 2 results from the two possible polarizations of the fields, while the factor
(4πρ2 /8)dρ counts the number of modes in a shell in the positive octant (n, m, l ≥ 0).
According to the equipartition theorem of statistical mechanics (see below) the average
energy of each oscillatory mode is: < E >= kB T . Using the relations: k = ω/c ≡ 2πν/c
(ν like f is the frequency), the energy density of the emitted radiation du = dNlnm <
E > /a3 becomes:
8πkB T 2
du = ν dν (2.1)
c3
hence the total energy, the integral of the energy density over all frequencies, diverges as
ν3 . This divergence became known as the Jeans’ (or ultra-violet) catastrophy. While the
data agreed with that formula at low frequencies, it differed at high frequencies (small
wavelengths).

2.1.1 Planck’s Black-body radiation
Rather than questioning the equipartition theorem which was verified in other contexts
or the possibility of atoms to emit light of arbitrarily high frequencies, Planck suggested
in 1900 that light was emitted by the cavity walls in very small discrete quantities,
quantas of energy: e = hν, where h, the Planck constant is:

h = 6.626 10−27 erg sec = 4.135 10−15 eV sec

, so that light of energy En is made up of n quantas: En = nhν. In that case the average
energy emitted at frequency ν is the sum over all possible energies En , weighted by their
Boltzmann probability (see below the Chapter on Statistical Mechanics):

e−En /kB T
P(En ) = −En /kB T
ne

So that the average energy is:


hν
< E >= En P(En ) =
n
ehν/kB T − 1

When hν kB T , one recovers the previous result: < E >= kB T , however at large
emission frequencies the average energy decays as < E >∼ hν exp(−hν/kB T ). Planck
therefore suggested to modify the previous result, Eq.2.1 to yield, see Fig.2.2:

8πhν3 1
ρ(ν) ≡ du/dν = (2.2)
c3 ehν/kB T − 1
Where ρ(ν) is known as the spectral density of radiation. Identifying the smallest quanta
of energy with a light particle (a photon of energy hν), Eq.2.2 states that the density of
photons in a Black-body is:

dN p ρ(ν) 8πν2 1
= = 3 hν/k T (2.3)
dν hν c e B −1
Notice that the total energy density in the cavity is now ﬁnite:
∞
8π(kB T )4 ∞
x3 8π5 k4 4
Utot = ρ(ν)dν = dx = B
T (2.4)
0 h3 c3 0 ex − 1 15h3 c3
∞
Where we used the equality: 0 dx x3 /(e x − 1) = π4 /15. The total radiated power per
unit area trough a small hole in the cavity becomes,
1
cUtot cUtot
Irad = k · ndΩ =
ˆ ˆ cos θd(cos θ) = σS B T 4 (2.5)
4π 2 0

which is known as Stefan’s law and where the Stefan-Boltzmann constant:
2π5 k4
σS B = B
= 5.67 10−5 erg sec−1 cm−2 ◦ K−4 = 5.67 10−8 W m−2 ◦ K−4
15h3 c2
Therefore by measuring the total intensity of the radiation leaking out from a cavity
(for example an oven) one can measure the temperature of that cavity. One can test
the validity of Planck’s law (actually how close to a black-body the cavity really is)
by measuring the dependence of the intensity on the radiation wavelength. From the
wavelength λmax at which the intensity is maximal an other estimate of the temperature
can be deduced: kB T hc/5λmax . For example at 300K (which corresponds to a thermal
energy kB T 25 meV), the maximum of emission is at λmax ∼ 10µm. The thermal cam-
eras that visualize humans and warm animals (see Fig.??) must therefore be sensitive to
far-infrared light.

The temperature of the Sun and the Earth
The Sun is to a very good approximation a black body, see Fig.2.2. The radiations
emitted by the fusion reactions occuring at its core (at temperature of 13 106 K) are

10 QUANTUM MECHANICS

F. 2.2. The emission spectra of the sun and the universe. The sun emission spectra is
pretty well fit by the spectrum of a lack body at 5770K, however notice the existence
of some specific adsorption bands in the visible and UV spectrum. The universe on
the other hand presents a spectrum that is perfectly matched by a black-body at
2.726K.
at thermal equilibrium with the reacting nuclear particles and diffuse out to the Sun’s
surface which is much cooler. By fitting the spectrum of the sunlight to Planck’s for-
mula one can determine the Sun’s surface temperature: T S = 5770K. The total power
generated at the Sun’s core and emitted at its surface is:

PS = 4πR2 σS B T S = 3.85 1026 W
S
4

where RS = 6.96 108 m is the Sun’s radius. Since the radius of the Sun’s core is es-
timated to be ∼ RS /5 the volume of the core is: Vcore = 1.13 1025 m3 and the average
power per unit volume generated in the Sun’s core is: PS /Vcore = 34 W/m3 . This is less
than the power generated by our body to keep warm!!
There are a few ways to verify that. Let us assume an average daily calory intake of
3000kcal 1.2 107 J, which comes to a power consumption of 150W. Approximating
a man as a cylinder of height L = 2m and radius r = 0.2m, the power consumption per
unit volume is 600 W/m3 of which about half goes to metabolic activity. Alternatively
one can use Stefan’s law to estimate the losses between a body at 37◦ C (T b = 310◦ K)
and an environment at 27◦ C (T e = 300◦ K) (this is a crude estimate since other effects
such as perspiration regulate our temperature): ∆I = σS B (T b − T e ) 64W/m2 which
4 4

yields a power per unit volume 640W/m3 . From these consistent estimates we de-
duce that our power consumption per unit volume is much larger than the Sun’s!! What
makes the Sun so bright and hot is its huge mass, not its rather inefficient thermonuclear
reactions.

Let us now estimate the temperature of the Earth T E resulting from its adsorption of


the Sun’s radiation and its own radiation at T E . The sunlight impinging on the Earth at
a distance from the Sun RS E = 1.496 1011 m has an intensity:

IE = PS /4πR2 E = (RS /RS E )2 σS B T S = 1.37 kW/m2
S
4

Of that radiation a fraction (known as the Earth’s albedo) α ∼30% is reflected, mostly
by the clouds, snow and ice-caps. The Sun radiation power arriving at the surface of the
Earth is thus about 1 kW/m2 . It is an important number to remember when designing
solar energy plants: its sets the maximal power per unit area available from the Sun.
Notice that by measuring the radiation arriving on Earth and the angle sustained by
the Sun: θS = (RS /RS E ) one can also get an estimate of the Sun’s temperature: T S =
(IE /σS B θS )1/4 . The energy absorbed by the Earth heats it and is reradiated (to a good
2

approximation) like a black body at temperature T E . We can compute the temperature
of the Earth by a simple energy balance. At steady-state the energy radiated is equal to
the energy absorbed:
σS B T E 4πR2 = (1 − α)IE πR2
4
E E

From which we get T E = ((1 − α)IE /4σS B )1/4 = 255K = -18C. The Earth is actually
slightly warmer because of the green house effect that reflects part of the emitted energy
back to Earth.

