Simple explanation of quantum mechanics Can you please describe quantum mechanics in simple words? When ever I read this word (quantum computers, quantum mechanics, quantum physics, quantum gravity etc) I feel like fantasy, myth and something very strange that i can never understand.
So what is quantum mechanics?
 A: Deepak's answer covers the question, except if "simple" means "in terms an english major would understand". I have been trying to formulate such an answer, fwiw:
Mechanics covers the macroscopic world, the world we see even with glass microscopes and simple telescopes. It explains the motion and interaction of bodies, from large, to quite small. Wave mechanics covers the macroscopic behavior of waves as observed in the sea and the behavior of light. The theories by the beginning of the 20th century were beautifully established so that some physicists thought that real physics was finished, and engineering was all that was left.
Then came the quanta. They came from various fronts. Chemists studying the elements and measuring atomic weights came up with a table of elements that had discrete numbers. The photo electric effect showed that light was not always behaving as a classical wave, because light could hit material and kick out a single electron. Then lines were found in the light spectra of elements. This forced physicist to think of energy coming in discrete numbers instead of a continuum, quanta of energy. 
The first model of an atom, the Bohr model, had the nucleus  like a miniature sun and the electrons as planets around  it. The discrete lines observed though, meant that the orbits had fixed positions. Electrons could only occupy certain orbits, get kicked to a higher one and release a photon falling back, with a specific line. Classical mechanics could not solve this conundrum: why there were discrete orbits and why the electrons did not fall into the nucleus anyway since the nucleus is positive and the electrons negative and the attraction inevitable classically. Physicists were forced to postulate quantum mechanics and developed a whole new set of theories of how the microscopic atomic world worked: in quanta of energy. 
In this new mathematical theory the electrons stay in orbit around the nucleus because  they can only change orbits by quanta of energy. They  cannot fall into the positive nucleus  because there is a lowest stable ground state, which is the lowest energy an electron can have. An atom can gain a quantum of energy and the electron can jump to a higher energy state; it can not go lower than the ground state. 
One can never take a micro photo of the electron, only a probability distribution where it might be, can be computed by the new theories. Studying the probability distributions coming out of the solution of quantum mechanical problems, it was found that in the microscopic world particles sometimes behave as waves, and waves (light) sometimes behave as particles, depending on the circumstances under study. Experiments confirmed all this.
Quantum mechanics explains beautifully the light spectra of atoms, the periodic table of elements and nuclear interactions and a lot of phenomena, from transistors to lasers. The price payed is a loss of intuitive understanding of  "particle" and "motion" , new intuitions have to be developed to understand the predictions of a quantum mechanical world that need long study and perseverance.
Quantum mechanics manifests macroscopically too, in superconductivity, where kilometers of wires run with currents controlled by quantum mechanical equations; in superfluidity; crystals themselves are built up by  a solution of a quantum mechanical system, classical physics could never contain this type of matter manifestations.
Specifically :
quantum physics includes 
 a) quantum mechanics: solutions of problems with known potential energy using the Schrodinger or Klein Gordon equations. The problems are treated as particles moving in a potential well 
b)second quantization, where particles are treated as creation and annihilation operators acting on the vacuum, and 
c)  quantum strings where one has quantized strings and the particles are energy levels on these strings.
d) whatever new coming down the theoretical pike.( It is turtles all the way down :) )
 To make any sense of this you have to read further.
quantum computers,  utilize the knowledge gained by quantum mechanics to create  computers, and my knowledge is covered by the wikipedia article
quantum gravity  is an attempt to extend quantum mechanics to general relativity, which is a classical theory. There are computational difficulties in doing this. Theorists are aiming at a unified theory of everything, called TOE. To get that, one has to quantize gravity, which means that the gravitational field should be coming in quanta, called gravitons. This is ongoing research and connected with string theory research, since, up to now, string theories are the only ones that have come up with both the quantum levels needed to describe particles and also to quantize gravity.
