Where do quantum numbers come from and how were they determined?
1 Answer
Quantum numbers are defined when determining the eigenvalues of energy and angular momentum, arising from their quantization. Calculating them is probably beyond the scope of general chemistry. You just need to know how to use them.

The principal quantum number is
#n = 1, 2, 3, . . . # , describing most directly the atomic energy levels. 
The angular momentum quantum number is
#l = 0, 1, 2, . . . , n1# describes the shape of the orbital. 
The magnetic quantum number
#m_l# is in the set#{l, l1, . . . , 0, . . . , l1, l}# .
Since the minimum
#m_l# is#l# and the maximum#m_l# is#l# , there are#2l+1# such values of#m_l# , corresponding to the number of orbitals in each subshell (given by#l# ).
 The electron spin quantum number
#m_s# is#pm1/2# because the spin of the electron was experimentally determined to be#pm1/2# , just like other particles that are classified as fermions.
Nevertheless, here is a brief origin of these quantum numbers from quantum chemistry.
An example of atomic energy quantized to the quantum number
#E_color(blue)(n) = (2pi^2me^4Z^2)/(color(blue)(n^2)h^2)#
Here, the energy
The square of the orbital angular momentum,
#ul(hat(L)^2) [Y_(color(blue)(l))^(m_l)(theta,phi)] = ul(color(blue)(l(l+1))ℏ^2)" "Y_(l)^(m_l)(theta,phi)#
where
(an operator acts on a function.)
Basically, the angular momentum you observe spits back a result that is given by
The zcomponent
#ul(hatL_z) [Y_(l)^(color(blue)(m_l))(theta,phi)] = ul(color(blue)(m_l)ℏ)" "Y_(l)^(m_l)(theta,phi)# where
#hatL_z# is the operator for the zcomponent of the orbital angular momentum,#L_z# .
Similarly, the zcomponent
#ul(hat(S)_z) [color(blue)(vecalpha)] = ul(color(blue)(+1/2)ℏ)" "vecalpha#
#ul(hat(S)_z) [color(blue)(vecbeta)] = ul(color(blue)(1/2)ℏ)" "vecbeta# where
#hat(S)_z# is the operator for the zcomponent of the spin angular momentum,#S_z# .
And it was defined that