The Universe as a perfect black-body
While it is difficult to design a perfect black-body, since as we shall see below
bound electrons adsorb at their resonance frequency (as is the case for the Sun’s spec-
trum for example), the Universe as a whole turned out to be the best known example
of a black-body, see Fig.2.2. The Universe is bathed in a uniform radiation field of
very low frequency whose spectral distribution is perfectly matched by a black-body
at 2.726K. This phenomena was predicted by George Gamow in 1948 and observed
serendipitously by Arno Penzias and Robert Wilson in 1964 when measuring the noise
of a microwave antenna they had built. It was higher than they had expected as they
were actually detecting the 3K radiation of the Universe. This background radiation is
the most striking evidence for the existence of the Big-Bang. According to this scenario,
the Universe began as a big explosion of matter and radiation. At the beginning light
and matter interacted continuously and were in thermal equilibrium (as they are in the
Sun’s core). But then as the Universe expanded it cooled. When it reached a temper-
ature of ∼ 3000K Hydrogen atoms started to form that could not absorb non-resonant
light: radiation decoupled from matter. At that point the radiation spectrum was that of a
black body at the temperature of decoupling. It is the relics of that original radiation that
we are observing today as an isotropic cosmic background radiation. Let us see why it
exhibits a black-body spectrum at a temperature of 2.7K.
Since once hydrogen atoms formed radiation largely stopped to interact with matter,
the number of photons at frequency ν (see Eq.2.3): a3 dN p remained constant. But as the
universe continued to expand to a size a > a so did the radiation wavelength (recall
that in a box k = 2π/λ is a multiple of π/a), i.e. the frequency of the radiation decreased


by the expansion factor αe = a /a. So that the energy density du of the background
photons at frequency ν obeys now:
8πν2 1
(a )3 du = hν a3 dN p = hν a3 3 ehν/kB T − 1
dν
c
8πν 2 1
= (a )3 α−3 hν α3 3 hν /k (T/α )
e e dν (2.6)
c e B e − 1

which is the energy density of a black-body at a temperature T , smaller than the tem-
perature at decoupling T by the expansion factor αe : T = (T/αe ):
8πhν 3 1
du = 3 hν /kB T − 1
dν
c e
Because the Universe expanded by a factor αe ∼ 1100 since the decoupling time, one
obtains a current temperature for the background radiation of T = 2.72◦ K. The precise
agreement on the value of that temperature is not very important as is the observation
that the cosmic background radiation is the best Black-Body ever observed. It is also
highly isotropic in the rest frame of the Universe. As our galaxy the Milky Way moves
at about 600 km/sec with respect to the Cosmic background, the Doppler effect red-
shifts the radiation in one direction and blue-shifts it in the opposite one. This effect can
be subtracted from the measured distribution of radiation intensities. One also needs to
subtract the contribution from the stars in the galaxy (which fortunately emit at much
higher frequencies, in the visible mostly). The measured variations in the temperature of
the Universe at different angular positions are then smaller than 10−5 K, yet these small
fluctuations served as the nucleation points for the galaxies and can account for their
observed distribution, see Fig.2.3. As E.Wigner wrote it is a ”miracle ... that we neither
undersand nor deserve” that a theory devised to explain (approximatively) the radia-
tion of hot bodies has turned out to provide such an amazingly precise and powerfull
description of the Universe!
2.1.2 The photo-electric effect
Besides the emission spectrum of atoms and the black body radiation, an other ex-
periment stood in apparent contradiction with Maxwell’s electromagnetic theory: the
photo-electric effect which observed that electron were emitted from a conducting ma-
terial with an energy that depended on the color (the frequency) of the radiation not on
its intensity. This was at odds with Maxwell’s electromagnetic theory that asserted that
the energy of radiation was related to its intensity (see Eq.??) not its frequency! Ein-
stein knew the solution for Black-body radiation for which Planck had to assume that
the radiation emitters in the walls’ cavity could only emit light in small quantas. In 1905
Einstein went further and assumed that all light actually comes in small bunches, pho-
tons, which energy is proportional to their frequency: E = hν. When such a photon is
absorbed by an electron its energy is used to tear the electron from the binding potential
Φ of the material and move it at velocity v:
1 2
hν = mv + Φ (2.7)
2


F. 2.3. The temperature of the cosmic microwave background measured across the
sky by the COsmic Background Explorer (COBE) satellite. The top image is the
raw data which is red/blue shifted due to the movement of our galaxy through the
universe at ∼ 600km/s. Correcting for this Doppler shift yields the middle image
which is still ”polluted” at the equator by the light emitted from the stars in our
galaxy. Substracting that measurable emission yields the bottom image where the
temperature fluctuations of the microwave background across the Universe are as
small as 10µ◦ K. These small fluctuations nonetheless served as nucleation points
for the formation of the stars and galaxies shortly after the decoupling time.

Hence an electron can only be observed if light of high enough frequency is used to
remove it from the material. The kinetic energy of the electron increases then linearly
with the illumination frequency. The current emitted is however proportional to the
number of adsorbed photons, i.e. to the light intensity. At the time Einstein proposal
was revolutionary since it assumed that energy came in discrete packets that could not be
infinitely divided and it appeared to contradict Maxwell’s equations. It took 16 years and
confirming experiments to establish the validity of his model, for which he got the Nobel
prize in 1921 (and not for his more profound and revolutionary theories of relativity
and gravitation). Incidently from Einstein’s relation between energy and momentum:
E 2 = (mc2 )2 + (pc)2 (see Appendix) one deduces that if the energy of the photon (of
mass m = 0) is quantized so must its momentum be: p = E/c = h/λ ≡ k ( ≡ h/2π).


Einstein’s understanding of the photo-electric effect has had enormous technologi-
cal impact. All digital cameras are based upon it. These CCD (Charge Coupled Device)
cameras consist of an array of small capacitors (a few micron in size) each defining
a pixel (= picture element). When light (with frequency in the infrared or higher) im-
pinges on a given pixel it kicks off an electron from one side of the capacitor to the other
and charges it with an amount which is proportional to the light intensity. The charges
in a given row of capacitors are then read out by transfering them from one capacitor to
the next along the line like in a ”bucket brigade”. Thus is the image read and stored. By
covering the array of pixels with a mask-array that filters different colors (Red, Green or
Blue) the device can be transformed into a color camera where adjacent pixels respond
to different colors.
Similarly all of today’s solar cells are based on the photo-electric effect using light
to generate a current by transfering electrons in a semiconductor material from the so-
called valence band (and leaving a positively charge ”hole” behind) into the conduction
band (on which more below).

2.1.3 Bohr’s atom
Following on the footsteps of Planck and Einstein who proposed that energy and mo-
mentum were quantized: E = n ω and p = n k, Niels Bohr suggested in 1913 that
the angular momentum of electrons in an atom was similarly quantized: L ≡ mvr = n
thereby explaining their paradoxical stability (see above). Indeed given the balance of
electrostatic and centrifugal forces: mv2 /r = e2 /r2 and the assumed quantization of an-
gular momentum one deduces that in the hydrogen atom the orbits of the electron are
quantized with a radius: r = n2 2 /me2 ≡ n2 r0 (r0 ≡ 2 /me2 = 0.53Å is known as the
Bohr radius of Hydrogen), velocity v = e2 /n and energy:
1 2 e2 me4 e2 1 13.6eV
En = mv − =− 2 2 =− =− (2.8)
2 r 2 n 2r0 n2 n2
Thus the energy to ionise a hydrogen atom, i.e. kick off its electron from its ground state
at n = 1 is 13.6 eV. Because of energy quantization an electron orbiting the nucleus will
not radiate continuously, but emit (or absorb) radiation in quantas of energy:
1 1
hνnm = Em − En = 13.6eV( 2
− 2) (2.9)
n m
Hence the emission or adsorption spectra of atoms consists of discrete lines correspond-
ing to electronic transitions between states with different quantum numbers. For the hy-
drogen atom only the lowest energy transitions to n = 2 (the so-called Balmer series) lie
in the visible range with wavelengths in the red: 656.3nm (m = 3); in the blue 486.1nm
(m = 4) and in the UV range: 434.1nm (m = 5) and 410.2nm (m = 6), see Table 2.1 and
Fig.2.1. The explanation of the stability of atoms and the emission lines of hydrogen
was a major success of the Bohr model for which he was awarded the Nobel prize in
1922.


m= 2 3 4 5 6 7 8
Lyman series (n=1) 121.6 102.6 97.2 94.9 93.7 93.0 92.6
Balmer series (n=2) - 656.3 486.1 434.1 410.2 397.0 388.9
Pashen series (n=3) - - 1870 1280 1090 1000 954