Deepak's answer is a good  beginning and also sb1's answer. Otherwise start with a quantum mechanics course.
A: Feynman explained a whole lot in plain language in "QED: the strange theory of light and matter". But as Zee takes pains to point out, one must pay PARTICULAR attention to what Feynman says, because he is not popularizing or dumbing it down, except for not getting too much into the notion of spin.
A: Consider a cavity or a black body in thermal equilibrium. According to classical notions, every harmonic oscillation mode in that system contains on average the same amount of energy, in equilibrium. But the density of modes rises proportionally to the frequency. Thus the system contains an infinite amount of energy, which obviously doesn't happen in reality.
Quantization resolves this. If the modes of the oscillation come at discrete energy levels at much larger steps than the average equilibrium thermal energy, that energy can't occupy every mode equally any more.
This however comes with the price of uncertainty. Consider the interaction of a photon and electron in a measurement process. Because of the wave particle duality, energy of the photon is proportional to the frequency of it's wave. If you want a sharp picture of the electron, a point particle, you need high frequency photons to scatter with it. The electron gains kinetic energy in the scattering and so taking an infinitely sharp picture of the electron is practically impossible, you end up with a fuzzy picture no matter what.
A: "Once more unto the breach ..."
The aspect of quantum mechanics that distinguishes it the most (IMHO) from classical mechanics, is that of superposition of states - that at any given moment a system is described by a state which can be a linear combination of various, physically realizable, outcomes:
$$ |\Psi\rangle = c_1 |\psi_1\rangle + c_2 |\psi_2 \rangle + \dots $$
Indeed one can give a simple visual analogy. Think of that prototypical classical mechanical system - the pool (or "billiards" depending on which time-zone you happen to be in) table. The pool balls obey simple classical trajectories, assuming the absence of non-uniformities on the table, which consist of straight lines interrupted by reflection off of the sides of the table. In classical mechanics one can enumerate all such trajectories and construct the resulting phase space for the system.
If you were to ask me to characterize the state of the pool table at any instant, I would do so by choosing a point in phase space whose co-ordinates specify the locations and momenta of all the balls on the table at that instant. Given this information and Newton's second law I can determine, to arbitrary accuracy , the future trajectories of all the pool balls. 
Quantum Mechanics introduces the entirely new possibility that I could specify a state which is a linear combination of eigen (or "physically realizable") states. In other words a given pool ball could be at location $x$ and at $x'$ at the same time. Even to a laymen this possibility open up great new vistas. Engineers can only imagine what sort of machines they could construct if they could build a transmission that could operate at more than one speed simultaneously. Computational scientists wonder at the thought that a quantum system could exist in a superposition of $0$ and $1$ simultaneously - this possibility is in fact being realized in the exploding field of quantum computation.
In return for all these nice possibilities that superposition opens up we must accept significant trade-offs - there is an inherent uncertainty in whatever physical quantity we measure and we must accept restrictions on which observables can be measured simultaneously.
To illustrate the first trade-off (uncertainty in measurement outcomes) I can - in theory - construct a quantum computer which employs "massive parallelism" and solves a problem by considering all possible solutions simultaneously, but the trade-off is that I can never be absolutely certain that the solution it spits out at the end is the right one.
The second trade-off is perhaps more significant in terms of its implications for our notions of "reality". Coming back to the pool ball, superposition allows me to say that it is at location $x$ and $x'$ simultaneously. However, the more accurately I try to localize the ball at these two locations, the less accurately I can localize the ball's momenta. Or as when a cop pulled Heisenberg over for speeding and asked "Do you know how fast you were going?", he replied "No. But I know exactly where I am"
[Note for the purists and experts: The language of this answer is directed at a beginner and not for a conference proceeding, therefore I have avoided technicalities and mathematics. Please keep this in mind when pointing out errors and such.]
