Need In quantum mechanics
The need of quantum mechanics arrises in century 19.
The atomistic theory already had been established, and there was need to describe properties of various atoms; mainly, in the chemical reactions.
Experiments with the the cathode beams required concept or an electron, as elementary carrier of negative electric charge. Place of electron in the atoms needed to be analyzed,
There was need to describe the spectral lines in the spectra of Sun and then those of artificial sources of light.
The need in quantum theory becomes evident at the beginning of century 20, since, discovery of atomic nuclei. The Classical mechanics and the Classical electrodynamics showed obvious failure in description of optical properties of atoms and molecules. the theory predicted falling of electrons to the nuclei, contracting observations of stable atoms.
Various pseudo-classical, semi-classical, quasi-classical attempts to describe the atoms are observed since beginning of century 20. The most notable is the Bohr-Zommerfield quantization; the action on a periodic orbit is postulated to be integer factor of the Planck constant. Such a description showed tremendous agreement with observed spectra of Hydrogen atom.
During century 20, a lot of discussions happens around postulates of Quantum mechanics and their interpretation. Many researchers still try to interpret the Quantum mechanics in terms of the classical mechanics. Such interpretations leaded to paradoxes, contradictions and philosophic speculations. In the USSR, these contradictions are used by the Soviet fascists for terror, repressions, genocide against researchers, who work with quantum theory; many Soviet physicists were killed, murdered by bolshevics; many other were converted to slaves, in order to make the nuclear weapon for the soviet gensek at the special prisons of class sharashka.
In century 21, it is mainly accepted, at least by the scientific community, that axioms of quantum mechanics cannot be interpreted in terms of the classical mechanics. These axioms can be only postulated. Then, the Cclassical mechanics can be interpreted in terms of Quantum mechanics.
Axioms of Quantum mechanics
In the narrow sense of the term, Quantum mechanics deals with isolated systems.
Axioms of quantum mechanics are requirements, that are applied to any physical model in order to qualify it as "quantum theory".
State of the physical system
The state of the physical system is described with wavefunction (or wave function), that is element of the space. It is called also "vector of state". It is called also "ket-vector", for example $|\psi\rangle>$.
In such a way the wave functions can be added ("addition of vectors"), multiplied by numbers, and the scalar product for two wavefunctions should be defined.
Conjugation is defined (and indicated with "dagger"),
$\psi\rangle^\dagger = \langle \psi |$
the result of conjugation is called "bra-vector".
For any two states $|\varphi\rangle$, $|\psi\rangle$, and any complex numbers $a$, $b$ the linear combination
$a |\varphi\rangle + b |\varphi\rangle$,
is defined and it is element of the same space of states of the system.
The scalar product
$\langle \varphi | \psi \rangle$
should be defined in such a way, that
$\big(\langle \varphi | \psi \rangle\big)^*=\langle \psi | \varphi \rangle$
Linear operations on the space of wave functions are called "operators". The results of these operations should be elements of the same space.
The special kind if the linear operators, namely Hermi
Time and time evolution