Table 2.1 The major emission lines in the hydrogen atom. The wavelength (in nm) is
tabulated for various values of the initial (m) and ﬁnal state (n)

Bohr’s approach to the hydrogen atom was generalized in 1915 by Arnold Som-
merfeld who proposed that for any bound particle (atom, harmonic oscillator, etc.) a
quantity known in classical mechanics as the action was quantized:

p · dq = nh (2.10)

where p, q are the momentum and coordinate of the particle. The Wilson-Sommerfeld
quantization rule, Eq.2.10, reduces to Bohr’s quantization of the angular momentum in
the case of the hydrogen atom since the integral pdq = 2πmvr. But the same rule also
explains why the oscillators of frequency ω assumed by Planck to exist in the walls of
a back-body would emit radiation in quantas of energy ω. For a harmonic oscillator
√
with frequency ω = k/m, the energy is:
p2 kq2 p2 mω2 q2
Eosc = + = +
2m 2 2m 2
from which one derives:
qm qm
pdq = 2 2mEosc − (mωq)2 = 2mω q2 − q2 dq = πmωq2
m m
−qm −qm

with q2 ≡ 2Eosc /mω2 . Hence Eq.2.10 implies: Eosc = n ω, which is the assumption
m
made by Planck.
The Wilson-Sommerfeld quantization rule can also be used to ﬁnd the energy level
of a quantum rotator rotating at frequency ω about one of its major axes with moment
of inertia I . In that case the angular moment Iω = l (l = 0, 1, 2, ...) and the angular
energy is El = Iω2 /2 = 2 l2 /2I.
In 1924, Louis de Broglie (in his Ph.D thesis!) generalizing Sommerfeld’s idea sug-
gested that all matter are described by waves. So, just as a photon possesses a momen-
tum p = k, thus an electron is characterized by wavevector: k = p/ . The Wilson-
Sommerfeld rule is thus equivalent to the requirement that the electron wave in a bound
system interfers constructively to form a standing wave. As we shall see in the following
de Broglie’s analogy opened the way for the formal development of quantum mechan-
ics from its analogy with optics: classical mechanics becoming to quantum mechanics
what geometrical optics is to electromagnetic waves.


F. 2.4. (A) The absorption of radiation by an atom in its ground state. (B) the stimu-
lated emission of a photon in presence of radiation by an atom in its excited state:
notice that this process is the time reversal of adsorption. (C) The spontaneous emis-
sion of a photon (in absence of radiation) by an atom in its excited state.

2.1.4 Adsorption, stimulated emission and the laser
Based on the Bohr-Sommerfeld model, Einstein proposed in 1917 a simple theory of
light-matter interaction which could account for Planck’s formula and would be (40
years later) the basis for the invention of the laser. First he pointed out that since micro-
scopic processes are reversible the adsorption of a photon is indistinguishible from the
process of stimulated emission, see Fig.2.5. In other words in presence of an external
electro-magnetic ﬁeld the transition rate B21 to state 2 from 1 should be the same as the
transition rate B12 to state 1 from 2. The transition being due to the interaction between
radiation and matter, the overall rate T i j (i, j = 1, 2) will depend on the spectral density
of radiation at the transition energy ρ(ν = ∆E/h) and on the density of states 1 and
2: n1 = N1 /V and n2 = N2 /V (where Ni is the number of atoms in state i and V the
volume): T i j = Bi j ρN j /V. Einstein also recognized that in abscence of interaction with
the external ﬁeld, an atom in an excited state 2 could spontaneously return to the ground
state 1 by emission of a photon of energy hν. That process depends on the lifetime τ s
of the excited state and occurs at a rate A12 = 1/τ s . To summarize in steady state the
transition rates to and from each state should balance:
B21 ρN1 /V = B12 ρN2 /V + A12 N2 /V
From which since B12 = B21 we derive:
A12 /B12
ρ(ν) =
N1 /N2 − 1
Since at thermal equilibrium the probability of being in the excited state is smaller than
in the ground state by the Boltzmann factor N2 /N1 = exp(−∆E/kB T ) one obtains:
A12 1 8πhν3 1
ρ(ν) = =
B12 e hν/kB T − 1 c3 ehν/kB T − 1


F. 2.5. Principle of operation of a laser. An amplifying medium is pumped by an
external energy source (e.g. a flash lamp) to generate a higher density of excited
states than of ground states (population inversion). The medium is placed in a cavity
with reflecting mirrors, one of which lets a small fraction of the light out. The light
reflected in the cavity is amplified by the stimulated emission of the excited states.
When the threshold for lasing is achieved the losses in the cavity are balanced by
the gain from the amplifying medium.

with the identification: A12 /B12 = 8πhν3 /c3 or:

c3 λ3
B12 = 3τ
=
8πhν s 8πhτ s

Einstein model could account for Planck’s Black-body radiation if atoms capable
of absorbing radiation at all frequencies are uniformly present. It also made possible
decades later the invention of the laser, acronym for Light Amplification by Stimulated
Emission of Radiation. A laser consists of a light amplifying medium inside a highly
reflective optical cavity, which usually consists of two mirrors arranged such that light
bounces back and forth, each time passing through the gain medium, see Fig.2.5. Typi-
cally one of the two mirrors is partially transparent to let part of the beam exit the cavity.
To achieve light amplification the excited state in the medium of a laser cavity emitting
at frequency ν0 = ∆E/h has to be more populated than the lower energy state. Since at
thermodynamic equilibrium low energy states are always more populated than higher
energy ones, energy must be injected in the medium to ”pump” (i.e. excite) atoms from
the ground state into the light emitting state.
Let us therefore consider light of intensity I(z, ν) = ρ(z, ν)c (0 < z < l) propagating
in a cavity of length l and cross section S . Due to stimulated emission of power dPemit
in a volume dV = S dz the increase in the light intensity is:


dI
= dPemit /dV = hν0 (n2 − n1 )B12 ρ(ν)δ(ν − ν0 )
dz
c2
= (n2 − n1 ) I(z, ν)δ(ν − ν0 ) (2.11)
8πν0 τ s
2

For various reasons that we shall discuss later, the transition frequency between two
states is never infintely sharp and one replaces the δ-function in the preceeding equation
by a function g(ν) which approximates it: g(ν)dν = 1. Quite often the response g(ν)
of a resonant oscillator is appropriate (see Appendix on Fourier transforms):
γ/π
g(ν) =
(ν − ν0 )2 + γ2
One then obtains:
dI c2
= (n2 − n1 ) g(ν)I(z, ν) ≡ γ(ν)I(z, ν)
dz 8πν0 τ s
2

As argued above, the light intensity grows exponentially when the population of atoms
in the medium is inverted, i.e. when n2 > n1 . As light propagates in the cavity it suffer
losses (due to adsorption for example) of magnitude α cm−1 . Since light has to come
out of the cavity there are also losses due to the fact that only a portion R of the intensity
is reflected back into the cavity. For a laser to operate the losses must equal the gain:
R exp[(γ(ν) − α)l] = 1, which imply that the population inversion at threshold has to
satisfy:
8πτ s 1
(n2 − n1 )t = 2 (α − ln R)
λ g(ν) l
Hence the longer the wavelength the smaller the required population inversion for las-
ing. That is one of the reasons that masers (lasers in the microwave range) were the
first to be invented while X-ray lasers, even though of great utility, have been difficult
to develop. During steady-state laser operation the balance of losses and gain imply that
the population inversion remains at threshold. The more the ground state is pumped, the
more the excited state is induced to emit by the increased light density in the cavity, thus
keeping the difference between the density of the two states fixed at its threshold.

