# How to use extra term in radial equation

1. Apr 28, 2007

### valtorEN

1. The problem statement, all variables and given/known data

Due to the modification of inner-shell electrons of a multi-electron system,

the outer shell electron can feel an effective electrostatic potential as

V(r)=-e^2/(4*π*eo*R)-lambda*(e^2/(4*π*eo*R^2)) ; 0<lambda≤1

Find the energy eigenvalues and wavefunctions of the outer shell electron

and compare to those of the hydrogen atom

2. Relevant equations

-hbar^2/2*m(d^2u/dr^2)+[(V(r)+(hbar^2/2*m)*l(l+1)/R^2]

Eigenenergies for hydrogen atom

En=-[(m/2*hbar^2)*(e^2/4*π*e0)^2]*(1/n^2)=E1/n^2

3. The attempt at a solution

i plugged the effective potential into the radial equation, divided by E

(E=-hbar^2*k^2/(2*m)) where k=sqrt(2mE)/hbar

and get an equation with the same general solutions as in my book

u(rho)=(rho^(l+1))*(e^(-rho))*(v(rho))

still, how does this term effect the energy compared to that with hydrogen's

eigenenergies? i assume a const term with lambda in it finds it way into the

eigenenergies, but how to solve this is beyond me!

how do i calculate the new energy eigenvalues and wave functions?

how do they compare to that of hydrogen?

cheers
nate

2. Apr 28, 2007

### lalbatros

My feeling is that you could rescale the r-axis to get this problem solved.
The additional term in the potential is similar to the angular momentum term.
Therefore, the effect should be equivalent to the same problem with r rescaled and V(r) rescaled correspondingly.
But that is only my guess ...
Hope it helps.