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If you do any problems in sections 2-1 and 2-2 and if you have issue with them, please let us know. If it is judged general enough, we will address them in class. Otherwise, someone will help you by answering your questions.

Otherwise, we will be spending most of Friday on SHO.

On pages 52-53 of Griffiths: I understand why we drop the B term from Equation 2.75, but then I can't figure out why we need to throw on <math>h(\xi)</math> in Equation 2.76, and then proceed with all of the math on the following page.

From what I understand about the material, in Footnote 23 on page 52 it says that stripping off the asymptotic behavior is merely to provide an alternative method for solving the differential equations of the wave. Of course this is based on estimations that <math>h(\xi)</math> will provide a simpler wave function to work with since we want to be able to analyze the wave at all values of <math>\xi</math>.

On today's lecture, I noticed when Yuichi was talking about stationary state how Hamiltonian is time independent, he wrote <math> <\hat{H}>=\int\Psi^\ast E_n \Psi dx=E_n</math> here <math>E_n</math> is an eigenvalue instead of an operator <math> ih\frac{\partial }{\partial t} </math> . Does that mean when the subscript shows up, it is always an eigenvalue?

<math>E_n</math> is associated with the eigenfunction (or eigenstate if you will) of the Hamiltonian, and since energy is quantized there can only be n eigenvalues corresponding to specific eigenstates (note that the eigenfunctions have subscripts as well). I'm sure there's a more lucid, mathematical way to express this but I tried.

I still have a question about stationary states and how to use and apply Schrodinger's equation: Since the solution to Schro's equation is a linear combination of stationary states, doesn't that mean that a particle that is described by the equation can have any amount of energy? How does this connect to the fact that the energy we measure from, say, an electron in hydrogen, is quantized?

I am not sure whether I am correct, but according to my understanding, once you make a measurement to a particle, you can get only the eigenvalues of energy. Although the state is a combination of several stationary states, the result of measurement is among the different energy eigenvalues E1,E2,E3,… The probability of each result is related with the coefficient before the steady states. For example, if the state of particle is combined with two steady states, <math>\Psi=c_1\Psi_1+c_2\Psi_2</math> their eigenvalue of energy E1 and E2, the measurement can never get a result between E1 and E2, the result may be E1, or may be E2, and the probability of E1 result is <math>\|c_1|^2</math> and probability of E2 result is <math>\|c_2|^2</math> So the energy is quantized.

Okay, thanks, that makes some sense, in that it explains why the energy level you measure will always be quantized, but isn't it true that particles will only absorb certain amounts of energy based on what energy levels are available for them to jump up to? Your answer seems to imply that my hydrogen electron could absorb any amount of energy, not just specific amounts.