Responsible party: Captain America, David Hilbert's hat

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Main class wiki page: home

Please try to include the following

• main points understood, and expand them - what is your understanding of what the points were.
  • expand these points by including many of the details the class discussed.
• main points which are not clear. - describe what you have understood and what the remain questions surrounding the point(s).
  • Other classmates can step in and clarify the points, and expand them.
• How the main points fit with the big picture of QM. Or what is not clear about how today's points fit in in a big picture.
• wonderful tricks which were used in the lecture.
  

Class input on main points of the beginning of Chapter 4:

• <math>\nabla^2</math> spherical coordinates
• <math>L^2, L_z</math>, and why they have discrete values
• The Hydrogen atom model
• How spin affects this
• Degenerate energies and states
• Quantum numbers
  • Where are they from?
  • What do they do?

Using 3-D Coordinates:

From the one-dimensional Schrodinger Equation: <math> [-\frac{\hbar^2}{2m}\frac{ \partial^2}{ \partial x^2} + V(x)]\psi=E \psi</math>

The kinetic energy term, <math>-\frac{\hbar^2}{2m}\frac{\partial^2}{ \partial x^2}</math>, must model the 3-Dimensional kinetic energy of the system, and therefore turns into:

<math> [-\frac{\hbar^2}{2m}\nabla^2+ V(x)]\psi=E \psi</math>

Where <math>\nabla^2</math> is equal to <math> \frac{ \partial^2}{ \partial x^2} + \frac{ \partial^2}{ \partial y^2} + \frac{ \partial^2}{ \partial z^2}</math>

Separation of Variables

Using spherical coordinates will be useful for future problems that we will be solving, so it is necessary to transform the Schrodinger equation into spherical coordinates. The Laplacian will take the form of: <math>\nabla^2=\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) + \frac{1}{r^2 sin(\theta)} \frac{\partial}{\partial \theta} (sin(\theta) \frac{\partial}{\partial \theta}) + \frac{1}{r^2 sin^2(\theta)} (\frac{\partial^2}{\partial \phi^2})</math>

Plugging this into the Schrodinger equation we get:

<math>-\frac{\hbar^2}{2 \m} [\frac{1}{r^2} \frac{\partial \psi}{\partial r}(r^2 \frac{\partial \psi}{\partial r}) + \frac{1}{r^2 sin(\theta)} \frac{\partial}{\partial \theta} (sin(\theta) \frac{\partial \spi}{\partial \theta}) + \frac{1}{r^2 sin^2(\theta)} (\frac{\partial^2 \psi}{\partial \phi^2})] + V \psi = E \psi</math>

To Be Finished:

• Discussion on The Angular Equation
• Legendre Polynomials

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