Quantum Final Thur. : Need Hydr./L/Spherical Help

[TARSIER]

I have had a extremely busy couple weeks where I could not focus on my quantum reading, and so I am a little lost conceptually. I am using Griffiths, and I am having trouble understanding what is going on in sections 4.1-4.3, which deal with the Hydrogen atom, angular momentum, and the Schrödinger equation in 3D (no spin problems, thank God). I have done pretty well in the course so far, and can conceptually connect (hopefully) anything you say with the previous concepts explored in chapters 1-3.

So, I am having an especially large problem understanding the notation and how it applies, specifically: R(n,l), c(j) as it applies specifically in this chapter, Y(m,l), P(m,l), and how this connects with calculating Psi.

You do not have to write the equations out as I have those, but if you could please help me conceptually grasp what is going on.

 

[jdstokes]

What in particular are you having trouble with? The idea is that you want to solve the time-independent Schrödinger equation \hat{H} |\psi\rangle = E| \psi \rangle in a system of coordinates in which the potential has some trivial shape. For example by adopting spherical coordinates when the potential is spherically symmetric.

Once you set up the equations you will find that the partial differential equation can be split into three ordinary differential equations (one for each of the angular variables) using a method called separation of variables, where you basically assume that the dependence of the wavefunction on each of the coordinates is not `intertwined'.

When you solve the angular part, you find that the possible wavefunctions dependent on \varphi can be indexed by the integers m. The part which depends on theta can be indexed by the nonnegative integers l, and that for each l there are 2l + 1 possibilities for m. The spherical harmonics are what you get when you multiply these angular bits together and normalize the resulting wavefunction.

 

[TARSIER]

Thats basically what I was looking for, a summary of what was going on. Now what do the that I listed above mean? Thank you for the above post, it was very helpful.

 

[turin]

R(n,l) are probably the radial factor of the wavefunction (probably in terms of Laguerre polynomials).

Y(m,l) are probably angular factor of the wavefunction (probably spherical harmonics).

P(m,l) are probably generalization of Legendre polynomials, called associated Legendre polynomials (a factor of spherical harmonics).

Not sure about c(j); could just be the coefficient in the expansion.

So, you should know that you can find eigenfunctions of the Hamiltonian. Well, it turns out that the spherical polar coordinates are quite convenient for spherically symmetric problems like the ideal, infinitely massive proton Hydrogen atom, and the eigenfunctions for such a system are exact in terms of these factors.

Spherical harmonics are on my top ten list of concepts to learn in QM, so definitely read about them and understand as much as you can about why they arise in the Hydrogen atom potential and how they are related to each other and to other functions. It took me several years to get comfortable with them, so I suggest to get started as soon as possible.

 

[TARSIER]

Thank you for posting this. I assumed those concepts I guess, but now I have a better understanding of what's going on going into the test.