Wavefunction and Electron Configuration (Toughy of the Day)

[Sapper6]

The wave function for a particular electron is given by:

Psi= 4/(9√(4π)) * (6/a)^(3/2) * (r/a)^2 * e^(2i(phi) - (2r)/a) * sin^2 (θ)

a) This is an electron in which subshell?
b) This is an electron in an atom of which element?
c) What is the ionozation energy for this electron, assuming H-like behavior?
d) In a neutral atom (not H-Like) can this electron be in the ground state?
e) What is the probability of finding this electron within Bohr's radius of the nucleus?


I am not sure where to start here, I am assuming I would normalize the wave function by squaring it, but then how do I pull out quantum number data? I am very confused here.. Could someone please walk me through this or point me in the right direction.

 

[fzero]

The problem wouldn't make much sense if you haven't already seen the general solution of a hydrogen-like atom. You will want to write this solution down in terms of the general solutions $$\psi_{nlm}$$ in order to answer parts a,b and d.

 

[Sapper6]

that is where I am confused. I see the hydrogen like wavefunction tables that contain R and (Theta Phi) values, but my function here does not fit any of them. Am I suppose to do something to the wave function first? If I am suppose to normalize then what would my boundaries be? and do I need to separate terms before integration. i haven't been able to find an example like this problem... and i have been stuck for 2 days now.

 

[fzero]

that is where I am confused. I see the hydrogen like wavefunction tables that contain R and (Theta Phi) values, but my function here does not fit any of them. Am I suppose to do something to the wave function first? If I am suppose to normalize then what would my boundaries be? and do I need to separate terms before integration. i haven't been able to find an example like this problem... and i have been stuck for 2 days now.

There is the possibility that this wavefunction is a linear combination of the solutions in your list, but I don't believe that it would be since this solution has definite $$nlm$$ quantum numbers. Compare the degrees of this wavefunction to the degrees of the ones in your list in order to determine the $$nlm$$ quantum numbers. The relevant parts of the equation are

$$\psi = A r^2 e^{-2r/a} \sin^2\theta e^{2i\phi}$$

in order to determine the $$nlm$$ quantum numbers.

 

[Sapper6]

sorry i am a pain in the butt here, but now I am even more confused... how did you simplify down to that? all of functions in my table retain a Z^(3/2) term, but none have exponential like this or sin^2 theta

 

[JesseC]

He simplified it down by putting all of your constants into A and splitting up your exponential into its two separate parts. Otherwise its the same as what you wrote down in your first post (except of course its in tex code).

The whole point of the question which you put in your first post, is to make you realize how you can determine the quantum numbers just from looking at wavefunction. You don't need to compare it to any specific wavefunctions you may have in a table and hope you find the matching one.

The only thing you really need to know is the general form of the wavefunction, then you can compare the exponents in the wavefunction you've been given to the general form.

 

[fzero]

Just to add, while comparing the degrees of this solution to the general solution would be the easiest way to determine the quantum numbers, you should also know that you can always act on a solution with the $$L^2$$ and $$L_z$$ operators to find the $$\ell, m_\ell$$ quantum numbers. The $$n$$ quantum number would come out if you substituted the solution into the radial equation. That's a lot more work, but it's fundamental knowledge that you will need at some point.

 

[Sapper6]

Does this mean I can just use the sin'2 in angular part to assign l and m equal to 2 and then determine n from there? Even if the functions don't match exactly.

 

[fzero]

Does this mean I can just use the sin'2 in angular part to assign l and m equal to 2 and then determine n from there? Even if the functions don't match exactly.

It's easier to read off m=2 from the $$\phi$$ dependence, but this is an l=2,m=2 state. The angular part of any l=2, m=2 state has to be

$$\sin^2\theta e^{2i\theta},$$

so I'm assuming that when you say "the functions don't match exactly" you might be talking about needing to compute a change of variables in the radial function or normalization coefficient.