QUANTUM MECHANICAL FORMALISM 19

2.2 Quantum Mechanical formalism
The early 20th century investigations by Planck, Einstein, Bohr, Sommerfeld, de Broglie
, etc. revealed a picture of matter that was different from the infinitely divisible contin-
uum of energy and momentum that prevailed untill then. Many of the properties of
matter could be explained by assuming that these quantities were discrete rather than
continuous. Thus could Planck explain the radiation spectrum of black-bodies, Einstein
the photo-electric effect and Bohr the stability of atoms and the emission spectrum of
hydrogen (though not of other elements). These early efforts suggested that matter and
radiation shared similar properties: light came as photons, particles of zero mass but
possessing definite energy and momentum. Similarly electrons had wave-like proper-
ties and could interfere with themselves, as in the orbitals of Bohr’s atom. What was
missing was a conceptual framework that would unite these observations and models
with classical mechanics. The breakthrough came with the works of Werner Heisenberg
and Max Born in 1925 and Erwin Schroedinger in 1926. The later in particular wrote an
equation for the probability of finding a particle at a given position that was inspired by
the analogy pointed out by de Broglie between waves and particles. As we shall see later
the eigenvalues of the famed Schroedinger equation yield the energy levels of a bound
system, much as one determines the resonant modes of electromagnetic radiation in a
cavity (in both cases one solves a Helmholtz equation, see Appendix).
Heisenberg proposed a matrix formulation of Quantum Mechanics that was later
shown by Schroedinger to be equivalent to his own formulation. Heisenberg’s approach
however inspired the mathematically rigorous and clear formulation of Quantum Me-
chanics presented in 1930 by Paul A.M.Dirac in his landmark book (”the Principles
of Quantum Mechanics”, Clarendon press, Oxford). In the following we shall follow
Dirac’s lead.

According to Dirac, a physical system is characterized by its state |Ψ > (also called
wave-function by Schroedinger). These states are complex unit vectors in a so-called
Hilbert space, defined so that their scalar product < Ψ|Ψ >= 1. When an observable
O is measured, the system is perturbed and ends up in an eigenstate |n > of O with
eigenvalue On : O|n >= On |n > (in linear algebra the eigenstates are the vectors that
diagonalize the matrix O). As in linear algebra one can expressing the vector-state |Ψ >
in terms of the eigenvectors |n > of O:

|Ψ >= αn |n > (2.12)
n

Since |Ψ > is a complex vector its conjugate is: < Ψ| = n α∗ < n|. The physical
n
interpretation of the amplitudes αn is at the core of Dirac’s formulation: the probability
of observing a system in state |n > after the measurement of O is: |αn |2 . So that the
probability of measuring a value On when the system is in state |Ψ > is:

P(On ) = | < n|Ψ > |2 = |αn |2 (2.13)


Notice that |Ψ > being a unit vector: n |αn |2 = 1 as it should for |αn |2 to be interpreted
as a probability. This is the physical interpretation of the wave-function and the intrinsic
indeterminism of QM that so annoyed Einstein. The average value of O in state |Ψ > is:

< Ψ|O|Ψ > = (< n|α∗ n )O(αm |m >)
n,m

= Om α∗ n αm < n|m > (2.14)
n,m

= |αn |2 On = P(On )On =< O > (2.15)
n n

where we assumed the eigenstates to be orthonormal < n|m >= δnm . Quantum me-
chanics in Dirac’s formulation is thus reduced to linear algebra: the states are complex
vectors and the observables complex matrices with real eigenvalues, i.e. Hermitian ma-
trices satisfying Amn = A∗ . If the eigenstates of one operator (observable) A are also
nm
the eigenstates of an other operator B then A and B commute:

AB|n >= Abn |n >= an bn |n >= bn an |n >= bn A|n >= BA|n >

If the operators commute they can be both measured simultaneously: they are diagonal-
ized by (i.e they share) the same eigenstates. If on the other hand the eigenstates of A
and B are not identical, their simultaneous measurement is not possible. Let {|n >} be
the eigenvectors of A, then

BA|n >= Ban |n >= an B|n >= an |m >< m|B|n >= an Bmn |m >
m m

On the other hand:

AB|n >= A|m >< m|B|n >= am Bmn |m > an Bmn |m >= BA|n >
m m m

hence the operators do not commute [A, B] ≡ AB − BA 0. We shall see later that
the non-commutability of operators (which is quite common with matrix operations)
is at the core of Heisenberg uncertainty principle which states that the position and
momentum of a particle cannot be simultaneously determined.
The energy being an important observable in physics, the energy operator (or Hamil-
tonian, H) plays an important role in Quantum Mechanics. Its eigenvalues are the pos-
sible measured energies of the system (which can be discrete or continuous) and its
eigenmodes are like the resonant modes of an oscillator or the speciﬁc orbits of the
electron in Bohr’s atom:
H|n >= n |n >
From Planck and Einstein, we know that there is a relation between energy and fre-
quency: n = ωn , so that an eigenstate with given frequency ωn evolves in time as
exp(−iωn t) = exp(−i n t/ ). Notice that if the initial state of the system is one of the

SIMPLE QM SYSTEMS 21

eigenstates |n >, it remains there with probability P = | exp(−iωn t)|2 = 1. If however the
initial state is |Ψ(0) >= n αn |n > then:

|Ψ(t) >= αn e−i n t/ t |n >
n

From which one derives Schroedinger’s equation:

∂ 1 1
|Ψ(t) >= αn n e−i n t/ t |n >= H|Ψ(t) >
∂t i n
i

or in the more common notation:
∂
i |Ψ(t) >= H|Ψ(t) > (2.16)
∂t
This is essentially all of Quantum Mechanics: a definition of physical states as vec-
tors in a complex space, measurements as matrix operations on these vector states, the
outcome of the measurements as eigenvalues of those matrices and a description of the
time evolution of the physical states by Schroedinger’s equation. The rest is application
of this linear algebra formalism!

2.3 Simple QM systems
2.3.1 The chiral amonia molecule
Our first application of the QM formalism described above will be the amonia molecule
and the amonia maser (the microwave equivalent of the laser discussed earlier). We will
consider here the chiral amonia molecule NHDT (D and T stand for the isotopes of Hy-
drogen: Deuterium and Tritium), rather than the achiral NH3 considered by Feynman (in
Vol3 of his ”Lectures on Physics”). This choice allows us not to care about rotational
motions and it exemplifies the queer nature of QM better than the achiral molecule.
NHDT is a tetrahedron with the four atoms sitting at the four apexes. The molecule
possesses distincts enantiomers, i.e. distinct states |1 > and |2 >, which are mirror im-
ages of each other depending on whether the Nitrogen atom is on the right or the left
of the HDT plane, see fig.2.6. These states are not eigenstates of the Hamiltonian as
the Nitrogen in state |1 > can end up in state |2 > by passing through the HDT plane,
like a left-handed glove can be transformed into a right-handed one by turning it inside
out. Even though this energetically costly transition is classically impossible there is in
QM always a small probability for such a process to happen (this tunneling through an
energetically forbidden zone (a wall, see below) is one of the oddities of QM).
Due to the symmetry of states |1 > and |2 >, the Hamiltonian of this two-state
system can thus be written as:
E0 −A
H0 =
−A E0


F. 2.6. The chiral amonia molecule, NHDT consist of a nitrogen atom bound to the
diﬀerent isotopes of hydrogen(H): deuterium(D) and tritium(T). This molecule exist
with diﬀerent chirality: left-handed or right-handed which are mirror images of each
other. Since nitrogen is slightly more electrophilic (negatively charged) than the
hydrogen isotopes the molecule possesses a small electric dipole moment µ.

