From WikidChem
[edit]Slide 15
Great mathematicians were also working on Chladni's Problems. Schrodinger did not actually discover the equations that he used. He used the work of mathematicians such as Lagrange, Bernoulli, Euler, etc.
[edit]Slide 16
Contains the equations used to create Ψ for H-like Atoms. Because an equation defined in terms of x, y, and z would be very complicated, we use the distance formula to derive an equation that uses the radius (r). This equation contains three simple, smaller equations that are multiplied together to find the overall Ψ equation. R refers to the radius, theta refers to the angle with the z-axis, and phi refers to the angle with the x-axis. We then name the Ψ function by either using the quantum numbers associated with the function or by the nickname, such as 1s, 2s, 2p, etc.
[edit]Slide 17
This slide shows why ρ is used instead of simply using r for the radius. By using ρ, we are able to take Z and n into account as well, and therefore use the equations for many different atoms. We also notice, then, that all of the ψ functions contain an e^(-ρ/2) term, which is not surprising becuase as the distance, r, approaches large numbers, that part of the equation gets smaller, and smaller, eventually approaching 0. This relates to the fact that the farther away from the nucleus one is, the less likely it becomes to encounter an electron (if we square ψ) (Think Coulombic potential). In this slide one also notices the introduction of constants, which are used to normalize the ψ function.
[edit]Slide 18
This slide contains examples of what happens to the e^(-ρ/2) term of the function when Z and n are changed. The end result is that we notice the curve expands as n increases and shrinks as Z increases.
[edit]Slide 19
Shows very similar information to slide 18. it shows that an increasing Z sucks the standard 1s function toward the nucleus. Renormalization then keeps the probability density constant and it also allows for easier comparison- the same scale is angstrom scale is used to plot both graphs.
[edit]Slide 20
This slide tells us that ψs for one electron atoms involve Spherical Harmonics, which are the 3D analogues of Chladni figures. The Spherical Harmonics are the sets of equations based on the variables XXX and XXX that determine where the spherical nodes are located.
[edit]Slide 21
This slide shows us that ψ is dependent on three variables p (the Assocaited Laguerre Function), θ, and Ф (the Associated Legendre Polynomial). As mentioned earlier, Shroedinger did not discover these functions. They were available because of the work of older mathematicians.
[edit]Slide 22
Slide 22 shows us that to create the ψ functions for various orbitals, all one must do is find the appropriate terms in the ψ table for H-like Atoms and multiply them together.
