PHY4604 Problem Set 10: Quantum Mechanics - Hydrogen Radial & Harmonic Oscillator, Assignments of Physics

Download PHY4604 Problem Set 10: Quantum Mechanics - Hydrogen Radial & Harmonic Oscillator and more Assignments Physics in PDF only on Docsity! PHY4604–Introduction to Quantum Mechanics Fall 2004 Problem Set 10 Nov. 17, 2004 Due: Dec. 1, 2004 Reading: Griffiths Secs. 4.1-4.3 1. Hydrogen wave functions. (a) Plot and label the first six (i.e. n=1,2,3, all allowed values of `) radial functions Rn`(r), as functions of r/a0. These functions are found to be Rn`(x) = √√√√ ( 2 na0 )3 (n− `− 1)! 2n[(n + `)!]3 e−x/2x`L2`+1n−`−1 (x) (1) where Ln` are associated Laguerre polynomials and x ≡ 2r/(na0). • (Option 1.) You may use Maple to calculate and plot these func- tions. In this case note that the built-in functions L(n, a, x) Maple calls generalized Laguerre polynomials are normalized differently from the associated Laguerre polynomials. To generate associated Laguerre polynomials, type: Laguerre:=(q,xi)->expand(exp(xi)*diff(xi^ q*exp(-xi),xi$q)): This is the def. of Laguerre polynomials, Lq(ξ) = e ξ(d/dξ)q(e−ξξq). Now type ALaguerre:=(q,p,xi)->(-1)^ p*diff(Laguerre(q,xi),xi$p): to define the associated Laguerre polynomials, Lpq−p(ξ) = (−1)p(d/dξ)pLq(ξ). To get the required L2`+1n−`−1 occurring in Eq. (1), we need p = 2` + 1 and q = n + `. So define Lnl:=(n,l,xi)->ALaguerre(n+l,2*l+1,xi): Then you should get, e.g. Lnl(3,0,xi); 18− 18ξ + 3ξ2 Next define a radial function Rn` using the substitute command subs: Rnl:=(n,l,xi)->subs(xi=2*r/n,expr); where expr is your expression for the form of the Rn`. This is necessary because if you try to define the Laguerre polynomial Laguerre(q,2*r/n), Maple tries to differentiate with respect to the variable 2 ∗ r/n and can’t, whereas subs substitutes 2 ∗ r/n after differentiation. Plot your 6 Rn` vs. r/a0 on a scale of r/a0 = 0..20 and R = −0.2..0.6. 1