ATOMIC PHYSICS


The three dimensional Schrödinger Equation can be solved for the hydrogen potential U(r) = -ke2/r, in spherical polar coordinates, and one notices right away that for any and all three-dimensional systems in a potential that only depends on r, the θ, φ portion of the solution is always the same. These solutions are the spherical harmonics, Yℓ,mℓ(θ,φ). The energies turn out to be the same as those found by Bohr, and the quantum number set needed to label the states is [n, ℓ, mℓ].


Here are some of the normalized radial states ψn,ℓ,mℓ(r). The actual radial probability distribution goes like 4πr2ψ2, as seen below:


The notation s, p, d, etc., signifies the value of ℓ. Thus ℓ = 0 is s, ℓ = 1 is p, etc.






Because the electron has an intrinsic spin of s = 1/2, we also need the quantum number ms. The relation between the various quantum numbers that's required in order to get solutions to the Schrödinger equation are that n can be any integer from 0 to infinity, ℓ can be any integer from 0 to n - 1, mℓ can take on [2ℓ + 1] values from -ℓ to +ℓ in steps of 1, and ms can only be  ±1/2. We can define vector operators L and S such that L2 = ℓ(ℓ + 1)ℏ2 and S2 = (3/2)ℏ2. A similar operator can be defined for the z-components, giving Lz = mℓℏ and Sz = msℏ = ±(1/2)ℏ.


Spin had been discovered experimentally when it was seen that electrons have a magnetic moment. This moment is a vector operator that can be expressed in terms of the spin operator by μ = (-e/m)S. As usual in quantum physics then, the experimenters found only two possibilities, μz = ±(eℏ/(2m)), when they put a beam of particles with spin 1/2 through a magnetic field.


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