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I think the two angular equations on page 136 (<math>\frac{1}{\Theta}\left[\sin\theta\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)\right]+l(l+1)\sin^2\theta=m^2</math> and <math>\frac{1}{\Phi}\frac{d^2\Phi}{d\phi^2}=-m^2</math>) and the radial equation (<math>\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)-\frac{2mr^2}{\hbar^2}\left[V®-E\right]=l(l+1)</math>) are important since we can use them to find probabilities of finding a particle in a sphere. also since the wave function is a separable functions we should be able to find radial probability density and angular probability density; for example the probability of finding the particle in the volume between r and r+dr.I think for this reasons and the fact that quantum mechanics is all about probabilities, these three equations are of importance.

definitely understand spherical representation of Shrod. eq and understand how to arrive at important eq. listed by Andromeda.

Also from chapter 3 i would put down equation 3.71(<math>\frac{d}{dt}\left<Q\right>=\frac{i}{\hbar}{\left<[\hat H,\hat Q]\right>+\left<\frac{\partial\hat Q}{\partial t}\right></math>). there was a few homework problem that we had to use this for.

I think, the equation 3.54 on page 108, the momentume space wave function might be important. <math>\displaystyle \Phi(p,t)= \frac1 sqrt{2\pi\hbar} \int exp(\frac{-ipx} {\hbar})\Psi(x,t) dx</math> We can use this equation to obtain the probability that a measurement of momentum would yield a value under a specific condition.

And also the position space wave function on page 108, <math>\displaystyle \Psi(x,t)= \frac1 sqrt{2\pi\hbar} \int exp(\frac{ipx} {\hbar})\Phi(p,t) dp</math> might be easily forgotten.

I think the matrix operators of H and P we did in discussion are worth for the reviews. Especially how to find eigenvalues ,which corresponds to the energy and probility and eigenvectors, which corresponds to the eigenstate of wave function.

I agree with Can about the discussion problem. Also, I think the Schwarz inequality may be useful, I know it has been used a few times in derivations in the book. Presently, it's important for the derivation of the general uncertainty principle. Here is the Schwarz inequality:

<math>\mid\int_{a}^{b}f(x)*g(x)dx\mid\leq\sqrt{\int_{a}^{b}\mid f(x)\mid^2 dx \int_{a}^{b} \mid g(x) \mid^2 dx}</math>

Question, Iam looking for the general consensus on should we be concerned with Legendre polys, Rodrigues formula,spherical harmonics, Bessel function and/or Nuemann functions. There was not a lot of emphasis in the HW, but considerable time in lecture was devoted to these. I feel like most of these are to envolved for a one hour exam. What does anyone else think?

This might be naive, but I think just a general understanding of what they do and how they act should suffice. In lecture yesterday (or monday maybe?), he spent a lot of time on the Bessel function graph. I think he wants us to really understand what the graph means and why it acts that way.

Making that equation sheet made me worry even MORE about the test… The most important equations , I felt, were the results of tedious derivations. These main equations were fairly large (larger than I could memorize with ease)… So I am wondering how Yuichi is going to approach writing this test? Obviously everybody has opinions about which equations were “most important” so asking a question stemming from an important equation, say 4.63, is either hit or miss. If you have the right equation you'd be in luck, but if not, it'd be painful to know that 20 (or even 40!) points is swirling down the toilet….

SOO long story short, I know Yuichi doesn't want to write a test where a few lucky people wrote down the right equation. I'm thinking everything we need will be given in the test problem, and we will be asked to build off of it with the methods learned in lecture & discussion. Something along the lines of today's lecture (writing out the basic steps taken to find equations for the radial wave function.)

I'm worried because it feels like I'm throwing away a bulk of the chapter's equations, just to commit to understanding the basic methodology of “how-to-get-there” Yikes! Is this a fast track to nowhere?

I wouldn't be too worried. The derivations Griffiths did, he pulled a bunch of definitions out of nowhere that were pretty arbitrary unless you already knew what you wanted to get. So I would be surprised if we were expected to pull that sort of stuff out of the ether.

Been doing the practice test and I need to know if problem 2 is as easy as plugging the change of coordinates into the operator equation.

Generally, yes, but since the x and y momentum operators involves derivatives of x and y respectively you need to change the derivatives from Cartesian to spherical coordinates. You can do this by exploiting the chain rule for derivatives and obtaining

<math>\frac{\partial}{\partial x}= \frac{\partial \rho}{\partial x} \frac{\partial}{\partial \rho} + \frac{\partial \theta}{\partial x} \frac{\partial}{\partial \theta} + \frac{\partial \phi}{\partial x} \frac{\partial}{\partial \phi} </math>

Then just use the definitions for the change of coordinates to compute each respective component derivative and you'll be left with spherical coordinate derivative operators multiplied by variables on to the left. If you do it correctly you'll obtain equation 4-129.

Doing problem 3.38 for review, part b for A, I get <math><A> = \lambda (c_1*c_2 + c_2*c_1 + 2|c_3|^2)</math>. Is it possible to cancel the <math>c_1*c_2</math> and <math>c_2*c_1</math> terms by kroenecker delta leaving <math><A> = (2\lambda|c_3|^2)</math>? Thanks!

I'm not sure that you can. I think when Griffiths proved that in chapter 2.2, he used two different states of wave function, but the c's {<math>(c_1*c_2 + c_2*c_1 + 2|c_3|^2)</math>} are, I think, more like dimensions of a single state rather than different states.