A: In the early part of the twentieth century, physics was revolutionized. The foundation of the subject changed once and for all. It was discovered that nature in its deeper scheme of things does not operate according to the deterministic rules of classical Newtonian physics. Heisenberg discovered his famous uncertainty principle in 1926. Physicists came to know that values of certain pairs of classical dynamical variables like position and its conjugate momentum can not be measured to arbitrary precision. The more you are certain about one of them the less you know about the value of the other. The simultaneous precise knowledge of these pairs of variables (at least in principle) are essential to predict the outcome of any experiment in classical physics. Classical mechanics used to assert that in principle nature is completely deterministic. However the uncertainty principle put an end to that dream. The uncertainty principle is a consequence of the fact that there is a wave particle duality for all the so called "particles"(like electron) and "waves" (like electromagnetic wave) in nature. One can say that some times under certain conditions, it is helpful to think a particle like electron as waves or a wave like em wave as particles. Schrödinger discovered his famous equation for particles with non-relativistic speed which describe the evolution of a characteristic function $\psi$ in space with time. Quantum mechanics is based on certain basic postulates.
1) It is assumed that to every observable physical quantity there exists a hermitian operator. The measurable values of the observable is the various eigen values of the operator.
2) To every physical system there corresponds an abstract Hilbert space. A state of the system is represented by a vector in this space on which the operator corresponding to the observable acts.
3) The outcome of a measurement of an observable in a particular state is given by the expectation value of the corresponding operator in that state.
4) The operator corresponding to a dynamical variable is obtained by replacing the classical canonical variables by corresponding quantum mechanical operators.
5) Any pair of canonically conjugate operators will satisfy the Heisenberg commutation rules.
The edifice of quantum mechanics is based on these basic postulates. What quantum mechanics really implies is that light, electrons, protons etc. can not be described by exclusively in particle terms or wave terms. They are like neither particles nor waves. Particles or waves are approximate classical concepts and to describe quantum reality these classical concepts should be viewed as complimentary concepts rather than contradictory. In the language of Neils Bohr, "However far the phenomena transcend the scope of classical physical explanation, the account of all the evidence must be expressed in classical terms". This is because "by the word 'experiment' we refer to a situation where we can tell others what we have done and what we have learned" so that "the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of the classical physics".
A: In classical mechanics the state of a particle is described by a point in phase space. We can directly measure that state. But in the microscopic world this not possible. Then we postulate that we can describe the state by  a ket vector and operator in this space. This operator when acting on this state gives the physical quantities as the eigen value.
A: I'd like to try offering my understanding. I'd post it in a comment but it's just too long.
The double-slit experiment offers a paradox because when you shoot electrons 'one at a time' at the slits they form an interference patter whereas if they really acted like real-world 'things' like a marble there would be no interference.
The point is that at the quantum level the 'things' being observed are of the same scale as the thing used to observe. In the macro level of marbles it's not an issue, in the natural world, because a light-wave is so much smaller than a marble that you don't see the interaction between the two.
But at the quantum level there is no matter per-se ... it's waves ... and you know waves are constantly changing so the 'state' of the electron (which is a packet of energy, a wave, not a 'thing') is constantly in flux and when you observe it you see ALL of the possible states at once because they happen so quickly.
It would be like if I gave you one half of water. In the quantum description it's not really a half-glass of water ... it's a glass of water with different amounts that you observe as a half glass ... could be 60, then 40, then 30, then 70, whatever ... some number of possible states, NOT an infinite number as they are "quantized" - the number of states depending on the energy of the system.
This really does mean that nothing truly exists until we observe it ... a philosophical perception. And this also means it makes no sense to say 'the system exists in some state until it's observed'. What's technically correct is to say 'the system exists is ALL states until it's observed'.
In other words, when we say "exist" or "observe" we mean whatever the state of the system is we're observing. Lets say it's a river. Each 'moment', whatever that is, is one state despite the fact that the river is constantly in motion. When we say we observer the river flowing one is actually saying it's a large group of events just as we say "there's a marble" thinking it's just one ... but it's not one, it's the sum-observation of all the states of all the atoms of the marble.
