Bragg diffraction of atoms by light standing wave

When a monochromatic electromagnetic wave incidents at some angle on a crystal lattice, it is reflected by the crystal lattice planes and a maximum intensity can be observed if the path difference of the reflected waves from the adjacent planes is an integral multiple of the wavelength of the wave. This phenomenon is called Bragg diffraction, after William Henry Bragg and his son William Lawrence Bragg, who were jointly awarded the 1915 Physics Nobel Prize for their services in the analysis of crystal structure by means of X-rays. Interestingly enough, diffraction can be observed when an atomic beam is directed through a light standing wave. A laser standing wave can be produced by reflecting a laser beam between two mirrors. When a beam of atom is passed through the standing wave, an atom absorbs a photon from a beam of laser and it does stimulated emission of a photon of the same momentum into the next beam, imparting a net momentum to the atom which is two times the momentum of a single photon. This is similar to the Bragg diffraction of electromagnetic waves as the laser standing wave is analogous to the crystal lattice planes.

Note: This posting is on progress.

What is the Poisson's Spot?

When a small opaque circular disc is placed in front of a point source of light, it casts a shadow on the screen placed behind it. But, the most amazing observation will be that there will be a bright spot at the center of the geometrical shadow. This spot is called the Poisson's Spot, after the famous physicist Simeon Denis Poisson. Poisson in the early nineteenth century, mathematically derived from the diffraction theory that there should be a bright spot at the center of the geometrical shadow of the disc, which he disliked as it was against an intuition. But when his mathematical results were experimentally verified, he became a strong follower of the wave theory of light.

The formation of the Poisson's spot can be explained using the Huygens' Theory. When light from a point source hits a circular disc, each point on the circumference of the disc acts as a secondary source of light. Since the waves from the points at circumference reach the center of the geometrical shadow in phase, they interfere constructively. As a result of this, the intensity there is a maximum which gives rise to the Poisson's Spot.

Noise-supressed large area atom interferometers

A paper-review

Atom interferometers can use just cold atoms or a BEC. The difference is that a BEC is a coherent source of cold atoms like a LASER unlike a beam of cold atoms which is at temperatures much above the BEC temperature. The researchers all around the world are working to realize more sensitive and more efficient atom interferometers by using either of these sources. One reason for that is - an atom interferometer is much more sensitive than an optical interferometer in the sense that the signal-to-noise ratio of an atom interferometer is greater than 10^11, compared to an optical interferometer of comparable area and the particle flux. Therefore, atom interferometers can prove to be more effective in precision measurements.
Atom interferometers are of trapped-atom-type or free-space-type. Recently, Herrmann et. al. have shown that a pair of simultaneous conjugate Ramsey-Borde atom interferometers can be operated at large momentum transfer to cold atoms from the optical pulses to supress the vibrational noise and to enhance the enclosed space-time area by a factor of 2500 of the area of the existing atom interferometers. They have used this interferometer to measure the fine structure constant more precisely.
A cold atom beam of Cs-133 shot vertically upward in a space of about a meter high is split by using a pi/2 laser pulse. It is split again after an interval of time by using another pi/2 pulse and two simultaneous interferometers are formed by using further two consecutive pi/2 pulses before they are finally recombined. Each atom can be given a momentum as large as that of the momenta of twenty optical photons (20*hbar*k). The total phase gained by the atoms during the interferometric cycle time has three contributions - the phase due to recoil velocity of the atoms, the phase due to free-evolution of the wave packet between the beam splitters and the phase due to the interaction of the wave packet with beam-splitting pulses. By operating both interferometers simultaneously, the phase due to free evolution and the effects of noise/vibrations can be subtracted off, leaving the contributions of the recoil velocity and the splitting pulses only. This is how the simultaneous conjugate Ramsey-Borde atom interferometers can be operated to get a large area without losing contrast.

What and why is normalization of wave function?

