Copasetic Flow

OK, so let's say you're assigned the problem of determining the mean value, (the expectation value), for n, (the number state), in a harmonic oscillator with a coherent state.  You go back to your favorite coherent state reference by Nieto and Carruthers[1] and get the probability for finding your coherent oscillator in the nth level almost immediately, (picture 1), You're looking for the expectation value for n though, so you need to multiply the probabilty by n and sum the whole mess over all possible values of n, (zero to infinity).  Here's what you get, (excuse my sloppiness in picture 2).  Also, the favored notation for coherent states around here happens to be lambda instead of alpha. So, that looks like a mess. How do you make it more tractable and get down to a single value?  Enter our genius quantum mechanics professor.  He points out that if you just factorize and relabel things a bit, you wind up with (picture 3) Cool! D...

I think I'm finally hitting my studying stride for finals.  I can tell because my thoughts on quantum mechanics are starting to merge with the text from Jr.'s Little Golden Books[1], (picture 1).  Sure, sure, to some this might mean that I'm studying too much or sleeping too little, but I see it as a sign of some sort of Zen integration of my personal and quantum mechanical lives :) hehehe  Hoo Boy! I hadn't realized it on Friday, but looking back on the whiteboard where my professor kind of wowed me, his solution uses most of the basic tools we were taught this semester in an integrated form rather than as disparate facts which is how they'be been rattling around in my head.  So, in the tone of Jr.'s Sesame Street book on helping each other, here goes the solution of "Show that a time dependent force applied to a harmonic oscillator will produce a coherent state.  Here's the original white board in all it's glory (picture 2).  By th...

I'd hoped I was going to be able to make an authoritative swoop through the oft-mentioned, (here anyway), AJP article by Shubaya and Wulfman[1] wherein they explain how the accidental degeneracy in the hydrogen atom energy solutions can be visualized by looking at the accidental degeneracy in the corresponding two dimensional problem of orbits around a Coulomb potential.  Unfortunately, about half-way through, I realized I'm still not quite there.  Here's what I have so far.  I've cleared up exactly what the definition of the accidental degeneracy is.  There's a more complete grasp on the skeleton of Shubaya and Wulfman's argument here, as well as what 'projection to a +1 dimensional space' actually means. The Accidental Degeneracy  In the hydrogen atom there are two kinds of degeneracy with respect to energy.  The first kind is related to the quantum number m and is expected.  It has to do with...

By the end of the day today, I'll have two practice presentations for the APS TX Section meeting this weekend that I'd really appreciate any feedback on, I'll be building them between some pretty cool meetings today.  Here's some more on one of the meetings... I get to see Dr. Schleich's talk on his gravitational red shift vs. gravitational accelerometer calculation[3][4] with respect to the KC interferometer, (picture 1), [2].  The authors of KC interferometer experiment claimed that it measured the gravitational redshift, or time dilation due to curved space time.  In his PRL paper, Dr. Schelich points out that by analyzing the KC interferometer by looking at the commutators of it's time evolution operators, one can avoid choosing a representation and show that the shift in phase of the atoms in the KC interferometer is due to the acceleration of the atom caused by the gravitational potential, and not due to the gravitational red s...

Today is electricity and magnetism midterm day, so I'm just going to jot down a skeleton of a thought process about the quantum mechanical phase operator research I've been reading for the last few days, and then I have to run. In matrix rperesentation, the derivative of a polynomial can be represented as[1]: for a third degree polynomial and extended for higher degrees.  Integration looks like this[ 2 ]: and can again be extended.  In the article by Nieto [3], he quotes Louisell as saying this about the discrete cosine and sine functions in quantum mechanics. In the Fourier domain where functions are represented by series of sine and cosine functions, derivatives are constructed simply by multiplying by i, (the square root of negative one), times frequency, and integrals are constructed by dividing by i times the frequency. Also, in relation to the EE discrete signal analysis, these two figures from the Nieto RMP article [4], (pictures 4 and 5)...

Synchronicity was defined as Jung as the occurrence of two unrelated events that combined in the mind of the observer created a significant feeling of connectedness.  For the purposes of physics research, it might be something that belongs in the purview of an institute like Jack Sarfatti's "Physics Consciousness Research Group".  It might also just be explained away as an initial ignorance of the underlying history of the events, kind of like a 'hidden variable' theory.  Keep all these things in mind, as they'll relate to the story below in various ways. Here are my two seemingly unrelated events.  First, in quantum mechanics this week, we've been assigned a set of problems on coherent states.  Second, my adviser suggested I go to the colloquium being given here at Texas A&M this week by Wolfgang Schleich.  For those of you well versed in physics history, you've probably already gleaned the hidden variables.  For everyone el...

A quick review of what I'll be looking at over the course of the upcoming week.  This is as much to get my own thoughts in order as anything else. Quantum Mechanics: I'll be working on still more uncertainty and harmonic oscillator problems in QM this week.  What a surprise right :)  Specifically, this week, I'll be calculating matrix elements for both position and momentum squared using both the Hermite polynomial recursion operators and the ladder operators.  These are covered in chapters 5 and 10 in Merzbacher.  I was playing around with one of the recursion relations (picture 1)  for Hermitian polynomials earlier in the year and wound up with the following kind of interesting table.  You can see the n level of the wave function moved out of the way by the successive application of the recursion formula which amounts to the successive application of the x operator, or a sum of the raising and lowering operators (picture 2). I...

My quantum professor always knows exactly the right approximations to make, (I suppose it helps to be the person who wrote the homework).  Regarding the recent homework questions involving boundary conditions in shallow square potential wells which are just oozing with tangents of ratios of wave numbers, he pointed out a trig approximation that's often overlooked, (at least by students like myself).  At small angles (picture 1) the tangent of x is just x. Yup, small angles aren't just for sine anymore!  The tangent of a small angle is also equal to it's argument.  I had to go through the additional little thought process of (picture 2) Here's a graph of tan and sin overlayed with x so you can see what's going on. (picture 3) The Levi-Civita Symbol Another mathematical tool that has come up repeatedly in the last few weeks is the Levi-Civita symbol.  If you've studied the cross product or the vector curl you've seen it.  Wikipedia d...

I'm studying for our pre-midterm practice exam in quantum mechanics II this weekend, hence I've developed a bit of a fascination for all things, (even trivial things), quantum.  On our last homework, we had to decompose a wave function to its eigenfunction components, (sines and cosines in the infinite potential well that we were assigned), and then write down the time evolution for those components.  It was a bit of an exercise to remember how to normalize the components once I had them, so I'm recording the process here to hopefully make it easier to remember next time. The wave function that had to be decomposed was As you can see, a trig identity arose to make simple work of the problem once again.  No need to do a Fourier decomposition to sines and cosines when a simple little power-reduction formula [1] is readily available. That leaves us with two components and two leading factors, or weights.  The mechanical normalization process is t...

While working on QM homework yesterday, I came across a very nice article [1] in the American Journal of Physics about a new perspective on solving for the energies of the bound states in a potential well.  The article points out that waves reflecting back and forth in a potential well have to remain self consistent (picture 1). Another way to say this is that they have to catch their own tails, or wind up back where they started after a round-trip transit of the well.  Using this as a basis, the following equations are quickly derived (picture 2) Equation 4 describes the particles wave function after traversing the well once immediately after the reflection.  Rho is the coefficient of reflectivity, the first exponent is the phase picked up by the waveform after travelling distance L, and the second exp is the wave function itself.  Equation five is the wave function immediately after the second bounce.  Note the extra reflection coefficient and phase g...