Solving a time independent Schrodinger equation

[collpitt]

Homework Statement

Hello!
I am currently stuck with a time independent Schrödinger equation where the potential "V(x)" is hyperbolic in nature. I was wondering if anyone could give me a hint as to how I should approach this problem in order to get an analytical solution (without using numerical techniques).

Homework Equations

The equation is of the form,
d²/dx²(Ѱ) + 2m/ħ²{E + δp²/x(p-x)}(Ѱ) = 0
where the voltage V(x) = -{δp²/x(p-x)}

The Attempt at a Solution

Thank you!

 

[AuraCrystal]

Stupid question: is \delta just a number here?

 

[collpitt]

Yes, δ, E, p are all constants.
I am trying to solve it by defining z = y(x) and changing the differential equation. But it is getting quite complicated. :(

Any help would be gladly appreciated.

 

[darkxponent]

Did u try series solution for diffrntial eqn

 

[collpitt]

Yes. But I am unable to reduce the equation to any of the forms I know (bessel, legendre, laguerre, hermite, chebyshev).

 

[genericusrnme]

Yes. But I am unable to reduce the equation to any of the forms I know (bessel, legendre, laguerre, hermite, chebyshev).

I believe the solution is a series, I'll give you a clue; hyper

 

[collpitt]

Yesss! Thank you! I believe it can be reduced to the hypergeometric form. Thank you very much [bold]genericusrnme[/bold] !
I will post the solution as soon as I finish.