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In quantum mechanics, we study the physical and chemical properties of particles of size micrometers or less.  We study the mechanics or dynamics of molecules, compounds, ions, atoms, particles like electrons, protons, neutrons, photons etc.
  The behaviour of the particles is entirely different from the classical mechanics.  In classical mechanics, we study the properties of a mass, substantial gross and aggregate amount of a substance.  In quantum mechanics, individual behaviour of a particle is studied.

The energy levels of the particles are expressed in terms of discrete numbers (levels).  The particles cannot exist with any energy value.The energy states allowed are called the quantum number.  

For an electron in an atom or an ion, it takes values = 1, 2, 3, 4.   There cannot be an electron, for example, with a quantum number 1.8 (or energy level = 1.8).  So electron exists only in 4 states.  This is called the principal quantum number.

Similarly, the quantities for other properties of the particles also exist only as some discrete values. and cannot be continuous.  Such discrete values are called quantum numbers.

For example,  the angular momentum of an electron in an atom is a quantum number. It is the second quantum number.  This quantum number tells us the shape of the electron's orbital, in its energy level.   For an electron in p orbital, it is equal to 1.

There is also a spin quantum number for an electron.  The electron can only spin in two ways +1/2  (spinning up)  or  -1/2  (called spinning down).

The behaviour of an atom in a chemical reaction, emitted radiation, spectrum can depend on the quantum numbers.    

Quantum numbers are the numbers which define the location of electrons in the orbitals around the nucleus of an atom.There are four quantum numbers, viz,Principal quantum number (n) : It defines the main shell to which an electron belongs. The values of n =1,2,3,4,........Azimuthal quantum number (  : It tells us the number of subshells present in the main shell and the orbital to which an electron belongs. Its value is equal to 0 to (n-1).Magnetic quantum number (m) : It informs the level of degeneracy of the orbitals. Its value is equal to -,0,+.Spin quantum number (s) : It tells us whether the electron in an orbital is spinning in upward direction or downward direction. s has two values +1/2 for upward spin and -1/2 for downward spin.