Quantum mechanics (QM - also known as quantum physics, or quantum theory) is a branch of physics dealing with physical phenomena where the action is on the order of the Planck constant. Quantum mechanics departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales.
Planck was as steeped in traditional physics as his colleagues, but he had an open mind. The older way wasn't working. So he changed one basic assumption: energy, instead of being continuous, comes in distinct particles. These were later called "quanta," from the Latin for "how much?"
The most important properties of atomic and molecular structure may be exemplified using a simplified picture of an atom that is called the Bohr Model. This model was proposed by Niels Bohr in 1915; it is not completely correct, but it has many features that are approximately correct and it is sufficient for much of our discussion. The correct theory of the atom is called quantum mechanics; the Bohr Model is an approximation to quantum mechanics that has the virtue of being much simpler.
The basic feature of quantum mechanics that is incorporated in the Bohr Model and that is completely different from the analogous planetary model is that the energy of the particles in the Bohr atom is restricted to certain discrete values. One says that the energy is quantized. This means that only certain orbits with certain radii are allowed; orbits in between simply don't exist.
Light exhibits wave-particle duality, because it exhibits properties of both waves and particles. Wave-particle duality is not confined to light, however. Everything exhibits wave-particle duality, everything from electrons to baseballs.
All these ideas, that for very small particles both particle and wave properties are important, and that particle energies are quantized, only taking on discrete values, are the cornerstones of quantum mechanics. In quantum mechanics we often talk about the wave function of a particle; the wave function is the wave discussed above, with the probability of finding the particle in a particular location being proportional to the square of the amplitude of the wave function.
A good equation for a physicist is like a hammer for a carpenter. Schrodinger's equation, more akin to a marvelous new toolbox, gave the means to solve whole varieties of problems. Even today, much of physics is summarized in Newton's relation between force and acceleration, F=ma; Maxwell's four equations for electricity and magnetism; and Schrodinger's equation for quantum mechanics.
The physics community, in a state of both shock and excitement, was soon informed that matrix mechanics and wave mechanics were the same theory, but the scientists required several years to fully its implications. The starting points of the two mathematical approaches were different and seemingly unrelated, but the endpoints were the same.
... the famous Heisenberg Uncertainty Principle. It says that although we can measure position as accurately as we want or momentum as accurately as we want, we cannot measure both as accurately as we want. The equation gives the combined uncertainty. The less uncertainty in one, the more uncertainty in the other.
The physical meaning of the equation is that if we measure the position of a particle and then its momentum, we get a different answer than if we measured the momentum first and then the position. Each measurement unavoidably disturbs the particle, in a way that depends on which quantity is measured, so that the results of a subsequent measurement are altered.
The extraordinary success of quantum mechanics in applications did not overwhelm everyone. A number of scientists, including Schrödinger, de Broglie, and -- most prominently -- Einstein, remained unhappy with the standard probabilistic interpretation of quantum mechanics.
"It is hard to sneak a look at God's cards. But that he would choose to play dice with the world... is something I cannot believe for a single moment."
To Cornel Lancszos, March 21, 1942, expressing his reaction to quantum theory, which refutes relativity theory by stating that an observer can influence reality, that events do happen randomly.