Laguerre polynomials and the hydrogen wave function

Transcription

1 Laguerre polynomials and the hydrogen wave function April 3, The radial equation: aymptotic limits We begin by writing the radial wave equation, ψ me r r me 1 l l 1 r ψ = 0 and finding the limiting forms as r and at the origin. For large r, since ψ me ψ = 0 Since E < 0, the limit has exponential solutions which we write in the form ψ = Ae 1 κr Be 1 κr where κ = 8mE For the wave function to vanish at infinity, we require B = 0. As r 0, the equation reduces to is bounded, and we set ψ = r α. Then ψ r 1 l l 1 ψ = 0 r 0 = r α r α 1 l l 1 rα r αr α 1 1 r l l 1 rα u = α α 1 r α u r = α α 1 l l 1 r α u so we have solutions We require the positive powers, α = l. α = l, l 1 Transformation First, simplify the variables. Starting with ψ me r r me 1 l l 1 ψ = 0 r 1

2 the first two terms may be written as 1 r r ψ me r me 1 l l 1 ψ = 0 r Let κ = 8mE κr = x λ = me κ Then multiplying by 1 κ, the radial equation becomes 0 = 1 κ 1 r = 1 x x r ψ x ψ x 1 me κ r λ x 1 l l 1 4 x 1 me κ 1 1 κ r ψ l l 1 ψ Now let Then 1 x e x/ x l Zψ x x x ψ = e x/ x l Z x = 1 x 1 x x e x/ x l Z le x/ x l 1 Z e x/ x l Z = 1 1 x x e x/ x l Z le x/ x l1 Z e x/ x l Z = 1 1 x 4 e x/ x l Z 1 le x/ x l1 Z 1 e x/ x l Z 1 x 1 l e x/ x l1 Z l l 1 e x/ x l Z l e x/ x l1 Z 1 x 1 e x/ x l Z le x/ x l1 Z e x/ x l Z = 1 4 e x/ x l Z 1 le x/ x l 1 Z 1 e x/ x l Z 1 l e x/ x l 1 Z l l 1 e x/ x l Z l e x/ x l 1 Z 1 e x/ x l Z le x/ x l 1 Z e x/ x l Z so, cancelling the common exponential, the radial equation is transformed to 0 = 1 4 xl Z 1 lxl 1 Z 1 xl Z 1 l xl 1 Z l l 1 x l Z l x l 1 Z 1 xl Z lx l 1 Z x l Z λx l 1 Z 1 4 xl Z l l 1 x l Z

3 Collecting terms, 0 = x l Z l x x l 1 Z 1 lxl 1 1 l xl 1 λx l 1 l l 1 x l l l 1 x l Z Dividing by x l 1, Z must satisfy Let Then = x l Z l 1 x x l 1 Z l 1 λ x l 1 Z This is the associated Laguerre equation. 3 The Laguerre equation xz l 1 x Z λ l 1 Z = 0 k = l 1 α = λ l 1 xz k 1 x Z αz = 0 A useful set of polynomials, the Laguerre functions, is given by the solutions to the Laguerre equation, x d L α and for α = n, the associated Laguerre polynimials, 1 x dl α αl α = 0 which satisfy L k n x = 1 k d k x d L k n k L nk x k 1 x dlk n αlk n = 0 Exercise: Derive the associated Laguerre equation by differentiating the Laguerre equation k times. For the Laguerre equation, we assume a solution of the form L α = a s x s s=0 Then dl α = d L α = sa s x s 1 s s 1 a s x s s= 3

4 so that s= x s s 1 a s x s 1 x sa s x s 1 α a s x s = 0 s= m m 1 a m1 x m m=1 The m = 0 term is For all m > 0, and therefore s s 1 a s x s 1 sa s x s 1 sa s x s α a s x s = 0 m 1 a m1 x m a 1 αa 0 = 0 m=1 s=0 s=0 ma m x m α a m x m = 0 m m 1 a m1 m 1 a m1 ma m αa m x m = 0 m 1 a m1 α m a m = 0 a m1 = α m m 1 a m For m = 0 this formula also gives a 1 = αa 0 so we may extend this formula to all m. Iterate this series: so that for k = m, where the Γ function satisfies a m = α m 1 m a m 1 = 1 α m 1 α m m m 1 a m = 1 k α m 1 α m α m k m m k 1 a m k a m = 1 m α m 1 α m α m 1 a 0 = 1 m Γ α 1 m!m!γ α m 1 a 0 Γ m = m 1! Γ α 1 = αγ α The solution is L α x = a 0 Γ α 1 m!m!γ α m 1 1m x m 4 Quantization Consider the large m limit of our solution for L α x. As α 1 becomes negligible in the numerator, the coefficients become a m = m α 1 m α m α 1 1 m a 0 m!m! 4

5 so asymptotically the series approaches m m 1 m 1 m!m! 1 = α 1 a 0 m! L α x α 1 a 0 α 1 a 0 1 m! xm e x This means that if the series extends to large m, the radial wave function becomes ψ = e x/ x l Z x e x/ x l and diverges. The only way to avoid this is by taking α i to be a non-negative integer so that the series terminates a m = 1 m i m 1 i m α m 1 a 0 a i = 1 i 1 i! a 0 a i1 = 0 and L n x is a polynomial. Returning to our definitions for the radial wave function and κ = 8mE κr = x λ = me κ i = λ l 1 We therefore get a quantization condition, λ = i l 1 n where Solving for the energy n = me κ = = me 8mE me 8mE 8mE n = 4m e 4 n E n = me4 n 5

6 Neglecting fine structure, these are the energy levels of hydrogen. Notice that in order for i to be a positive integer, we must have n l 1 The final polynomials are: so that the complete wave function is where A gives the normalization. L k α x = L l1 n l 1 κr Ψ r, θ, ϕ, l, m l, m s = Ae κr/ κr l L l1 n l 1 κr Y lm θ, ϕ χ α, β 5 Appendix: The associated Laguerre equation The associated polynomials solve a related set of equations given by differentiating the Laguerre equation for L nk, k times: 0 = dk k x d L nk 1 x dl nk n k L nk = dk k x d L nk dk k 1 x dl nk n k dk L nk k For the first term, The pattern is emerging: d x d L nk d d 3 3 x d L nk x d L nk d k x d L nk k = x d3 L nk 3 = x d4 L nk 4 = x d5 L nk 5 = x dk L nk k d L nk d3 L nk 3 3 d4 L nk 4 k dk1 L nk k1 Check one more derivative to complete the induction: d k1 k1 x d L nk = d x dk1 L nk k 1 dk11 L nk k1 k11 so the form is correct. For the second term, we need Look at the last part, d k k 1 x dl nk d x dl nk d x dl nk = dk dlnk k x dl nk = dk1 L nk k1 = x d L nk = x d3 L nk 3 6 dk k dl nk d L nk x dl nk

7 so we guess that the generic term is d k k x dl nk = x dk1 L nk k1 k dk L nk k and check one more: d k1 k1 x dl nk = dk1 L nk k1 x dk L nk k k dk1 L nk k1 = x dk11 L nk k11 k 1 dk1 L nk k1 Therefore, returning to the equation, 0 = dk k x d L nk dk k 1 x dl nk n k dk L nk k = x dk L nk k k dk1 L nk k1 dk1 L nk k1 x dk1 L nk k1 k dk L nk k = x d d k L nk k k 1 x d d k L nk d k L nk k n k n k dk L nk k and, inserting a minus sign, the associated Laguerre equation is x d L k n k 1 x dlk n nlk n = 0 7