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For a free particle of mass \(\mu\) and energy \(E > 0\) in spherical coordinates, we start with the eigenfunction of angular momentum \((l, m)\) \(\psi_{Elm}(r, \theta, \phi) = R_{El}(r)Y_{lm}(\theta, \phi)\), \(R_{El}(r) = \frac{U_l(r)}{r}\) With \(\rho = kr = \sqrt{\frac{2\mu E}{\hbar^2}}r\), we define the \"raising\" and \"lowering\" operators \(A_l^+ = -\frac{d}{d\rho} + \frac{l}{\rho}\), \(A_l = \frac{d}{d\rho} + \frac{l}{\rho}\) \(A_l^+ + A_l = \frac{2l}{e}\) \(A_l^+ - A_l^+ = \frac{1}{i}\frac{d}{de}\) (a) Write the radial equation for \(U_l\) in terms of \(A_l^+\) and \(A_l\). (b) What are \(A_l^+U_l\) and \(A_lU_l\) related to? Justify your answer with appropriate derivations. (c) Use the results obtained for (a) and (b) to find the radial wave functions \(R_{El}\) for all \(l\) which are regular at the origin.

          For a free particle of mass \(\mu\) and energy \(E > 0\) in spherical coordinates, we start with the eigenfunction of angular momentum \((l, m)\)
\(\psi_{Elm}(r, \theta, \phi) = R_{El}(r)Y_{lm}(\theta, \phi)\), \(R_{El}(r) = \frac{U_l(r)}{r}\)
With \(\rho = kr = \sqrt{\frac{2\mu E}{\hbar^2}}r\), we define the \"raising\" and \"lowering\" operators
\(A_l^+ = -\frac{d}{d\rho} + \frac{l}{\rho}\), \(A_l = \frac{d}{d\rho} + \frac{l}{\rho}\)
\(A_l^+ + A_l = \frac{2l}{e}\)
\(A_l^+ - A_l^+ = \frac{1}{i}\frac{d}{de}\)
(a) Write the radial equation for \(U_l\) in terms of \(A_l^+\) and \(A_l\).
(b) What are \(A_l^+U_l\) and \(A_lU_l\) related to? Justify your answer with appropriate derivations.
(c) Use the results obtained for (a) and (b) to find the radial wave functions \(R_{El}\) for all \(l\) which are regular at the origin.

For a free particle of mass μ and energy E > 0 in spherical coordinates, we start with the eigenfunction of angular momentum (l, m)
ψElm(r, θ, ϕ) = REl(r)Ylm(θ, ϕ), REl(r) = (Ul(r))/(r)
With ρ = kr = √((2μ E)/(ħ^2))r, we define the r̈aisingänd l̈oweringöperators
Al^+ = -(d)/(dρ) + (l)/(ρ), Al = (d)/(dρ) + (l)/(ρ)
Al^+ + Al = (2l)/(e)
Al^+ - Al^+ = (1)/(i)(d)/(de)
(a) Write the radial equation for Ul in terms of Al^+ and Al.
(b) What are Al^+Ul and AlUl related to? Justify your answer with appropriate derivations.
(c) Use the results obtained for (a) and (b) to find the radial wave functions REl for all l which are regular at the origin.

Added by Thomas C.

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