Its eigenvalues (eigen-energies) are: E I,II = E0 ± A and its eigenvectors are:
0

1 1
|I >= 1
√ |II >= 1
√
2 −1 2 1

You can check that H0 |I >= E I |I > and H0 |II >= E II |II >. In the eigenvector basis the
0 0

Hamiltonian is diagonal:
E +A 0
H0 = 0
D
0 E0 − A
It is a general result from linear algebra that the matrix of eigenvectors:

1 1 1
Λ= √
2 −1 1

diagonalizes the original matrix: H0 = ΛT H0 Λ.
D

Notice that the energy eigenstates for this chiral amonia molecule consist of a co-
herent superposition of a left- and a right-handed state with probability 1/2! This is a
classically absurd situation akin to Schroedinger’s famous cat paradox, see Fig.2.7. In
this gedanken experiment he proposed to couple a cat enclosed in a box to a two-state
QM system: the cat is dead if the system is in state |1 > and alive if in state |2 >. Ac-
cording to QM, before one looks into the box the cat exists as a superposition of the two
states: dead and alive; just like our chiral amonia molecule is described as a superpo-
sition of left and right-handed states. However once a measurement is made the cat is
either dead or alive; just as the chiral amonia molecule is -when observed- either left-
or right-handed.


F. 2.7. The gedanken (thought) experiment that Schroedinger proposed to test the va-
lidity of QM. In a closed box isolated from the external world there is a cat and a
radioactive source of say β−particles (electrons). If a particle is emitted and detected
by a Geiger counter a poison vial is broken that kills the cat. Otherwise the cat is
alive. As the state of the particle is a superposition of bound and emitted particle,
one must consider the cat to be in a superposition of a live and dead animal. For Ein-
stein that thought experiment demonstrated that QM was incomplete since it gave
rise to absurd assumptions. Yet, when isolated from the external world (a tall order
for macroscopic systems) all experiments so far are consistent with this ”absurd”
superposition of states.

So did Einstein ask: how was the cat ”really” before we looked into the box? He
claimed that the superposition is non-sensical and the cat is either dead or alive. He then
argued that some unknown factors (hidden variables) in the description of the micro-
scopic two-state sytem that determines the cat’s observed state results in the QM proba-
bilistic and verified predictions. However as mentioned earlier, the experimental viola-
tion of Bells’ inequalities suggest that Einstein was wrong and that such ”Schroedinger
cats” exist as a superposition of dead and alive states, even if we have no clue what that
means! We will discuss again that point more quantitatively below.
A crucial ingredient enters into the QM picture and it is the measurement process.
For a system to exhibit the interference effects resulting from QM state superpositions
it must not interact with the environment. These interactions are like independent mea-
surements of the system and they destroy its coherence (e.g. the two-state superposition)
by implicating (entangling them with) many external uncontrolled states. Now it is very
easy for a macroscopic system like a cat to interact with the world outside the box (for
example through the radiation it emits or adsorbs). Hence quantum experiments with
large objects are notoriously difficult to perform. So far the largest molecule for which
quantum interference effects have been demonstrated is the buckyball: C60 (see below).
Since the chiral states are given by:


√
|1 > = ( |I > + |II >)/ 2
√
|2 > = (−|I > +|II >)/ 2 (2.17)
if the molecule has left-handed chirality to begin with: |Ψ(0) >= |1 >, it will evolve as:
0 0
e−iE I t/ |I > + e−iE II t/ |II >
|Ψ(t) >= √
2
The probability of finding the system with a left-handed chirality (state |1 >) after a
time t is: 0 0
|e−iE I t/ + e−iE II t/ |2
P1 (t) = | < 1|Ψ(t) > |2 = = cos2 (At/ )
4
and the probability of finding the system with right-handed chirality is P2 (t) = sin2 (At/ ).
When a given molecule is measured its chirality is well defined (either left or right), but
measurements over many molecules yield the oscillating probability distribution just
computed. Since cos2 (At/ ) = (1 + cos 2At/ )/2 the oscillation frequency is related to
the difference between the energy levels of the eigenstates ω0 = 2A/ . For NH3 that
frequency ν0 = ω0 /2π = 24 Ghz is in the microwave range. For NHDT it is slighly
lower due to the higher mass of Deuterium and Tritium.

The essence of Bell’s inequalities may be grasped from this simple example. Imag-
ine that a NHDT molecule prepared in a definite chiral state (|1 > or |2 >) is observed
a time δt later so that the probability of finding it in the same state is 99%. If it is
measured again a time δt later it will be observed in the same state as previously with
probability 99%. One may now ask: what is the probability of observing the system
in its initial state if we look at it a time not δt but 2δt later? If as Einstein believed
the system is at δt in a definite state that is only once in a hundred times different
from the initial state, then at 2δt the state of the system would be at worst twice in
a hundred times different from the initial state, namely the probability of observing
the system in its initial state would be at worst 98%. However the QM prediction is:
P(2δt) = cos2 (2Aδt/ ) = 1 − 4(Aδt/ )2 = 0.96 (since we assumed that P(δt) = 0.99,
i.e. Aδt/ = 0.1). The QM mechanical prediction violates the lower bound (the Bell
inequality) set by ”realistic” theories which assume that the system is in a definite state
which we have simply no way of determining, not a ”meaningless” superposition of
left and right-handed molecules. As mentioned earlier the experimental results (mea-
sured not on chiral amonia molecules but some other two-state system) vindicate the
QM prediction and rule out the ”realistic” theories.

2.3.2 The amonia molecule in a constant electric field
Since nitrogen is more electrophilic than hydrogen, it tends to be slightly more nega-
tively charged than the hydrogen isotopes and the molecule ends up with a permanent


electric dipole moment µ, as shown in Fig.2.6. In presence of an electric field E the en-
ergy of a dipole is W = −µ · E, (Eq.??). If the electric field is along the x-axis in Fig.2.6,
then we expect state |1 > to have higher energy than state |2 >. The Hamiltonian of the
molecule in an external electric field is thus:
E0 + W −A
H=
−A E0 − W
Notice that in the eigenbasis representation (where the unperturbed Hamiltonian H0 is
diagonal), the perturbed Hamiltonian can be written as:

E0 + A 0 0 W
H = ΛT HΛ = H0 + δH =
D
+ (2.18)
0 E0 − A W 0
√ √
The eigenvalues of H (and H ) are: E I,II = E0 ± A2 + W 2 . Defining tan θ ≡ ( A2 + W 2 −
W)/A the eigenvectors of H are:

cos θ sin θ
|I >= |II >=
− sin θ cos θ

When W = 0 one recovers the previous result. When W A: |I > |1 > and |II >
|2 >, in which case the enantiomers are also eigenstates. In practice however W =
µE A and the energies vary as: E I,II = E I,II ± µ2 E 2 /2A. State |I > (with its dipole
0

essentially opposite to the electric field) has higher energy than state |II >. Notice that
we can write the energies E I,II in the following form (we will see later that this is a
general result when the diagonal Hamiltonian is perturbed by an amount δH)
δH12 δH21
EI = EI +
0
0 0
(2.19)
E I − E II
δH21 δH12
E II = E II + 0
0
0
(2.20)
E II − E I
These results suggest a way to separate the eigenstates by passing a beam of amonia
molecules through a strong electric field gradient. This field gradient generates a force
on the molecules:
µ2
F I,II = − E I,II = E2
2A
which separates them: state |I > is deflected to regions of small electric fields, while
state |II > is deflected to regions of high electric fields. Thus can a sub-population
inversion be generated where high energy amonia molecules are separated from lower
energy ones. This high energy sub-population can then be used to amplify microwave
radiaton by stimulated emission. The resulting device, known as a maser, was the first
implementation of a stimulated radiation amplification device and served as the first
atomic clock.


