The equation of Schroedinger for the hydrogen atom in the time-independet, non-relativistic form is a partial differential equation involving the position coordinates (x, y and z). The potential energy term for the proton-electron system is spherically symmetric of the form -1//4pi in_(0) xx (e^(2)//r) . THus it is advantages to change over from the cartesian coordinates (x,y and z) to the spherical polar coordinates, (r, theta and phi ). In this form the equation become separable in the radial part involving r and the angular part involving theta and phi . The probability of locating the electron within a volume element d tau = 4pi r^(2)dr is then given |Psi|^(2)(4pir^(2)dr) , where Psi is a function of r, theta and phi . With proper conditions imposed on Psi , the treatment yields certain functions, Psi , known as atomic orbitals which are solutions of the equations. Each function Psi correspods to quantum number n, l and m, the principal, the azimuthal and the magnetic quantum number respectively, n has values 1, 2, 3,...., l has values 0, 1, 2, ....(n-1) for each value of n and m (n-1) for each value of n and m (m_(l)) has values =1, +(l+1),...1,0,-1,-2...-l i.e., (2l+1) values for each value of l. In addition a further quantum number called pin had to be introduced with values +-1//2 . Any set of four values for n, l , m and s characterizes a spin orbital. Pauli.s exclusion principle states that a given spin orbital can accomodate not more than electron. Further the values l = 0, l=1, l=2, l=3 are designated s,p,d and f orbitals respectively. It is a basic fact that any two electrons are indistinguishable. 3 electrons are to be accomodated in the spin orbitals included under the designated 2p, conforming to the Pauli principle. Calculate the number of ways in which this may be done.
