From where do the properties of the magnetic quantum number #m_l# originate?

It just came from solving the Schrodinger equation. Erwin Schrodinger apparently was the first guy to introduce the quantum numbers.

I'll save you most of the math, and just skip to the part where #m_l# is introduced... if you want to see the whole thing, here it is.

At some point in solving the Schrodinger equation for the hydrogen atom, we would come upon this second-order linear homogeneous ordinary differential equation:

#(d^2Phi)/(dphi^2) + BPhi = 0#

We assume a solution to this ordinary differential equation with constant coefficients by choosing #phi# on the #xy# plane:

#Phi = e^(im_lphi)#

This gives the auxiliary equation:

#-m_l^2 cancel(e^(im_lphi))^(ne 0) + Bcancel(e^(im_lphi))^(ne 0) = 0#

#=> sqrtB = pm |m_l|#

The solution is then a linear combination of #Phi#'s,

#Phi(phi) = c_1e^(im_lphi) + c_2e^(-im_lphi)#

For one full rotation on the #xy# plane in one direction (say, clockwise), the wave function must coincide with itself (be cyclic), so #Phi(0) = Phi(2pi)#.

To eliminate variables, currently, #m_l >= 0#. So, let #c_1 + c_2 = A# and allow #m_l# to be negative, so that:

#Phi(phi) = (c_1 + c_2)e^(im_lphi) = Ae^(im_lphi)#

From Euler's relation,

#e^(i cdot m_l cdot 0) = cos(0m_l) + isin(0m_l) = e^(i cdot m_l cdot 2pi) = cos(2pim_l) + isin(2pim_l)#

This only holds true for integer values of #m_l#, which we define to be the magnetic quantum number.

The unnormalized #phi# component to the angular wave function is then:

#color(green)(barul(|" "stackrel(" ")(Phi(phi) prop e^(im_lphi))" "|))#

Note that since #cos# is even, #m_l# is even as well, and indeed, it takes on integer values symmetrically about #0#.

#m_l = { . . . , -l+1, . . . , -1, 0, +1, . . . , l-1, . . . }#

Note, however, that we don't get any restriction that #|m_l| <= l# until we solve the associated Legendre differential equation for the #theta# part of the angular wave function, so at this point we only know that #m_l# is an integer of some sort.

Although, this neat proof is an alternative way to do so.