F. 2.8. Principle of operation of a maser. An amonia beam (here the chiral molecule
NHDT) is sent trough a slit into a beam splitter that consists of a strong inhomoge-
nous electric field. At high field one can separate the different enantiomers which
are eigenstates of the energy. The high energy eigenstate (|I >) is sent into a cavity
where it goes into the low energy state |II > by emitting stimulated radiation at the
resonant frequency ω0 = (E I − E II )/

2.3.3 The amonia maser and atomic clocks
In an amonia maser, Fig.2.8, the high energy state |I > selected as described, enters
a resonant cavity (tuned to the transition frequency ω0 ). A population inversion of the
amonia molecules is thus generated in the cavity and amplification of radiation can be
expected. The emitted radiation stimulated by the presence in the cavity of an external
source generates a highly coherent microwave beam.
To analyse the operation of a maser, let us assume that amonia molecules enter a
cavity in which they experience a time varying electric field: E = Ee−iωt x. This field
ˆ
couples with the dipole moment of the molecules to modulate their energy by W(t) =
−µEe−iωt . The perturbed Hamitonian in the eigenbasis (|I > and |II >) is then, see
Eq.2.18:
EI 0 0 W(t)
H = H0 + δH(t) =
D
+ (2.21)
0 E II W ∗ (t) 0
Looking for a general solution: Ψ(t) >= C I (t)|I > +C II (t)|II >, Eq.2.16 yields:
i ∂t C I = E I C I + W(t)C II
i ∂t C II = W ∗ (t)C I + E II C II (2.22)
Looking for a solution C I,II = αI,II (t) exp(−iE I,II t/ ) yields the following equation for
αI,II :
i ∂t αI = W(t)e−i(E II −EI )t/ αII = −µEe−i(ω−ω0 )t αII
i ∂t αII = W ∗ (t)e−i(E I −E II )t/ αI = −µEei(ω−ω0 )t αI (2.23)
At the resonance: ω = ω0 , one obtains:


∂2 αI,II + Ω2 αI,II = 0
t with Ω = µE/

which solution is αI (t) = cos Ω(t − t0 ) and αII = sin Ω(t − t0 ). The probability of being in
state |I > (α2 ) or II > (α2 ) oscillates with frequency 2Ω: the molecule go periodically
I II
from stimulated emission in state |I > to adsorption in state II >. These oscillations
are different from the oscillations between the enantiomers |1 > and |2 > observed at
frequency ω0 in absence of electric field. This oscillation in the probability of observing
a specific enantiomer is not associated to emission of any radiation, it results from the
fact that the enantiomers are not eigenstates of the Hamiltonian: in absence of a time
varying electric field, if the molecule is in eigenstates |I > or |II > it remains there.
If the frequency of the electric field is slightly off resonance ω − ω0 0 and if the
molecule remains in the cavity for a short time t 1/Ω we may assume that αI 1
and integrate the equation for αII to yield:

ei(ω−ω0 )t − 1
αII (t) = iΩ
i(ω − ω0 )
The probability of transition from state |I > to state |II > is then:

sin2 (ω − ω0 )t/2
PI→II (t) = |αII |2 = (Ωt)2
[(ω − ω0 )t/2]2
The function sinc x ≡ sin x/x decays rapidly for values of |x| > π. Hence the transition
rate is significant only for frequencies which are very close to resonance: |ω − ω0 | <
2π/t. If the molecule remains in the cavity for 1 sec, the relative possible detuning :
|ν/ν0 − 1| = 1/ν0 t ∼ 4 10−11 . Only frequencies within that very narrow range can be
amplified by stimulated emission. This allows for very high Q resonators, Eq.??, i.e.
very precise frequency generators and clocks. Because of the sharpness of the function
sinc2 (ω − ω0 )t/2 one often rewrites the previous equation using the limit

2π
lim sinc2 (ω − ω0 )t/2 = δ(ω − ω0 )
t→0 t
to obtain the transition rate from state |I > to |II >:

dPI→II (t) 2π|δH12 |2
BI,II = = 2πΩ2 δ(ω − ω0 ) = δ(E I − E II − ω) (2.24)
dt
Notice that if we had assumed the amonia beam to enter the cavity in state |II >, i.e.
with αII = 1, then we could have similarly integrated the equation for αI to yield the
transition probability from state |II > to |I >, which comes to be:

2π|δH21 |2
BII,I = δ(E I − E II − ω) = BI,II

which was the intuitive assumption Einstein made: the rate of stimulated emission BI,II
is equal to the rate of adsorption BII,I . Since the energies of the excited states are never


perfectly sharp, due to the Heisenberg uncertainty principle that we shall see below and
due to thermal motion that leads to Doppler broadening of the emission lines (see sec-
tion??), the δ-function is replaced by the density of states with the appropriate energy:
ρ(E) (with E = E II + ω) and the normalisation ρ(E)dE = 1).
The amonia maser was the ﬁrst atomic (or molecular) clock. The present genera-
tion of atomic clocks use a microwave transition in Cesium (Cs) as the reference fre-
quency for the clock. The atoms are cooled to very low temperatures (µ◦ K) to reduce
the Doppler broadening of their emission lines and keep them for as long as possible
in the cavity, usually a Penning trap (see ????). As a result the record for the frequency
precision of atomic clock is: |ν/ν0 − 1| ∼ 10−16 .

2.3.4 The energy spectrum of aromatic molecules
A simple generalization of the two-state system that we considered previously is the
n-state system consisting for example of a circular chain of n identical atoms such as
benzene C6 H6 (n = 6) around which electrons can hop. If an electron has energy E0
when associated with a particular atom |n > and can hop only between nearest neighbors
the Hamiltonian for this system in the basis of the atom’s position |n > is:

 E0 −A 0 . . . −A 
 
 −A E −A . . . 0 

 

 0 
H= .
 
. 
 

 .  (2.25)
 .


 . 
. 



 
−A . . . 0 −A E0
 

The eigenstates of that system obey: H|Ψ >= E|Ψ >, with |Ψ >= n C n |n > From
which we derive the equations:
E − E0
C1 + C2 + Cn = 0
A
E − E0
C1 + C2 + C3 = 0
A
.
.
.
E − E0
C1 + Cn−1 + Cn = 0
A
Looking for a solution: Clm = exp i(2πlm/n) where l = 1, 2, ...., n we get:
El − E0
= −(ei2πl/n + e−i2πl/n )
A
Thus the eigen-energies of the system are:
El = E0 − 2A cos 2πl/n (2.26)
For benzene (n = 6) we have: E6 = E0 − 2A; E1,5 = E0 − A; E3 = E0 + 2A and E2,4 =
E0 + A. Energies E1,5,6 which are smaller that E0 are associated to so-called bonding


F. 2.9. Band-gap theory of material. (a) If the gap between the valence and conduction
band is large (a few electron-Volts) the material is an insulator. (b) Variation of the
band-gap energy with wave-vector k. Top curve: energy of electrons Ee ; bottom
curve: energy of holes Eh (increases towards the bottom). (c) A semi-conductor
which due to thermal excitation or illumination has some electrons in the conduction
band and some holes in the valence band. (d) a conductor is a material for which
the valence and conduction band overlap or equivalently for which an energy band
is not ﬁlled.
orbitals or wave-functions, whereas energies E2,3,4 > E0 are associated to anti-bonding
orbitals, on which we shall have more to say later. Notice that the eigen-state associated
√
to the maximally bonding orbital (l = 6) is fully symmetric: |Ψ6 >= (1, 1, 1, 1, 1, 1)/ 6,
while that associated to the maximally anti-bonding orbital (l = 3)√ antisymmetric to
is
the permutation of nearest neighbors: |Ψ3 >= (1, −1, 1, −1, 1, −1)/ 6.

