By James R. Barrante
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Extra resources for Applied Mathematics for Physical Chemistry (2nd Edition)
LAdding two more terms to an infinite sum is the same as adding zero The Schrodinger equation describing a simple one-dimensional harmonic oscillator is where n,m, h, E, and k are constants. Show that this equation can be solved using Hermite polynomials. 80 Chapter 6 Solution. Differential Equations Section 6-6 Let Special Polynomial Solutions to Differential Equations 81 leads to the Laguerre polynomials of degree n: E 8n2m~ h2 = -- and 4n2mk hZ ,T2 = ---- Substituting these into Equation (6-29)gives We now make a change of variables by letting A differential equation closely related to Laguerre's equation is the equation t*=a x .
Thus, v = Lk, and a particular solution to Equation (6-36)is =e-~12x(k-~l/2~k (6-38) This function is called the associated Laguerre function. Example - The radial part of the Schrodinger equation for the hydrogen atom is The series solution is therefore - where n, w, h, E Q , and e are constants. Show that solutions to this equation are the associated Laguerre polynomials. Again, suitable choice of a,, a,,=(-l)"n! Solution. To transform the radial equation into a form that resembles Laguerre's equation, let us first expand the equation.
If you can find a copy, grab it! 3. NAGLE, R. , Boston, 1996. 4. , Upper Saddle River, NJ, 1997. PROBLEMS 1. Solve the following linear differential equations: y 3y=o -d+ dx Notice that this isolates the @ term. By the same arguments used above, the @ term must equal a constant, call it -mZ. Therefore, d y- 3y = 0 1 d Z @ = -m2 -- @(@I dx d@Z dZY+2dy+y=0 dx2 dx and d2y dx2 6 -dy +9y=O dx d2y =O -+9y dx2 dx - k l (a - x ) - k2x; k,. kZ, and a are constants. df since the sum of these two terms equals zero.
Applied Mathematics for Physical Chemistry (2nd Edition) by James R. Barrante