Since the Heisenberg Uncertainty Principle tells us that we cannot know both the position and the momentum of a particle with absolute certainty, quantum mechanics makes no attempt to define the path that an electron follows. Rather, quantum mechanics works by determining the energy of the electron. Once the energy of an electron is known, the probability of finding an electron in a given volume of space can be determined. Further, the degree to which the electron will interact with other atoms can also be determined (bonding).

Quantum numbers and electrons

To describe the energy of an electron quantum mechanics tells us we need to know four things about the electron; that is, we need to use 4 quantum numbers to describe each electron. Additionally, no two electrons have identical sets of quantum numbers.

n, the principle quantum number is the shell (base energy level) in which the electron resides. This is similar to the value "n" in the Bohr model.

n is a counting number; i.e., n starts at 1 and counts upwards.

l, the angular momentum quantum number is the type of orbital (s,p,d..).

l is n-1 down to zero.

m_l, magnetic quantum number is the direction orbital points; additionally, the number orbitals that make up an orbital type (1 s orbital, 3 p's, 5 d's) is determined by counting the number of m_l's.

m_l varies from l to -l.

m_s electron spin quantum number describes the magnetic moment of the electron. Electrons are charged, and they are moving. Moving charged things create magnetic fields (e.g. electromagnets). Spin can be either +¹/₂ or -¹/₂,

a. discovered by putting atoms in magnetic fields

the magnetic fielding created two new energy levels. Since the electrons were affected by a magnetic field the electrons must be like little magnets. We know that spinning charges create magnetic fields, so the electrons must have "spin" (this does not necessarily mean the electron are spinning in the traditional sense).

b. There are two allowed spin states, up and down. Since only two new energy levels are observed then there are two possible spin states for an electron.

The size of the quantum state n determines how many orbital types are available

n=1 low energy quantum level, then l = 0 (1 orbital type)

n=3 (higher energy quantum level), then l = 2, 1, 0 (three orbital types)

and thus the number of orbitals in which the electrons can exist.

If you are familiar with the names of the orbitals it may help to point out that the names of the orbital types correspond to the different l levels.

quantum state l = "0" named s
quantum state l = "1" named p
quantum state l = "2" named d
quantum state l = "3" named f
quantum state l = "4" named g
and on and on....

So, an electron could be described with the quantum numbers n = 3, l = 2, ml = -2, and ms = -1/2	Incidentally, these two electrons are in two different orbitals. n = 3 and l = 2 = d means the electron is in a 3d orbital.
Another electron could be described with the quantum numbers n = 2, l = 0, ml = 0, and ms = +1/2	n = 2 and l = 0 = s means the electron is in a 2s orbital.

but an electron can not be described using the following designations:

n = 2, l = 0, m_l = 1, and m_s = +¹/₂

n = 2, l = 2, m_l = -2, and m_s = -¹/₂