Incorporating the spherically symmetric potential energy we need to deal with the radial equation that came from separation of space variables, which contains V(r): the goal here is to find the function R(r), which tells us how the wavefunction varies with distance from the origin it appears that the energy of the state will depend on l but this turns out not to be the case! a change of variables: let u(r) = r R(r) one thing is that when calculating probabilities the volume element, times the wavefunction squared, contains r 2 R 2 := u 2

Incorporating the spherically symmetric potential energy the radial equation becomes, in terms of u(r) this is exactly in the form of the 1d TISE what appears here is a modified potential energy, which contains both V(r) and an additional term, dubbed the centrifugal potential TISE has been reduced to an equivalent 1d problem, at the cost of introducing another contribution to V eff (r) now the fun is at hand! What kind of V(r) forms to play with?

The Coulomb potential at large r, both terms die off, but centrifugal one dies quicker: < 0 at small r, both terms blow up, but centrifugal one dominates: > 0 Coulomb potential for an electron of charge e in the presence of a nucleus of charge Ze, separated by a distance there is a stable equilibrium distance, where attractive Coulomb force is ‘balanced’ by ‘repulsive’ centrifugal force note that one could approximate the well by a harmonic potential for small disturbances from equilibrium bound states have E < 0

Introducing a dimensionless distance, and getting long-and-short distance behavior define an ‘inverse decay length’ must take B = 0 to avoid blow-up as   0 equation is interesting: must take D = 0 to avoid blow-up as   ∞ equation is trivial, with solution

Peeling off those two behaviors, and defining yet another function v(  ) life seems to have gotten very very ugly, but when one puts this in amendable to power series approach since it’s been peeled!

The power series approach to the radial equation doing the by-now familiar process we get rearranging, we arrive at the recurrence relation there is no odd/even setup here, by the way

Truncating the power series if one assumes the limiting form of the recurrence relation right from the get-go j = 0 one gets a nice approximation: thus, there must be a j max such that since j max is an integer, so must  0 /2 be one, which we call n we have arrived at a quantization condition on the energy! recall what  0 was:

More about the principal quantum number n, etc. if n = l + 1 (which is its smallest possible value), the v function is a constant, and so R is a single power of r (the l –1 power) times exponential dropoff as j max grows, a polynomial in r with more and more powers of r enters the game, times r l−1 times exponential dropoff normalization and probability calculations with R(r) requires the integration of r 2 R 2 (r) we see that given an l value, n must be at least 1 bigger, and that the closer l is to n, the simpler the polynomial in r the solutions for v(  ) are the Laguerre polynomials numbers: for electron and Z = 1 (H atom), Bohr radius is.053 nm and ground-state energy is 13.6 eV therefore, is quantized:  n = 2  n (we knew that!)

Structure of the radial solutions I The associated Laguerre polynomials are the v(  ) functions and they are derived by differentiating the Laguerre polynomials: The Laguerre polynomials are given by since l = 0, 1, 2…, n  1, and |m|  l  the superscript represents the possible number of m values (the degeneracy) and the subscript can range from 0 to n  1 (the range of l values, in reverse order) some Laguerre polynomials:

Structure of the radial solutions II Some associated Laguerre polynomials: Some normalized radial wavefunctions R nl :

Structure of the radial solutions III for probability calculations, use r 2 R 2 as probability density note that its units are m  1 as usual for a 1d case some pictures of these probability densities

Building up atomic orbitals the way the chemist’s like it done we’ll look at the 3 p orbitals: one is already up and running this one (p z ) is lobe-like along ± z, and azimuthally symmetric it is an eigenfunction of L 2 and of L z now consider two linear combinations: this one is lobe-like along ± y it is an eigenfunction of L 2 but not of L z what operator might it be an eigenfunction of, too? this one is lobe-like along ± x it is an eigenfunction of L 2, and of another operator, but not of L z