Quantum mechanics treats moving matter as a wave, called a 'matter wave'. A matter wave is always assigned a wave function, usually called a 'psi'. The wave function is usually complex and composed of two parts - an amplitude and a phase. The square of the modulus of the wave function gives the probability density of finding the particle somewhere in space. If this quantity is integrated around the whole space, it gives the probability of finding the particle in that space. Since the particle should exist somewhere in space, this probability should be a maximum. To fulfill this requirement, the wave function is normalized in such a way that the total probability in the whole space of our consideration is 1 (maximum). The fixing of the coefficient of the wave function by requiring to fulfill the above condition is called normalization. Therefore, to get a normalized wave function - (1) Write a suitable wave function with some coefficient to represent the quantum system of your consideration. (2) Integrate the modulus square of the wave function over the whole space of your consideration and get the value of the coefficient in terms of some known quantities. (3) Plug the coefficient back into the wave function, which now is a normalized wave function. It is more convenient to work with a normalized wave function than with a non-normalized one !

Matter waves and wave packet

Moving matter behaves like a wave in quantum mechanics. The wavelength associated with a matter wave can be obtained by dividing the Planck's constant (6.63*10^-34 Js) by its momentum. This wavelength is called the de Broglie wavelength and the matter waves are called the de Broglie waves after the pioneer on matter waves, Louis de Broglie, who was awarded the 1929 Physics Nobel Prize for his discovery of the wave nature of electrons. Since for a massive particle moving with some velocity, the momentum of the particle is large and since this quantity divides the Planck's constant which itself is very small, the de Broglie wavelength is very small and hence we can't visualize the wave nature of the moving matter in day-to-day observation. For example, if a ball of 1kg is rolling on a ground with a speed of 1 m/s, the de Broglie wavelength of the ball is just 6.63*10^-34 m, which is too small to realize as a wave but if we consider atomic and subatomic particles moving at some speed, the wavelength is large enough to visualize the wave nature of matter, at least, doing some precise experiments. For example, the electron microscopes work on the principle that a de Broglie wave is associated with the moving electrons. In atom interferometers, the moving atoms are considered to be waves. In fact, in the modern quantum world, there are a number of devices where matter waves are used.
A wave packet is the unit of a matter wave which carries matter and energy in its direction of propagation at the speed of the wave. It is a small region around a matter particle where there is a very high probability of finding the particle. These wavepackets show properties like that of optical waves ,e.g., diffraction, interference, etc. The following YouTube videos depict how matter waves look like !

God does not play dice with the universe !

No one can say for sure which face is turned up when a dice is rolled in a game of dice. One can say something like - the chance of getting a 6 in a single roll of a dice is 1/6 and so on. Quantum physics also deals with the probability of occurrence of some event or existence of something rather than the certainty of it. In most of the cases, we find the probability of occurrence of any quantum mechanical event or the uncertainty that exists in its measurement. Heisenberg, Born and other quantum physicists were in favor of the probabilistic nature of the atomic and subatomic phenomena which Einstein did not like. In his disagreements of such 'may be' kind of thing, Einstein said that - 'God does not play dice with the universe'.
But the winner were the quantum physicists in the sense that the probabilistic nature of quantum physics was shown experimentally and a new discipline was established !
You can also visit HERE and HERE !

What is Gross-Pitaevskii Equation?

A non-degenerate system can be explained by the Schrodinger equation, which is a linear second order differential equation: ( i hbar del/ del t ) psi = (- hbar^2/(2m) * laplacian^2) psi + V psi. But for a degenerate quantum gas like a Bose-Einstein Condensate(BEC), there is a non-linearity caused by the atom-atom interactions, which needs to be explained by a non-linear second order differential equation of the form: ( i hbar del/ del t ) psi = (- hbar^2/(2m) * laplacian^2) psi + V psi + NU|psi|^2 psi. This equation was first derived by Gross and Pitaevskii independently and is called the Gross-Pitaevskii equation. The last term in the equation is the nonlinear term introduced by the atom-atom interactions.

What is Schrodinger's Equation ?

