Interpreting Hydrogen Atom Wave Functions: A Question of Correctness?

[gfxroad]

Homework Statement

I solved the Schrödinger equation, obtaining a wave function in terms of Radial and the spherical harmonics as follows:

$$Ψ(r,0)= AR_{10} Y_{00} + \sqrt{\frac23} R_{21} Y_{10} + \sqrt{\frac23} R_{21} Y_{11} - \sqrt{\frac23} R_{21} Y_{1,-1}$$

Homework Equations

The Attempt at a Solution

The constant A is equal to i; is this result right or not?

 

[Simon Bridge]

Please provide a problem statement.
But I doubt the constant is going to be imaginary - what is your reasoning?

 

[gfxroad]

The states is more to write but I make a print screen.

http://www.gfxroad.com/print-wf

 

[Simon Bridge]

I can see why you didn;t want to write that down ;)
See the line below the equation where it says "where A is a real constant..."?
Your question:

The constant A is equal to i; is this result right or not?

... is answered.

You seem to be trying to answer part (b).
What is the condition that must be satisfied for $\psi(\vec r,0)$ to be normalized?

 

[gfxroad]

∫ψψ*dτ=1

 

[Simon Bridge]

Very good ... imagine you had $\psi = a\psi_a + b\psi_b$ ... where $\psi_a$ and $\psi_b$ are already normalized. In order for $\psi$ to be normalized, $a$ and $b$ need to satisfy a condition ... what is it?

 

[gfxroad]

a²+b²=1

 

[Simon Bridge]

Well done.
Technically: $a^*a+b^*b=1$ in case you have complex coefficients.

Now imagine you have:

$\qquad \psi = a\psi_a + \sqrt{\frac{2}{3}}\psi_b + \sqrt{\frac{2}{3}}\psi_c - \sqrt{\frac{2}{3}}\psi_d$

... now your problem is that to get $|\psi|^2=1$ it looks like you have $a^2+2=1 \implies a=\sqrt{-1}$

But you are told that $a$ is real so this is a contradiction.
Anyway, if $a=i$, then $a^*a= (-i)i = -i^2=1$ not the -1 you were looking for.

In fact, is there even a solution for $a^*a=-1$?

Therefore - what does this tell you about your approach?
Did you properly account for the R and Y functions?
i.e. is $\psi_{nlm}=R_{nl}Y_{lm}$ normalized already?
... did you do part (a) correctly?

 

[gfxroad]

Yes, I thought part a was done correctly like:
http://www.gfxroad.com/print2

 

[Simon Bridge]

Yeah, I'm getting the same thing ... I have a nagging feeling there's a wrinkle here I'm missing but on the face of it the textbook problem has no solution.

It may be that the text-book has a typo.

 

[gfxroad]

What's about e branch, is there any starting point or equation for this, because I don't know where can I starting. The other branches solved correctly.

 

[Simon Bridge]

For (e) $\text H\psi_{nlm}=E_n\psi_{nlm}$

What do you mean "correctly"? Do you have model answers?