What Is the Radial Schrödinger Equation for Hydrogen with Maximum l Value?

[tuomas22]

Sorry in advance my english, I tried to translate it to english as good as I can

Homework Statement

When $$l$$ has its maximum value, the hydrogen atom radial equation has a simple form of

R(r) = Ar^n-1e^-r/na₀, where a₀ is Bohr's radius.

Write the respective radial schrödinger equation.

Homework Equations

Radial Schrödinger Equation:
$$-\frac{\hbar^2}{2*\mu*r^2}*\frac{d}{dr}(r^2*\frac{dR(r)}{dr})+\left[-\frac{kZe^2}{r}+\frac{\hbar^2*l(l+1)}{2*\mu*r^2}\right]*R(r) = ER(r)$$

The Attempt at a Solution

I've attempted substituting the given R(r) to the radial equation, but I can't get anything out of it that makes sense. Too long to write it here with this hard latex thing :S

Should I get some nice reduced form or can I expect some equation monster?

 

[gabbagabbahey]

Sorry in advance my english, I tried to translate it to english as good as I can

Your translation is very good!

When $$l$$ has its maximum value...

This is the key assumption in the problem statement...What is the maximum value of $l$ for any given value of $n$ (remember, only certain values are allowed)?...Substitute that into your radial Schroedinger equation (along with the given function $R(r)$).

Radial Schrödinger Equation:
$$-\frac{\hbar^2}{2*\mu*r^2}*\frac{d}{dr}(r^2*\frac{dR(r)}{dr})+\left[-\frac{kZe^2}{r}+\frac{\hbar^2*l(l+1)}{2*\mu*r^2}\right]*R(r) = ER(r)$$

 

[tuomas22]

Thanks for answer!
The maximum value of $l$ is $n-1$ right?
I tried to substitute it but it doesn't get much prettier :)

I'm doing the math with Maple here, and it freaks me out to even think that I should be able to do it without computer, which is the case actually...That's why I think I'm doing something wrong, maybe I understand the equation wrong or something.

Here's a screenshot of my maple session (left side of the equation)
http://img526.imageshack.us/img526/5863/tryw.jpg

If I count the right side in too, I can reduce the Ar^n-1e-^r/na0 term from both sides but still I'm not so sure about it

 

[gabbagabbahey]

You also have an equation for Bohr's radius $a_0$ right?...And $Z=$____ for the Hydrogen atom?

When you substitute these things in, you should get a lot of terms canceling each-other out.

 

[tuomas22]

aaah the bohr radius equation, didnt even think about it :)
(and Z=1 ofcourse)

the next part of the question was to prove that the given R(r) is the solution to the schrödinger equation with $l=0$, and I got it right now with that bohr radius tip you gave! :)
$$- \frac{1}{2} \frac{k^{2}e^{4}\mu}{\hbar ^{2}} = E$$

But with $l=n-1$ I still get horrible monster equations...*cry* nothing seems to cancel out

there must be something I'm doing wrong since I can't be expected to do that kind of differentations without computer in a 2 hour exam ! No human could do it right! :D
(this is an old exam question)

 

[tuomas22]

Aaah I got it!
0 = 0 finally :)
There was some silly mistake in the equation.

The derivation is still inhumane to do with pen and paper though :)

Thanks much gabbagabbahey!