Inferring the Principal Quantum Number from a Given Wave Function

[Useong]

Hi, I am new here. I am a graduate student of department of physics at some university in Korea. If there is any wrong in my english, I will apologize in advance. I am preparing for my qualifying exam that is going to be held on next month.

Homework Statement

The question is very simple as I stated in the title. "Can you infer the principal quantum number from a given wave function of the hydrogen atom?" Not by memorizing but by logical deduction. I think no one can memorize all of the wave functions of the hydrogen atom.

Homework Equations

For example, you are given this wave function $$\psi _{nlm} = \frac{1}{{\sqrt {4\pi } }}\left( {\frac{1}{{2a}}} \right)^{3/2} \left( {2 - \frac{r}{a}} \right)e^{ - r/2a}$$
where$$a = \frac{\hbar }{{me^2 }}$$. Of cource, you may know the Hamiltonian that is composed of the kinetic term and the Coulomb potential.
$$H = - \frac{{\hbar ^2 }}{{2m}}\nabla ^2 - \frac{{e^2 }}{r}$$
The principal quantum number of above wave equation is 2. But how would you infer it?

The Attempt at a Solution

I tried this method. I know the energy eigenvalue is given by
$$E_n = - \frac{{e^2 }}{{2a}}\frac{1}{{n^2 }}$$

So, when I carried out the integration to find the energy eigenvalue, I could obtain

$$E_n = \iiint {d^3 r\psi _{nlm}^ * \hat H\psi _{nlm} } = - \frac{{e^2 }}{{2a}}\frac{1}{{\left( 2 \right)^2 }}$$
and therefore I could conclude n=2. But the method took me so long time that it may fail me if I meet this problem in exam time. So if you know any better methods to solve this problem, please let me know. Please enlighten me. Thanks a lot.

 

[Redbelly98]

The is one zero-crossing for the given function, at r=2a. So the principle quantum number is 2.

For n=1, there are no zeroes.
For n=2, there is 1 zero.
For n=3, there are 2 zeroes.
etc. etc

(At least, that's the case when L=M=0. It has been awhile since I had this, so nonzero L and M might or might not change the zero-crossing rule.)

 

[buffordboy23]

Yes, you can. There was a similar post about this not too long ago.

You only need to look at the radial wave equation, since by definition it has a term $$e^{-r/na}$$, where n is the quantum number.

Take a look at equation (11.136) here:

http://books.google.com/books?id=oc...X&oi=book_result&resnum=5&ct=result#PPA433,M1