In physics, quantum mechanics is “the study of the relationship between quanta and elementary particles”. Quantum mechanics is a fundamental branch of physics with wide applications in both theoretical and experimental physics. Quantum theory generalizes all classical theories, including mechanics, electromagnetism and provides accurate descriptions for many previously unexplained phenomena such as “black body radiation” and “stable electron orbits”. The effects of quantum mechanics are typically not observable on macroscopic scales, but become observable at the atomic and subatomic level.

An easier to understand definition is “Quantum mechanics is a mathematical theory that can describe the behavior of objects that are roughly 10,000,000,000 times smaller than a typical human being”. Quantum particles move from one point to another as if they are waves. However, at a detector they always appear as discrete lumps of matter. There is no counterpart to this behavior in the world that we perceive with our own senses. One cannot rely on every-day experience to form some kind of "intuition" of how these objects move. The intuition or "understanding" formed by the study of basic elements of quantum mechanics is essential to grasp the behavior of more complicated quantum systems.

The discovery that waves have separate energy packets, otherwise known as “quanta”, that behave in a manner similar to particles led to the branch of physics that deals with atomic and subatomic systems which we today call, quantum mechanics! It is the mathematical framework of many different fields of physics and chemistry, including condensed matter physics, solid-state physics, atomic physics, molecular physics, computational chemistry, quantum chemistry, particle physics, and nuclear physics. The foundations of quantum mechanics were established during the first half of the twentieth century by Werner Heisenberg, Max Planck, Louis de Broglie, and many others. Some fundamental aspects of the theory are still actively studied. The word “quantum” came from the Latin word which means "what quantity". In quantum mechanics, it refers to a discrete unit that quantum theory assigns to certain physical quantities.

The history of quantum mechanics began essentially with the 1838 discovery of cathode rays by Michael Faraday. The 1859 statement of the black body radiation problem by Gustav Kirchhoff, and the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete. Finally the 1900 quantum hypothesis, by Max Planck explains that any energy is radiated and absorbed in quantities divisible by discrete ‘energy elements’. E, such that each of these energy elements is proportional to the frequency v. In 1905, to explain the photoelectric effect, that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, as based on Planck’s quantum hypothesis, which light itself consists of individual quanta.

The year 1924 saw the publication of another fundamental paper. It was written by Satyendra Nath Bose and rejected by a referee for publication. Bose then sent the manuscript to Einstein who immediately saw the importance of Bose's work and arranged for its publication. Bose proposed different states for the photon. He also proposed that there is no conservation of the number of photons. Instead of statistical independence of articles, Bose put particles into cells and talked about statistical independence of cells. Time has shown that Bose was right on all these points.

Also in 1927 Bohr stated that space-time coordinates and causality is complementary. Pauli realized that spin, one of the states proposed by Bose, corresponded to a new kind of tensor, one not covered by the Ricci and Levi-Civita work of 1901. However the mathematics of this had been anticipated by Eli Cartan who introduced a 'spinor' as part of a much more general investigation in 1913. Dirac, in 1928, gave the first solution of the problem of expressing quantum theory in a form which was invariant under the Lorentz group of transformations of special relativity. He expressed d'Alembert's wave equation in terms of operator algebra. The uncertainty principle was not accepted by everyone.

Its most outspoken opponent was Einstein. He devised a challenge to Niels Bohr which he made at a conference which they both attended in 1930. Einstein suggested a box filled with radiation with a clock fitted in one side. The clock is designed to open a shutter and allow one photon to escape. Weigh the box again some time later and the photon energy and its time of escape can both be measured with arbitrary accuracy. Of course this is not meant to be an actual experiment, only a “thought experiment”.

However Niels Bohr had the final triumph, for the next day he had the solution. The mass is measured by hanging a compensation weight under the box. This is turn imparts a momentum to the box and there is an error in measuring the position. Time, according to relativity, is not absolute and the error in the position of the box translates into an error in measuring the time. Although Einstein was never happy with the uncertainty principle, he was forced, rather grudgingly, to accept it after Bohr's explanation. In 1932 von Neumann put quantum theory on a firm theoretical basis. Some of the earlier work had lacked mathematical rigors, but von Neumann put the whole theory into the setting of operator algebra.

There are four basic principles of quantum mechanics and they are:

Physical States
Every physical system is associated with a “Hilbert Space”, every unit vector in the space corresponds to a possible pure state of the system, and every possible pure state, to some vector in the space. In standard texts on quantum mechanics, the vector is represented by a function known as the wave-function, or function.
Physical Quantities
Hermitian operators in the Hilbert space associated with a system represent physical quantities, and their Eigen values represent the possible results of measurements of those quantities.
Composition
The Hilbert space associated with a complex system is the tensor product of those associated with the simple systems (in the standard, non-relativistic, theory: the individual particles) of which it is composed.
Dynamics (2 types)
Contexts of type 1: Given the state of a system at t and the forces and constraints to which it is subject, there is an equation, “Schrodinger's equation” that gives the state at any other time. The important properties of U for our purposes are that it is deterministic, which is to say that it takes the state of a system at one time into a unique state at any other, and it is linear, which is to say that if it takes a state onto the state, and it takes the state onto the state then it takes any state of the form onto the state.
Contexts of type 2 ("Measurement Contexts"): Carrying out a "measurement" of an observable B on a system in a state has the effect of collapsing the system into a B-eigenstate corresponding to the Eigen value observed. This is known as the Collapse Postulate. Which particular B-Eigen state it collapses into is a matter of probability, and the probabilities are given by a rule known as Born's Rule.

All in all quantum mechanics is a bunch of scientific nonsense. You have to be almost a genius to one hundred percent, fully understand it. If you have to the choice to never do anything on it, make the right choice and not get yourself mixed up with it, like gangs. Hopefully everything that I’ve blogged about will help you understand it if you are unfortunate enough to get stuck with it. A lot of the information I researched I don’t even completely understand, good luck!