# 4803-LECTURES

What is Quantum Mechanics ?
Quantum Mechanics is the theory describing the laws of nature governing the physics of very small systems. How small ? Smaller than simple molecules. Typically few Ångstroms (Å) for elementary systems.

But there are examples of macroscopic systems that behave like quantum objects: superconductors, superfluids, quantum Hall devices, Bose-Einstein condensates   (BEC).  Some stars, heavier than the Sun, also behave like quantum systems: white dwarfs, neutron stars .

The quantum systems liable to be used in quantum computing are : photons produced by laser, trapped ions,  cold atoms, electrons (e.g. in quantum dots), nuclear spins, (in NMR), Josephson junctions. Their typical size may vary enormously.

Typical sizes
atoms                        1-3 Å, namely 10 -10 m
nuclei                        1-3 Fermis (F) namely 10-15m,   i.e. 100,000 times smaller than an atom
electrons                       10-18 m ,  namely 1000 times smaller than a nucleus
photon                       size of its wavelength,
for visible light it varies between  0.4-0.8 µm, namely 1000 to 10,000 times the size of an atom.
Josephson junction : depends upon the device used; it can be as small as few µm

Particle-wave duality
Each quantum system can be seen from two points of view: as a particle, with mechanical properties, or as a wave, with propagation, diffraction and interference properties. Interferences or diffraction produce maxima and minima of intensity. Max Born (1926) showed that the intensity, at some point x, of the wave attached to a particle, should be interpreted as the particle density probability of being at x.

Particle   characterized by
mass                    m
electric charge     =  q
spin                     =  s
other "quantum numbers"  (isotopic spin, baryonnic number, ....)

Mechanical data :
energy             =  E
momentum      =  p = (px, py, p z )

Wave data   characterized by
frequency            = ν                           or pulsation  ω = 2 π ν
wave vector        =  k = (kx, ky, kz             |k| = 2  π / λ           λ : wavelength

Planck-Einstein-de Broglie relations  wave and mechanical data are related through Planck's constant

h = 6.02 x10-34 Joules.sec

E = hν =   (h/2 π ω     (Planck 1900, Einstein 1905)             p = (h/2π )  k       (Einstein 1905, de Broglie 1923)

Heisenberg uncertainty inequalities

If particles are wave, the minimal length scale accessible is the wavelength. Thanks to de Broglies's relation, the smaller the wavelength the bigger the momentum.
In 1927, Heisenberg remarked that, as a consequence of this remark, there is an inequality relating the accuracy for measurements of position and momentum ofthe form

Δ x . Δ p   >   h / 2 π

He then realized that such a relation applies to all pairs of conjugate variables occuring in Hamiltonian Mechanics. One way to get a mathematical framework liable to provide such an inequality automatically consists in representing the observables  by noncommuting objects called q-numbers (Dirac 1925). Heisenberg was the first to realized that in his seminal paper written in May 1925 ( Heisenberg, W.  "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen ",  Z. Phys., 33, (1925) 878--893) .  His proposal was soon interpreted by Max Born in July 1925: what Dirac later called q-numbers were nothing else than matrices .

Spin

The spin of a particle, like electron, provides the most elementary model for a quantum bit of information called the qbit . It is the quantum version of an angular momentum, characterizing the rotating motion of the particle on itself (spinning). However, when this angular momentum becomes very small (of the order of Planck's constant), thanks to Heisenberg's inequality, it becomes impossible to measure simultaneously the three components of this angular momentum. In particular the three spin components must be matrices

s = (sx , sy , sz ) =    (h /2 π )(σ x , σ y ,σ z )

where the σ i 's are the so-called Pauli matrices  and are given by

X = σx   =
 0 1 1 0

Y = σy  =
 0 - i +i 0

Z = σz  =
 +1 0 0 -1

Applications

Applications of this observation range from the explanation of the photoelectric effect (Einstein 1905) to the electronic microscop in which light is replaced by electrons, going through the structure of atoms, Compton's effect, etc.. Particle accelerators can be seen as an extrapolation of electron microscops.

In the photoelectric effect, the light must be seen as made of elementary particles called photons with energy given by the Planck relation. An energy threshold  is observed corresponding to the minimum energy the photon must provide to an electron of the photoelectric cell to be accelerated from one electrode to the other. The photon energy is transformed into an electric current. Thus only photon of frequency high enough can be detected by the cell. By measuring the relation between the light frequency and the electron energy (electric current), Planck's constant can be measured accurately (Franck & Hertz).

In electron microscops, or in particle accelerators, there is a relation between energy and momentum: for non relativistic particle it reads

E = p 2 /2m

The electron energy can be varied through applying a potential difference through the equipment: it increases the momentum thus, thanks to the de Broglie relation, it decreases the wavelength. This can be used to produce an optical image as well as a diffraction patterns (transmission electron microscopy T.E.M.). Such technics is currently used in crystallography to analyze the structure of materials, crystals, large molecules like proteines or DNA. By decreasing the wavelength, the optical imaging provides a way of seeing smaller and smaller objects. The cost is the increase of the incoming electron energy. In particle accelerators, the energy is so large that it is possible to see inside the nucleus, or even inside the protons or neutrons.

It is believed that the total energy of the Universe is finite. If true, there is a minimal wavelength below which it is impossible to distinguish between points in space: the Planck scale which of the order of 10-30m ! At this scale the Universe should be quantized: the  components of the position-time should become noncommutative with an uncertianty relation.

On the other side of the application are cold atoms. By slowing down the velocity of an atomic jet, like Cesium (Cs) or Rubidium (Rb),
the atomic wavelength can become very large if compared to the actual size of the atom. Typically for a velocity of few cm/s, the wavelength can be as big as few µm : that is 10,000 times the size of the original atom ! By exchanging energy between light and cold atoms, the information contained in photons can be stored, transmitted and read.  Moreover, such atoms can combine if they behave according to the Bose-Einstein statistics (this is the case for Rb) to condensate into a macroscopic quantum object, the BEC, that may be used eventually in quantum computers.