Eigenstates, eigenvectors, and eigenvalues

The eigenstates of  a system are its characteristic states, the eigenvectors are the characteristic vectors describing the states,  and the eigenvalues are the characteristic values representing the states of the system. For example, if we consider a one dimensional quantum harmonic oscillator, its ground state is one of its eigenstates which is represented by the eigenvector |g> and the eigenvalue is (one-half *hbar*omega), where (omega) is the frequency of the oscillator. The other eigenstates of the oscillator are the excited states |e> with the increase of energy by (hbar*omega) while climbing up from the ground state.

Quantum World

'The World of Small' has been given a new title 'Quantum World' to make its purpose obvious to the readers !

Happy reading !

Condensed Matter via Atomic Physics

Condensed matter systems are complicated systems  because they are ensemble of atoms and molecules held  tight by inter-atomic and inter-molecular forces. The forces arise as a result of various interactions - interactions between electrons in neighboring atoms, dipole interactions, overlap of the wave functions, etc. A condensed matter system may also have defects  and dislocations of various kinds. Therefore, studying the properties of  condensed matter systems like a chunk of gold or  a cup of water is not an easy subject. 

Because of  the discovery of new ideas and a rapid development of technology, there are a lot of new methods developed to study condensed matter systems. One such way is simulating condensed matter using an optical lattice. An optical lattice is a 'light crystal' formed by interfering laser light waves traveling in opposite directions. A pair of counter-propagating laser light waves form a standing wave - a wave with intensity maxima and minima at regular spatial intervals. This is a one dimensional optical lattice. Two orthogonal pairs of lasers form a sheet of optical lattice - a two dimensional structure. If three pairs of counter-propagating lasers along X-, Y- , and Z- directions interfere each other, they form a structure which is called a three dimensional optical lattice. 

An optical lattice can be controlled as you wish, by varying the wavelength of the component lasers. What is an optical lattice good for? An optical lattice is a clean system - free of defects. It is tunable, controllable and easily manipulated as desired and required. An optical lattice can simulate a condensed matter system. How? In an optical lattice, there are arrays of regularly spaced sites of low potential which can be filled with atom(s). One can fill with similar or dissimilar atoms in the sites. Therefore,  the optical lattice with atoms sitting in the potential minima simulates condensed matter systems. 

A condensed matter system as it stands is 'GOD-MADE', and so it can not be manipulated easily. For example, you can not change the lattice constant or the coupling strength in a block of silver. Therefore, a system which may represent widely distributed condensed matter systems - ranging from a water molecule to the complex structure like a human brain is extremely important to understand the properties of the condensed matter, and ultimately the Universe !

Note: This posting is in progress. Your comments are highly appreciated !

BOSE-EINSTEIN CONDENSATE vs EXAM

Read for fun !


A Bose-Einstein condensate(BEC) is like an Exam ! It seems funny, but please keep on reading, I'll prove it. A BEC is created out of atoms or molecules at low temperatures so that all atoms in an ensemble reach the ground state of the potential or a trap. A commonly used trap is a Magneto-Optical Trap (MOT). This is like an exam room with physical walls, exam questions, proctors, professors, and the career in a long term. They bind the test-takers in the exam room tightly.

The most energetic atoms leave an MOT quickly, the same way as a smart test-taker does. The low energy atoms go towards the lower and lower state of the trap and they are bound to form a BEC. Similarly, a less smart test-taker remains in the exam room for the whole time assigned to the exam or even more if the proctors or professors allow to do so ! If the trap is stronger or tighter, the atoms are more firmly trapped than in a weaker trap and the coherence is also maintained for a longer time. Analogously, if the questions in the exam are tougher, it makes a test-taker to remain in the exam room for a longer time; but if the questions are easier, the test takers escape right away ! A good exam will help making a good career of a test taker; it is a similar situation to a good BEC, which can be used for various purposes ranging from the study of its properties to the applications in inertial navigation systems and precision measurements. Oh, most importantly, all atoms in a BEC are in the same quantum state like all the test takers in an exam room; all of them are solving the problems the same way, with a goal of succeeding in the test; they are supposedly in the same state of their minds!

I hope I convinced you that a BOSE-EINSTEIN CONDENSATE is NOT different from an EXAM !

Spin-1/2 system is bizarre object !

Consider a spin-1/2 particle. Rotate it about an axis by a 2 pi angle. You are not getting it back into the same state but with sign flipped. If you rotate it further by another 2 pi angle, you will see the particle in the original state. This is a bizarre object, which is beyond our intuition !

If you ask physicists, why is this? The answer is simple : it is an SU(2) physics, not an SO(3) ! If you want to visualize it, please look at Feynman's description of the rotation of a coffee cup in the following links:

(1) WIKI: http://en.wikipedia.org/wiki/Orientation_entanglement

(2) DEMO: http://www.youtube.com/watch?v=Rzt_byhgujg

Your views on this matter are highly welcome !