2.3.5 Conduction bands in solids
An interesting generalization of the previous analysis is the case of a long chain of
n atoms a distance a apart. In that case we have: Clm = exp[i(2πl/na)ma]. Since the
position of atom m is x = ma we may write Ck (x) = exp ikx with k = 2πl/na. Like the
oscillation modes on a string of length na, the electron’s eigenstates are 1D transverse
waves of wavelength λ = 2π/k = na/l. The energy of such a mode is:

E(k) = E0 − 2A cos ka (2.27)

The energy of the electron is bounded: E0 − 2A < E(k) < E0 + 2A. If each atom
contributes two electrons, these 2n electrons will occuy all the n-energy states (we shall
see later that each state can accomodate 2 electrons) and electron hoping in this energy-
band will be impossible. If however the on-site energy E0 can possess discrete values
(El as it does indeed, for example in Bohr’s model), the coupling of the n−atoms will
generate energy bands around each energy value El into which electron may hop. This
forms the basis of the band-theory of conduction in materials, see Fig.2.9: a material
will conduct if there are empty states into which electrons can hop. If the low energy


F. 2.10. A p-type semiconductor consists of a material (usually Silicon, Si) doped
with an element (such as Aluminium) which having less electrons in its outer shell
than Si tend to trap electrons from the lattice, leaving a electron vacancy instead:
a hole. This depletes the valence bands of electrons allowing conduction through
motion of holes. A n-type semiconductor consists of a material doped with an elec-
tron donor, an element (such as Phosphate) which has more electrons in its outer
shell than Si. This injects electrons into the conduction band allowing the material
to conduct electricity. A pn-junction is formed when p-type and n-type semiconduc-
tors are brought in contact. Such a junction can be used as a Light Emitting Diode
(LED) when electron from the n-side of the junction recombine with holes from the
p-side. The wavelength of emitted light is determined by the energy-gap between
the conduction and valence and bands.
(valence) band is filled with electrons and the next (conduction) band is empty but
many electron-volts (eV) above it, then the electrons have no states in which to go and
the material is an insulator. If the conduction band overlaps with the valence band then
the electrons have empty states to hop to and current can flow in the material. The
interesting and technologically important case is the situation where the band-gap Eg
between the conduction and valence bands is small: Eg 1eV (the gap for Silicon
(Si) is: Eg = 1.1eV; for Germanium (Ge): Eg = 0.72eV). In that case electrons can be
transfered from the valence to the conduction band by thermal agitation or via the photo-
electric effect resulting in a material that can behave either as a metal or an insulator
depending on the external conditions (temperature, voltage, illumination wavelength
and intensity, etc.).
In particular the introduction of atomic impurities (doping) that donate electrons to
(or accept electrons from) a semiconductor lattice creates a situation where few elec-
trons occupy an almost empty conduction band (or few holes (electron vacancies) oc-
cupy an almost full valence band). As we shall see later the electrons in an atom oc-
cupy certain orbitals or shells around the nucleus. Atoms (such as Phosphate or Ar-
senic) that have more electrons in their outer shell than the bulk semiconductor (the
so-called majority carrier, usually Silicon) will usually donate an electron to the lat-
tice, whereas atoms that have less electrons in their outer shell (such as Boron or Alu-
minium) will accept an electron from the lattice. The doping creates so called n- and

MOMENTUM AND SPACE OPERATORS 31

p-type semiconductors that have electrons in their conduction-band (n-type) or holes in
their valence-band (p-type), see Fig.2.10. The energy of the electrons moving near the
bottom Emin = E0,c − 2Ac of the conduction band (i.e. when ka << 1 in Eq.2.27) is:
E(k) = Emin + k /2mc
2 2

where the effective mass of the electron in the conduction band is defined as: mc =
/∂k E(k) = 2 /2a2 Ac which can be quite different from the mass of a free electron.
2 2

Notice that if, as proposed by de Broglie, we identify the momentum of the electron
as p = k , then Eq.2.3.5 represents the kinetic energy of an electron with mass mc at
the bottom of the conduction band. Similarly the energy of the hole near the top of the
valence band is:
Eh (k) = Emax − 2 k2 /2mv
The movement of holes in the valence band is similar to that of air bubbles in water: as
the moving water displaces the bubble up, the electrons moving in the opposite direction
to the hole minimize their energy by displacing it to the top of the valence band. As a
result holes have minimal energy at the top of the valence band: their energy increases
as k increases.

Doped semiconductors are the basic ingredients of all the semi-conductor industry
(transistors, diodes, integrated circuits, etc.). For example the coupling of p-type and
n-type semiconductors, generate a pn-junction which acts like a diode (current flows in
only one direction). It can also be used as a powerful and efficient light source. Electrons
in the n-type part of the junction can recombine with holes in the p-type (i.e. transit to
the valence band) with emission of light (just as in the atomic or molecular transitions
discussed earlier in the context of stimulated emission and the laser). The advantage
of these Light Emitting Diodes (LED) is that by appropriate tuning of the energy gap
(appropriate choice of the semi-conducting material) one can tune the wavelength at
which the LED will emit light. For example in the red and infrared part of the spectrum
Galium-Arsenide (GaAs) is the material of choice for LEDs. The intensity of light is
controlled by the current flowing through the junction. These LED are used in all kind
of electronic displays, in new high efficiency spot-lamps and traffic lights, in the remote
control of various electronic device, in the laser diode of DVD players, etc.

2.4 Momentum and space operators
In the example above we considered the case of a QM system that could occupy only
discrete states |n >. It is easy to generalize that to free particles that can be found at
continuous positions |x >. In that case the general wave-function in position (or real)
space can be formally written as:
|Ψ >= Ψ(x)|x >
x


Where Ψ(x) is the probability amplitude of finding the particle at position |x >, so that:

P(x) = Ψ∗ (x)Ψ(x) (2.28)

Of course the probability of finding the particle somewhere is one, so that the wave-
function Ψ(x) is normalized:

dxΨ∗ (x)Ψ(x) = 1 (2.29)

The mean position of the particle is:

< x >= dxP(x)x = dxΨ∗ (x)xΨ(x) (2.30)

For example, the wavefunction can be:
1
e−(x−x0 ) /4σ
2 2
Ψ(x) =
(2πσ2 )1/4
for which the probability distribution:
1
e−(x−x0 ) /2σ
2 2
P(x) = √
2πσ
corresponds to a gaussian distribution with mean < x >= x0 and standard deviation
< (x− < x >)2 > = σ. In Fourier-space (see Appendix) we can write:

1 1
Ψ(x) = √ dkΨ(k)eikx = √ d pΨ(p)eipx/ (2.31)
2π 2π
where we used de Broglie’s relations: p = k. Conversely:

1
Ψ(p) = √ dxΨ(x)e−ipx/ (2.32)
2π
Ψ(p) is the wavefunction in the momentum (or Fourier) space: |Ψ >= p Ψ(p)|p >. For
the example chosen above the momentum wavefunction is:

e−ipx0 / ∞
dxe−(x−x0 ) /4σ e−ip(x−x0 )/
2 2
Ψ(p) = √
2π (2πσ2 )1/4 −∞
−ipx0 / −p2 σ2 / 2 ∞
e e
= √ dxe−(x−x0 +2ipσ/ )2 /4σ2
2π (2πσ2 )1/4 −∞
√
2σ/ −ipx0 / −p2 σ2 / 2
= e e
(2π)1/4
Being the Fourier transform of Ψ(x), Ψ(p) satisfies the normalization condition:

MOMENTUM AND SPACE OPERATORS 33

Ψ∗ (p )Ψ(p)
1= dxΨ∗ (x)Ψ(x) = d pd p dxei(p−p )x/
2π
= d pΨ∗ (p)Ψ(p) = d pP(p) (2.33)

where we used the identity (see Appendix on Fourier transforms):

dxeipx/ = 2π δ(p) (2.34)

Eq.2.33 is known in the theory of Fourier transforms as Parseval’s theorem. For a Gaus-
sian wavefunction Ψ(x) (see above example) P(p) corresponds to a Gaussian centered
on p = 0 with standard deviation < (p− < p >)2 > = /2σ. The mean value of the
momentum satisﬁes:

< p >= d pΨ∗ (p)pΨ(p) = d p d pΨ∗ (p )δ(p − p)pΨ(p)

