===== Sept 25 (Fri) Hopefully, wrapping up analytical solutions to SHO =====
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==== Anaximenes - 23:00 - 09/23/09 ====
In the last page, Daniel Faraday asked about the spring potential, specifically why we can look at the limit of x -> infinity and have it mean anything (see the bottom of page 51).  I have what I think is a good answer, but I'd like a second opinion on it.

The reason that we normally look at SHO potentials only with small displacements x is that the potential is always an approximation to something, whether that means a physical spring deforming at large x or looking at a small segment of a complicated potential.  However, there is still a solution to the SHO potential going to very large x; it isn't the Schrödinger equation failing at large x or the spring potential failing at large x.  Rather, the approximation that the potential we're working with is a SHO potential fails at large x.  Therefore, observing the behavior of solutions to an SHO potential in the limit of large x can still be useful.  As long as one doesn't assume that the system behaves the same way outside that limit (which Griffiths doesn't), there's no problem.

I rambled a bit, but hopefully, that was clear enough.

==== chavez 9:00 9/25 ====
My understanding of Griffiths reasoning was that he was looking at the limit of x-> infinity only in order to get the basic form of the differential equation, which he used to work backwards and get the exact solution.


==== Zeno 9/24 12:00PM ====

The only major issue I have with the SHO analytical solution is a seemingly-logical contradiction in the step from equation [2.72] to [2.74], culminating in equation [2.77] in conjunction with footnote 23: "Note that although we invoked some //approximations// to motivate Equation 2.77, what follows is //exact//." --How can an approximation used to remove complexity in an equation then be solved to an "exact" solution of the original equation? Doesn't an "approximation" to the DE, by definition, require that the //exact// solution to the //approximate// DE be an //approximate// solution to the //exact// DE? Or is there something very obvious or very tricky I'm missing here?

//**Yuichi**//
Even though 2.76 is approximate and a very good approximation only when <math>\xi\approx\pm\infty</math>, when you insert <math>h(\xi)</math> as in 2.77, we are looking for exact solutions again.  The solution in the form of 2.77 was subbed in 2.72, not 2.74, to figure our what equation <math>h(\xi)</math> must satisfy (resulting in 2.78 - that's why "K" is back in the business).  So when one finds solutions,  <math>h(\xi)</math>, and when you combine them with 2.76, they are exact solutions of 2.72, not 2.74.

==== poit0009 9/24 5:10PM ====

I was looking at problem 2.15, and I was wondering how would you go about solving this.  I haven't even been able to come up with a way to start it.  Any ideas?

===Green Suit 9/24 9pm===
The problem is asking about the ground state for the harmonic oscillator. So I would start with Equation 2.59. Also, since the problem is asking what the probability of finding the particle outside the classically allowed region I would try modifying 2.59 with the dimmensionless variable from Equation 2.71.

===chavez 9:22 9/25===
I would start by writing <math>E_{0} = \hbar\omega/2</math> and then substitute this for <math>E</math> in the given equation and solve for x. Given this and the probability density function the problem should be pretty straight forward.
==== Pluto 4ever 9/24 5:56PM ====
I was just wondering about the discussion problem. When we extend the barrier from L to 2L does it matter where the particle is along the ground path?


=== poit0009 9/24 6:35PM ===

The barrier is assumed to move fast enough that the wave function is unable to respond immediately.  Since the wave function is a description of the probability of where the particle may be, it does not matter where the particle is specifically.

=== Spherical Chicken 9/25 10:30 ===
Does it ever matter where the particle is on these scales?  I thought the idea of making it a wave was that we don't really concern ourselves with where it is "really" because 1 it's on such as mall scale we can't exactly know, and 2, ... well we're approximating it with a wave... so aren't we saying the wall moves so fast the __wave__ can't simultaneously react?   Am I misunderstanding all wave theory or isn't this where we approximate the particle //as// a wave (or a probability under the wave?), not as much a particle along a waves path?  Perhaps my perception is really off... haha now I'd really like to know for sure that I'm either right or wrong...  

=== Ralph 9/25 10:55 ===
Hold on, when you ask where the particle is, are you asking about its wave function, its expectation value for position, or its "real position"?  I had assumed that the original question was a question about where the wavefunction ended up being in the new 2L square well, rather than a question about the "actual" position of the particle (I thought this second interpretation was the classical way of thinking about the problem).  But it does make me wonder:  is the particle "interacting" with the walls of the potential well in the same way that we do when we "measure" something like position?  I think I'm confusing myself now!

=== Esquire 9/25 14:15 ===
It is my interpretation that you are rarely concerned where the particle exactly is, but rather the goal is to obtain the probability density function of the particle's position and work with expectation values.

=== Anaximenes - 21:55 - 09/25/09 ===
The official answer (as seen on pages 3 and 4 in Griffiths) is that the probability distribution //is// the location of the particle.  It does not describe where the particle might already be; the particle is at all of the points in the probability distribution (or, if you prefer, none of them until a measurement is made).  You can talk about where you would //measure// the particle to be, but the answer to where it //is// can only be the probability distribution.  I would suggest going back and reading pages 3-4 (or maybe all of chapter 1) again now that we've dealt with the problems a little.

For Ralph's question about whether the particle interact with the walls and whether any such interaction counts as measurement, I'm less sure, but I would say that we're assuming that the particle does not interact with the wall as the wall is moving and that there is no measurement involved.  If that were not the case, the probability distribution would change.  Whether this is a physically realistic situation, I don't know, but I'm dubious.  I see it as more of a mathematical exercise to familiarize us with the math.

==== Ralph 9/25 10:45 ====
I had assumed that the change in energy was the important part (rather than the specific initial position), and that the change in energy was directly related to the change in the dimensions of the infinite square well.  It seems like the final state would be the same regardless of position because the change in energy is the same.  The evolution of the probability distribution would be different (and the intervals of integration) , but the fundamental situation would remain the same.


==== time to move on ====


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