create a presentation to display the results of your research. The report should be a powerpoint or slide.
All work should be typed with double-spacing and 11 pt Arial font. You will be graded on the evidence of work, mathematical detail
and understanding, proper exposition and neatness, time management, and organization of ideas. Your work should also be
supported with properly labeled and embedded plots and equations. Any references used should be cited as well.
Structure:
A suggested outline of the final report include:
Title page (including Title, Names, Abstract), Introduction to Problem (Statement and Background)
.Formulation of Model and Method of Solution
Solution Results (Analytical, Numerical, Plots of Solutions)
The presentation includes: Discussion/Results (Summary and Discussion of Solution)
References
Fundamentals of Differential Equations In quantum mechanics one is interested in determining the wave function and energy states of an atom. These are determined from Schrödinger’s equation. In the case of the hydrogen atom, it is possible to find wave functions c that are functions only of r, the distance from the proton to the electron. Such functions are called spherically symmetric and satisfy the simpler equation (1) 1 r d2 dr2 1rc2 = -8mp2 h2 aE + e20 r bc , where e20, m, and h are constants and E, also a constant, represents the energy of the atom, which we assume here to be negative. (a) Show that with the substitutions r = h2 4p2me20 r , E = 2p2me40 h2 e , where e is a negative constant, equation (1) reduces to d21rc2 dr2 = - ae + 2 r brc . (b) If f J rc, then the preceding equation becomes (2) d2f dr2 = - ae + 2 r b f . Show that the substitution f1r2 = e-arg1r2, where a is a positive constant, trans- forms (2) into (3) d2g dr2 - 2a dg dr + a 2 r + e + a2bg = 0 . (c) If we choose a2 = -e (e negative), then (3) becomes (4) d2g dr2 - 2a dg dr + 2 r g = 0 . Show that a power series solution g1r2 = g∞k= 1 ak rk (starting with k = 1, as per Frobenius) for (4) must have coefficients ak that satisfy the recurrence relation (5) ak+1 = 21ak - 12 k1k + 12 ak , k Ú 1 . B Spherically Symmetric Solutions to Schrödinger’s Equation for the Hydrogen Atom (d) Now for a1 = 1 and k very large, ak+1 ≈ 12a>k2ak and so ak+1 ≈ 12a2k>k!, which are the coefficients for re2ar. Hence, g acts like re2ar, so f1r2 = e-arg1r2 is like rear. Going back further, we then see that c ≈ ear. Therefore, when r = h2r>4p2me20 is large, so is c. Roughly speaking, c21r2 is proportional to the probability of finding an electron a distance r from the proton. Thus, the above argument would imply that the electron in a hydrogen atom is more likely to be found at a very large distance from the proton! Since this makes no sense physically, we ask: Do there exist positive values for a for which c remains bounded as r becomes large? Show that when a = 1>n, n = 1, 2, 3, . . . , then g1r2 is a polynomial of degree n and argue that c is therefore bounded. (e) Let En and cn1r2 denote, respectively, the energy state and wave function correspond- ing to a = 1>n. Find En (in terms of the constants e20, m, and h) and cn1r2 for n = 1, 2, and 3.