Introduction to spectroscopyIntroduction: the Unity of ScienceElectromagnetism (textbook: Griffiths, Jackson)

Electrostatics (Coulomb’s Law) in vacuum and dielectricsMagnetostatics (Ampère’s Law)Electromagnetism (Faraday’s Law and Maxwell’s equations)The wave-equation, simple radiating systemsOptics (Fresnel’s Law)

Mid-term Exam 2/9/12 (30%)Quantum Mechanics (textbooks: Feynman, Bohm, Dirac,….)

The problem of atomic spectra and black-body radiationQM formalismTwo-body systems: the Amonia maserDirac’s and Schroedinger’s equationsAtoms and molecules: the Periodic TableThe chemical bond and perturbation theory

Final Exam (40%)

The Unity of Science

David Bensimon

To explain almost everything with almost nothing

Mathematics: the language of ScienceMathematics is a game with rules like chess, go, backgammon, …

These arbitrary rules are the rules of logics.

logics ~ arithmetics (Russel, Whitehead, Gödel)

arithmetics ~ computing (Türing, Von Neuman)

Within mathematics there exists sub-fields with their own arbitrary additional rules (axioms), for example Euclidian or Riemannian geometries that differ in their axioms of parallels. Contrary to Kant’s assertion no mathematical proposition is informative.

Science investigates the quantitative (measurable) relations between objects (mass, speed, position, charge, …). Mathematics by dealing with numbers provide a conceptual framework for their formulation.

The fundamental theories

Gravitation – Newton, Einstein(free fall, motion of planets, tides, expansion of the Universe, etc.)

Electromagnetic theory– Maxwell, Faraday, Ampère, …

(electricity and electrical motors and generators, radio waves, optics, rainbow, ….)

Quantum mechanics- Einstein, Bohr, Schrödinger, Heisenberg, …(atoms, molecules, chemistry, transistors, conduction, magnetism, etc.)

Statistical mechanics- Boltzman, Einstein, Gibbs, …(heat, engines, refrigerators, lasers, phase transitions, plastics, blackbody radiation of the Universe, etc.)

Gravitation

F = G m M / r2NGC4414

Every gravitational system is unstable → Big-Bang

Electromagnetic theoryMaxwell’s equation determine the electric and magnetic fields E and B as a function of the charges and currents densities: and J

These equations unify electricity, magnetism and optics (light). They predicted radio-waves and X-rays.

The force exerted by the electromagnetic field on a charge q is:

It explains the movement of electrons in a TV tube, electrical motors and generators, lightning, aurora borealis, particle accelerators, etc.

Electromagnetic spectrum

Quantum mechanics

Quantum mechanics describes the motion of isolated particles:

iћ ∂tΨ = H Ψ

Ψ is known as the particle wave-function. Only the probability |Ψ|2 of finding the particle at a given place and time can be known.

Quantum mechanics explains the chemical properties of elements, the origin of magnetism, light emission, electrical conduction, etc.

Quantum mechanics is the foundation of chemistry, electronics (transistors, computers), lasers, medical imaging (MRI), nuclear power, etc.,

Schrodinger’s equation

Electronic wave in an atom corral

Don Eigler, IBMRadius: 72Å

Direct observation of the wave-like behaviour of electrons

Fe corral on Cu

Quantum Mirage

Co corral on Cu

By placing a Cobalt atom in one of the focii of the ellipse, one creates a replica of that atom in the other focus.

The Bohr atom

The electrons move around the nucleus like planets orbiting the sun.Because the electron is also a wave only certain orbits allow for a stationary wave.The transition from one orbit to an other is accompanied by emission (or adsorption) of a photon of well defined energy.

Low energies

High energies

E = h

The Rosetta stone of the Universe

See: Omnilingual by H.Beam Piper (how to decipher a Martian civilization with the periodic table)

Chemistry and Quantum mechanics

The solution of Schrödinger’s equation for a hydrogen-like atom has the electron orbiting the nucleus on orbitals characterized by their quantum numbers: n = 1,2, ….; l= 0,…n-1; m=-l,…,l. Each orbital can be occupied by at most two electrons (with opposite spin). The maximal number of electrons on the orbitals with quantum number n is thus 2 n2. The chemical properties of atoms is determined by the number of electrons on its not fully filled outer orbitals.

n =1

n =2n =3

The chemical bond

The chemical bond between two atoms is established by the sharing of a pair of electrons that fill up the unfilled orbitals of the respective atoms.

