Eigenfunction problem, no solving necessary just a quick q

[IdealFugacity]

(to cite source of problem, this is problem 13.19 from Physical Chemistry, by Thomas Engel and Philip Reid, (c) 2006 Pearson Education, Inc.)

This is not a collected homework problem, it is an 'assigned' practice problem (I dont have to hand it in or even do it for any kind of grade)

I've solved through applying the operator to the function on different occasions and got the same answer, so I don't need anyone to actually do any math here, just tell me if my final answer is 'allowed' please/

The question:

Find the result of operating with:
( 1 / r^2 )( d/dr )( r^2 )( d/dr + 2/r )
on the function
A * exp( -br )

What must the values of A and b be to make ths function an eigenfunction of the operator?

Solving through, applying the operator results in:

A * exp( -br) * ( b^2 + 2/r^2 - 4b/r)

Because of the 2 / r^2 term, I think the answer has to be:
"A must be 0, and b can have any value"

If that is not allowed, I think the answer is "there are no values that would make this function an eigenfunction of the operator"

Is A=0 a cheap way out? It seems like 0 would be a common solution for many eigenfunction problems if it is allowed.

Thanks again for any input!

[AWoodside]

Well there are two solutions for the term in parenthesis for b, but it makes b a function of r...

b = (2 +/- sqrt(2))/r

I'm not sure if having b be a function of r is a problem, but A=0 does seem like it's probably not what they're looking for.

[IdealFugacity]

I'm pretty sure that having b as a function of r defeats the purpose.

[flipdeadshot22]

I haven't worked it out for myself, but if you solved it right, A * exp( -br ) cannot be an eigenfunction of this operator, regardless of values for A and b. The A=0 solution is a trivial one that works, but doesn't tell us much.

[djames]

[ QUOTE ]
Find the result of operating with:
( 1 / r^2 )( d/dr )( r^2 )( d/dr + 2/r )
on the function
A * exp( -br )

[/ QUOTE ]

By this operator do you mean
( 1 / r^2 )*( df/dr )*( r^2 )*( df/dr + 2/r )
where f(r) = A * exp( -br )

or do you mean this
( 1 / r^2 )*( d/dr [ ( r^2 )*( df/dr + 2/r )] )
where f(r) = A * exp( -br )

or something entirely different?

If this is directly from the text, I'm not sure why that notation was chosen. Perhaps it's only ambiguous to me.

[IdealFugacity]

I typed it as it is in the book, it's not ambiguous as the operator works linearly. However, ironically it turned out in the end that there was a typo in the text book and the solutions manual had the following function:

( 1 / r^2 )( d/dr )( r^2 * d/dr + 2/r )

which results in a clearly defined answer. So of the two posts I've made in this forum asking questions for this class, both of them have turned out to be invalidated when I gain further information (typo in the textbook here, the other thread was useless once my professor told me he intended for us to get the 'wrong' answer)

[IdealFugacity]

(To answer your question djames, I intended the former)