In quantum mechanics, the Schrodinger equation plays the same role as the Newton's second law does in classical mechanics. The Schrodinger equation was such a breakthrough made in the history of mankind which revolutionized the world of physics. The total energy of a particle is obtained by adding its kinetic energy to its potential energy. This is called the Hamiltonian of the particle. A simple way to get the most respected equation which was written by Erwin Schrodinger in the first quarter of the 20th century is to replace the energies by operators. The total energy is replaced by ( i hbar del/ del t ), the kinetic energy is replaced by ( hbar^2/(2m) * laplacian^2) and the potential energy is replaced by the potential energy operator, V. Then, we get the time dependent schrodinger equation, which can be operated to a wave function psi as :
( i hbar del/ del t ) psi = (- hbar^2/(2m) * laplacian^2) psi + V psi
This equation is the backbone of quantum mechanics. If we consider a free particle, the potential energy of the particle is zero. Then this equation explains how a free particle evolves with time. The solution to this equation under a suitable potential always gives the time evolution of a quantum system under that potential. If we replace ( i hbar del/ del t ), by the energy operator E, the above equation takes a form:
( E ) psi = (- hbar^2/(2m) * laplacian^2) psi + V psi
This is time-independent (stationary) Scrodinger equation. The solution to this equation gives the stationary state of a system depending upon the nature of the potential. For example, if we take the potential to be parabolic, V = (1/2)m (omega)^2 (x)^2, in the time-independent schrodinger equation, we will get the stationary states of this potential, which is popularly known as the stationary states of a harmonic oscillator.

What are GOOD QUANTUM NUMBERS?

In classical mechanics, we can assign any number of variables to specify the state of a system. For example, for a free particle, its position, momentum, etc,. can be known and can be assigned simultaneously to specify the state of the free particle. But the story of quantum mechanics is different. If the position of a particle at some instant is correctly known, its momentum will be highly/maximally uncertain. This means, we can't assign both the position and momentum to specify the quantum mechanical state of the particle simultaneously. In this sense, the quantities (observables) which can be assigned simultaneously to a quantum system at any time to specify its state at that time are called the good quantum mumbers of the system. A set of good quantum numbers is also called a complete set of commuting observables (C.S.C.O.). These coordiantes commute each other, which means that their measurements can be made simultaneously.

Is light a wave or a particle?

Sir Isaac Newton gave a theory of light- called the corpuscular theory of light. As soon as the light particles reach some material mediums like glass, water, etc., the particles are attracted by a force (F = ma) and the particles will be accelerated in the medium, resulting into an increase of speed. This means that light should travel faster in material medium than in a vacuum. But this theory failed when Olaf Roemer, Armand Fizeau, Leon Foucault and other experimentalists measured the speed of light, the result of which showed that the speed of light in a material medium be less than that in a vacuum. To comply with the experimental observations, the wave theory of light was proposed at about the end of the 18th century by Christiaan Huygens. A deeply-rooted concept is difficult to change at once. The world had to wait about a hundred years to accept that light was a wave until when Thomas Young first demonstrated the interference of light in his famous double-slit experiment in 1801. 'Light is a wave' was now established. It remained like that until the end of the 19th century. One of the three groundbreaking papers of Albert Einstein in 1905 was on the photoelectric effect. The phenomenon of photoelectric effect couldn't be explained if light behaved like a wave. Therefore, he proposed the particle theory of light. Light consisted of 'quanta' (packets) of energy-which he called 'photons.' His theory explained the phenomenon of the photoelectric effect without any discrepancy, proving that light shows a particle-like nature. Although the fame of Einstein spread all over the world by his work on the Theory of Relativity, he was awarded the Nobel Prize of Physics in 1921 for his discovery of the law of the photoelectric effect.
Thus, light shows a dual nature - wave like nature in the phenomena of reflection, refraction, diffraction, interfernece and polarization but particle-like nature in the phenomenon of the photoelectric effect.

Can ALICE be of BOB?

BOB and ALICE make a short talk over phone. Bob is desperate to get Alice and the same goes with Alice, too. But they have heard that there is Heisenberg Uncertainty Principle (?), which may be a problem(?) for their union.

HERE is their telephone conversation:

BOB : OH ALICE, YOU'RE THE ONE FOR ME !
ALICE : BUT BOB - IN A QUANTUM WORLD, HOW CAN WE BE SURE ?

Can ALICE be of BOB?