Quantum mechanics is a theory of mechanics that applies to small particles such as electrons and atoms. It was suggested as an alternative theory to classical mechanics after the failure of the latter to explain physical experiments such as the blackbody radiation.
At the beginning of the 19th century several experiments in physics were done and were not possible to explain based on classical mechanics. These experiments could not be explained according to Newton’s laws. This posed new questions that forced scientists to deviate from the then current theory of classical mechnics.
These experiments included the blackbody radiation experiment and the ultraviolet catastrophe. In addition the concept of heat capacity for thermodynamic systems has to revised in such a way as to conform to the new data that were later incorporated as quantum effect.
Other experiments and models that deviated from classical mechanics interpretation included the photoelectric effect in addition to Bohr model of the hydrogen atom. All these experiments were then theorized by using a similar concept of quantization of energy.
According to the photoelectric effect it was postulated that energy is quantized and was ascribed a name as a quantum of photon. A photon is an energy particle which has wave as well as particle identities. This is called in physics the wave-particle duality of light.
The theory of Bohr about the hydrogen atom, although was successful in predicting the structure and energetics of the hydrogen atom it could not be extended to other atoms such as the helium atom. His ideas of the this model, however, put forward the new concepts that formed the quantum theory later.
De-broglie is a french scientist that postulated that particles such as electrons possess wavelike characters. This assumption was confirmed experimentally by the observation of waves that are associated with particles.
The De-broglie postulate of duality of light and matter put forward the way for Schroedinger to formulate his famous wave equation which depicts the energetics of electrons in addition to atoms and molecules.
Schroedinger equation has two forms one of them is the time dependent wave euqation and the other is the time independent wave equation. The time independent wave eqation could be applied to atoms and molecules and many variations of it has been developed for atoms and molecules such as the Hartree-Fock equation.
The time dependent wave equation of Schroedinger could be applied to time dependent phenomena such as electronic transitions in atoms and molecules and in particular it finds uses in spectroscopy.
The Schroedinger equation despite its usefulness and its widespread use in quantum mechanics could not be solved accurately for systems other that the hydrogen atom. Even with the hydrogen atom where the solution to the equation is accurate the mathematics involved is very complex.
Therefore, many approximation methods were developed to approach the problems of many electrons systems such as atoms and molecules with electron correlation. The energy concept is a major concept in the quantum theory of matter. Other concepts that are also important are the momentum and displacement.
Quantum mechanics depicts physical observables with the assistance of quantum mechanical operators. For example, the energy of a given system can be depicted quantum mechanically using the Hamiltonian operator. Likewise happens with the momentum observable and the position observable in which case corresponding quantum mechanical operators depict their physics and chemistry.