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Measuring Carcinogens in Yuffa’s lungs
Daniel Gibson
June 2, 2008
Contents
1 Introduction 2
2 Theory 2
3 Experiment 3
4 Data Anal 4
5 Conclusion 5
6 About the author 6
List of Figures
1 The rectangular barrier potential of width 2` and height V 0.......... 3
2 The setup used. Not that the gamma ray emmitter looks like a CD player. 4
3 Density of carcinogens in Yuffa as a function of depth............ 4
4 Dan in the first Mines jazz concert featuring accordion............ 6
Abstract Yuffa’s lungs were bombarded with high-energy radation to detect the carcinogens present. He claims to have been smoking since he had been around the sun 12 times, but I thought he was full of crap (pardon the pun). The purpose of this is to analyse the data collected, and determine if Yuffa is, in fact, a liar or was just an unintelligible pre-teen.
List of Tables
1 Introduction
What is inverse scattering?
In the mid-1960s, an important new class of transformation techniques was discovered. This transform, called scattering transform, is capable of producing exact solutions for a small group of nonlinear evolution equations. The idea of this transform is to view the solution q(x, t) of the equation to be solved as the potential of the Schr¨odinger equation
−
d^2 u dx^2
For each fixed time t, we study the scattering problem and and the reflection and transmission coeffcients and the bound states. Then, as time evolves, the potential q(x, t) (and hence its scattering data) evolves as well. Now, the amazing thing we discover is that there are certain equations for which the evolution of the scattering data is diagonalized, that is, the scattering data for a given frequency evolves independent of all other frequencies. Thus, instead of studying directly the evolution of q(x, t), we study the evolution of the scattering data, which is described by separated, linear differential equations whose solutions are easily obtained, and from this information we can construct q(x, t). To make any transformation work[2], we must know how to transform functions and also how to invert the transform. For the Schr¨odinger equation, we know how, at least in principle, to transform potentials q(x) by ending scattering data. A natural question to ask is if one can reconstruct a potential q(x) from knowledge of its scattering data. This is an important problem, not only in the context of the Schr¨odinger equation, but in many other applications such as x-ray tomography, seismic exploration, radar imaging, medical imaging, etc., where the shape of an object must be determined from indirect information, such as the way waves are reflected or transmitted by the object. We will discuss only the problem of reconstructing the potential of the Schr¨odinger equation, with the realization that this is but a small part of the important field of inverse imaging problems in which there is still much active research and development.
2 Theory
ℏ^2
2 m
d^2 ψ dx^2
Vs =
V
ℏ^2 2 mb^2
, Es =
E
ℏ^2 2 mb^2
, xs = x b
d^2 ψ dx^2 s
Vs(xs) =
0 : xs ≤ −V 0 : − < xs < 0 : xs ≥
Figure 2: The setup used. Not that the gamma ray emmitter looks like a CD player
4 Data Anal
The graph below effectively illustrates the average amount of carcinogens in Yuffa’s chest as a function of depth.
20 40 60 80 100
Figure 3: Density of carcinogens in Yuffa as a function of depth
The density of carcinogens is high on the surface do to poor ventialtion of smoke around the time of inhalation, which explains certain smells that I often though emmi- nated from the computer. The density becomes low in the sternum, as it is not exposed
to any smoke, and high again in the lungs, where the smoke is absorbed and contained. The best fit equation describing this, which is somewhat accurate, is equation (6) with V 0 = 10.8.
5 Conclusion
Alex Yuffa indeed has a high probability of having smoked since he had traveled around the sun 12 times, meaning he is telling the truth for once. The rectangular barrier potential (Yuffa’s chest) could not be determined from the transmission coefficient, though the amount of tar within would probably stop and projectile from going all the way through him (that is, he would make a good shield).
6 About the author
Figure 4: Dan in the first Mines jazz concert featuring accordion
Dan is a narcisstic sesquipedlian. ’Nuff said.[1]
References
[1] James P. Keener, Principles of Applied Mathematics, (Perseus Books, 2000). [2] David J. Griffths, Introduction to Quantum Mechanics, (Prentice Hall, 1995).
