While working on QM homework yesterday, I came across a very nice article 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 gain. Equation six points out the condition that must be met if equation five is to remain consistent with the original wave function.
The authors then proceed to derive the necessary reflection and transmission coefficients for a number of helpful examples including the non-symmetric square well potential, (the problem of interest on my homework). Without giving too much away, there derivations are very easy to understand and conceptualize do to a clever representation of the depth of the potential well in terms of the k's of the wave functions.
Folks who are used to WKB and/or the Bohr-Sommerfeld quantization rule will find this article very easy reading. On the subject of WKB, if you want to know a lot more about it and how it relates to complex analysis, check out his 1947 article from W. H. Furry in the +American Physical Society's Physical Review.
No, he's not related to Felix Bloch as far as I know, so that's not the cool part. Here it is. Dr. Bloch works at the Grenoble Institute of Technology's Institute for Stationery and Graphics. That's right, he gets to work on all the pretty stuff! I quote, loosely translated by Google from their web site:
"From fiber to paper, screen printed"
Located in the heart of the university campus of Grenoble, the French School Stationery and Graphic Industries is the only engineering school in France which forms in three years future managers stationery, processing of paper and paperboard, and printing .
I had no idea, although it makes perfect sense, that you could work on the physics of paper! Recently he's been working on advances in paper based capacitative touch pads.
Mazzeo A.D., Kalb W.B., Chan L., Killian M.G., Bloch J.F., Mazzeo B.A. & Whitesides G.M. (2012). Paper-based, capacitive touch pads., Advanced materials (Deerfield Beach, Fla.), PMID: 22539155
Picture of the Day (from the mountains north of Grenoble!)