Using the identity Eq.2.34 and Eq.2.31 allow us to express the momentum operator in
real space as:
1
<p>= dxd p d pΨ∗ (p )ei(p−p )x/ pΨ(p)
2π
1 ∂
= dx d p Ψ∗ (p )e−ip x/ dp Ψ(p)eipx/
2π i ∂x
∂
= dxΨ∗ (x) Ψ(x) ≡ dxΨ∗ (x) pΨ(x)
ˆ
i ∂x
where we identify the momentum operator in real space as:
∂
p=
ˆ ≡ −i ∂ x (2.35)
i ∂x
Similarly by writing the mean position < x > in momentum space we come to identify
the position operator in momentum space as:
∂
x=i
ˆ ≡ i ∂p (2.36)
∂p
Notice that momentum and space-operators do not commute. In real space:

< x|[x, p]|x >= dxΨ∗ (x)(x p − px)Ψ(x) =
ˆ ˆ dxΨ∗ (x)[x∂ x Ψ − ∂ x (xΨ)] = i
i
We would have obtained the same result by computing the commutator in momentum
space: < p|[x, p]|p >= d pΨ∗ (p)( x p − p x)Ψ(p) = i . Since two observable cannot
ˆ ˆ
share the same eigenstates if they do not commute, the position and momentum of a
particle cannot be simultaneously determined with absolute precision. The same result
holds also for the commutator of time and energy:

< x|[H, t]|x >=< x|Ht − tH|x >= dxΨ∗ (x)(i ∂t t − ti ∂t )Ψ(x) = i


2.4.1 Heisenberg uncertainty principle
From the result that momentum and space do not commute we can derive the Heisenberg
uncertainty principle. Consider the action on state |ψ > of the operator xo +iλpo where λ
is a number and the operators xo and po are the deviation of the position and momentum
operators from their mean: xo = x− < x > and po = p− < p >:

|φ >= (xo + iλpo )|ψ >

Since [xo , po ] = [x, p] = i the positiveness of the probability implies that:

0 ≤ < φ|φ > = < ψ|(xo − iλpo )(xo + iλpo )|ψ >
= < ψ|xo + iλ[xo , po ] + λ2 p2 |ψ >
2
o
= < ψ|xo |ψ > − λ+ < ψ|p2 |ψ > λ2 = P2 (λ)
2
o

For the quadratic polynomial P2 (λ) to be non-negative for any real λ, its determinant
has to satisfy:
2
− 4 < ψ|xo |ψ >< ψ|p2 )|ψ >≤ 0
2
o

Or in terms of the position and momentum variables:

2
< ∆x2 >< ∆p2 > ≥ (2.37)
4

This is Heisenberg’s principle: it sets a limit on the precision with which one can mea-
√
sure both the position δx = < ∆x2 > and the momentum δp = < ∆p2 > of a physi-
cal system. The smallest uncertainty (the equality in Eq.2.37) is obtained for a Gaussian
probability distribution as can be verified from the example worked out above. Since
the Hamiltonian and time operators do not commute similar uncertainty relation can be
obtained for the energy and time uncertainties:

2
< ∆E 2 >< ∆t2 > ≥ (2.38)
4

Notice that Heisenberg principle is a direct mathematical consequence of the QM
description of physical systems by a complex wave-function Ψ(x) (Eq.2.13) and of de
Broglie’s relation between wavelength and momentum. Heisenberg uncertainty princi-
ple is a tautology: a consequence of the definition of Fourier transforms (see Appendix).
In the context of communication it has been known for a long time: a very short time
signal is spread over a very large frequency spectrum. In the context of optics we have
also encountered it in the diffraction pattern from a hole which is larger the smaller the
hole is.

SCHROEDINGER’S EQUATION 35

2.5 Schroedinger’s equation
We have determined the representation of the position operator in momentum space and
of the momentum operator in real space. Note that in its eigenspace the momentum
space p is diagonal, i.e. it is a number. We can now write the representation of the
ˆ
Hamiltonian in any of these Hilbert spaces. It is often easier to work in real space, in
which case the Hamiltonian which is the sum of kinetic and potential energy is written
as:
p2
ˆ 2 2
H= + V(x) = − (∂2 + ∂2 + ∂2 ) + V(x) = −
x y z
2
+ V(x) (2.39)
2m 2m 2m
and Schroedinger’s equation, Eq.2.16 can be recast as:
∂ 2
i Ψ(x, t) = − 2
Ψ(x, t) + V(x)Ψ(x, t) (2.40)
∂t 2m
Multiplying Eq.2.40 by Ψ∗ , its complex conjugate by Ψ and subtracting the two yields
a conservation law for the probability distribution P(x) = |Ψ(x)|2 :
2
i ∂t P = − (Ψ∗ 2
Ψ−Ψ 2
Ψ∗ )
2m
2
= · (Ψ∗ Ψ − Ψ Ψ∗ )
2m
which can be recast in the usual form (see Eq.?? for the charge distribution in EM):
Ψ∗ vΨ + c.c.
ˆ
∂t P + ·J=0 with J = Ψ∗ Ψ + c.c. = (2.41)
2im 2
where c.c stands for the complex conjugate and v = ˆ /im is the velocity operator.
Eq.2.41 is a very important self-consistency check of Quantum Mechanics, since for
P(x) to be interpreted as a probability distribution it must satisfy a conservation law.
This equation expresses the intuitive expectation that the change in the probability of
finding a particle at a given position is equal to the particle flux gradient.

2.5.1 Diffraction of free particles
If the potential is null V(x) = 0 then the eigen-solutions of Schroedinger’s equation,
Eq.2.40:
2
dΨ
i =− 2
Ψ (2.42)
dt 2m
are plane waves:
Ψ(x, t) = eik·x−iEk t/
with p = k and Ek = k /2m. These plane-waves are eigenmodes of the momentum
2 2

operator:
pΨ(x, t) = −i
ˆ Ψ(x, t) = p Ψ(x, t)


F. 2.11. The double-slit or Young’s experiment. (a) a wave passing through two slits in
a screen generates two wave-sources which interference creates on a far-away screen
a pattern of interference consisting of alternating minima and maxima of intensity.
(b) a particle in state |O > impinging on a double slit generates two states |I > and
|II > corresponding to its passage through slit 1 or 2. The phase of these states
evolves as exp ikl. If their coherence is maintained they can interfere on a screen a
large distance z from the slits, generating an oscillating pattern related to their phase
difference: φint = k(l2 − l1 ) = kd sin θ. (c) Observation of the interference pattern of
electron passing through a double slit and impinging on a camera. Each electron is
observed as a particle (white dot) with a well defined position on the camera. The
QM interference pattern is only visible when a sufficiently large number of particles
has been observed (A.Tonomura, Proc.Natl.Acad.Sci. 102, 14952 (2005).

One of the most striking confirmations of the QM mechanics picture is the obser-
vation of a diffraction pattern like the one seen with electro-magnetic radiation when
a free particle is passed through one or a few slits (see ??? and Fig.2.11(a)). Let the
particle be in an eigenstate

|0 >= Ψ(x, t) = eikz−iEk t/

as it impinges on a screen that is absoring except for two apertures of size a a distance
d apart. In the far-field, i.e. at distances z d2 /λ the wave amplitude is given by
Huygens’s principle, Eq.??:

eik(z+(x +y )/2z
2 2

Ψdi f f (x, y, z) = dx dy e−i(kx x +ky y ) (2.43)
iλz
Where k x = kx/z and ky = ky/z. Thus the probability of detecting a particle on a screen
a distance z from the slits is:

4a2
|Ψdi f f |2 = sinc2 k x a cos2 k x d/2 (2.44)
(λz)2
where sinc x ≡ sin x/x. If the distance between the diffraction slits is much larger than
their width (d a), the probability oscillates with a period: δx = λz/d. While each

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