- #1

MxwllsPersuasns

- 101

- 0

## Homework Statement

This is a (long) multi-part question working through the various stages of solving the radial Schrodinger equation and as such it would be impractical to type it all out here but I will upload the pdf (https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU) of the problem set (its problem 2b.) and then refer to the various questions by recalling their latin numerals (i, ii, iii, iv, etc...)

## Homework Equations

[/B]

The Radial ODE: d^(2)u/dx^2 + (2/x - l(l+1)/x^2 - b^2)u = 0

## The Attempt at a Solution

[/B]

So I just started this problem and already am running into issues (will be updating this thread as I progress through the problems). Problem section i) tells us to show the asymptotic behavior of u(x) as x tends towards +inf is u ~ e^(+/-)bx. I imagine this means solve the Radial ODE for u in terms of x. Looking at the ODE one notices quickly its separable. Once separated it looks something like this: {d^(2)u}/u = -(-2/x - l(l+1)/x^2 - b^2)dx^2. In this form its clear we must integrate twice with respect to each variable in order to get something of the form u(x) = ... ... ...

Now, integrating once I get: ln(u)du = {b^(2)*x - l(l+1)/x - 2ln(x)}dx

This doesn't feel write to me though; It's hard to see how I'll get a clean exponential out of this. Furthermore the hint for this section states

*To do this omit the terms in 1/x and 1/x^2 which can be ignored relative to b^2 when x tends towards +inf*. This leads me to believe I need to do some manipulation whilst the radial ODE is in its original form (as that's where the 1/x and 1/x^2 terms appear).

Any help would make my day, I would be so grateful. I have been struggling with this problem for a little bit and can't seem to make any headway trying anything