Thus Oxygen which lacks two electrons to fill up its n=2; l=1 orbital will bind to two Hydrogen atoms, each of which lacks one electron to fill up its n=0; l=0 orbital.

The resulting molecule H2O (H-O-H: water) is stable because it minimizes the energy of the Oxygen and Hydrogen atoms.

The unity of life

« What is true for a bacteria is true for an elephant», J.Monod

Life is a set of chemical reactions driven in fine by the Sun’s energy. In contrast with physical phenomena which unity is derived from general principles (EM, QM, StatMech), Life derives its unity from evolution.

Since all life forms evolved from the same source (ancestor), they share the same components (amino-acids, nucleotides, proteins), the same genetic code, the same chemical reactions, etc.

ADNDNA

Statistical mechanicsStatistical mechanics addresses the behaviour of a large number of particles. This large number and the collisions between the particles randomizes their motion. The fundamental concept is entropy.

The entropy S of a system measures the lack of information on the system: it is defined as the logarithm of the number of its possible states. For a regular dice: S = log(6) For a loaded dice: 0 < S = -∑ pm log pm ≤ log(6)

S = log(6) S = log(37)

Law of ideal gasesIf one assumes that Nature is not « loaded » (ergodic hypothesis) then the dynamics of a system of many particles will tend to maximize its entropy S under some fixed constraint (total energy E, volume V, number of particles N, etc.).

Mathematically this is done by maximizing the function:

F= S – (E + V + N)

For a system of m states of energy Em occupied with probability pm with the constraints ∑ pm = 1 and total energy E = ∑ pm Em :

F= S – E – = -∑ pm log pm – ∑ pm Em – ∑ pm

Maximizing F: ∂ F/ ∂ pn = 0 → pn = eEn / ∑ eEm Boltzmann distribution

Law of ideal gases

The temperature is defined as: kBT = -1 = ∂S/∂E , which is the parameter that is equaled when the vessels are brought in contact.

Similarly the pressure is: P = = -1 ∂S/∂V = kB T N/V

This is the law of ideal gases, where we used the fact that for an ideal gas of N particles in a volume V, the entropy S ~ N log(V).

V1, T1,P1 V2, T2,P2 V’1,T P, V’2 V’1,T,P T,P

F1= S1 – (E1 + V1)

F2= S2 – (E2 + V2)

F= S – (E + V) = S’1+S’2 – (E’1+E’2)

+ V’1+V’2)=

F’1= S’1 – (E’1 + V’1)

F’2= S’2 – (E’2 + V’2)

Temperature of a gas of photons

According to the laws of QM, electromagnetic waves (e.g. light) are also particles: photons.

In contrast with atoms their energy (E) is not linked to their speed (the speed of light c which is fixed), but to their frequency ().

Like the atoms of a gas, confined photons thermalize at the temperature of their container. Their energy (and thus their frequency) are proportional to that temperature: E = h ~ T

By observing the light emitted by a body one can thus deduce its temperature, be that a human body (infrared), the Sun (visible) or the Universe (radio waves).

E = h

Sun’s spectrum and temperature

The universe is expanding

The Doppler shift to the red indicates that galaxies are moving away from us: the Univers is expanding.

The temperature of the Universe is 2.7°KIn

tens

ité

The Universe is the best example of a black body

Spatial distribution of primordial radiation

After susbtracting radiation from the Milky Way

T ~ 0.00001 °K The residual radiation inhomogeneities served as nucleation points for the formation of galaxies

T ~ 0.001 °K (as a result of the galaxy’s motion (Doppler effect) at à 600 km/sec. )

After susbstracting the galaxy’s motion. The equatorial contribution is due to the stars in our galaxy (the Milky Way).

N

S

0° 360°

Temperature map T ~ max

The mind-boggling enigma“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“ E.Wigner.

Why can we predict the temperature of the Universe but not next week’s weather? Why can we predict the behaviour of an electron but not our dog’s?

The mathematical equations of our fundamental theories (EM, QM) are soluble because they are linear (the field of a sum of particles is the sum of the particles’ fields).

This might not have been so (hydrodynamics). The Universe would then have remained mysterious and unpredictable, like the weather.