Solution to the Schrodinger Equation

[bradx888]

Homework Statement

Consider the full 1-electron hydrogen wave function.
[/B]
Prove that

ψ =A(6r –r2/a0) exp[-^r3/a0] sinθ exp[+iφ],

is a solution to the Schrödinger equation H|ψ> = E|ψ>, where H is the Hamiltonian operator.

Hence show that it's energy E= -1.51 eV and its principle quantum number n = 3.

Homework Equations

[/B]
So as far as I understand the Schrödinger equation is

-ħ2/2m[1/r2 δ/δr(r2 δ/δr) + 1/(r2sinθ) ∂/∂θ(sinθδ/δθ) + 1/(r2sinθ) δ2/δφ2]ψ - Vψ = Eψ

3. The Attempt at a Solution

I have subbed the solution into the equation, differentiating where needed and so on. I'm unsure as to what actually proves it is a solution, do I need to get one side equal to zero?
I also have no idea how to show that its E = -1.51 eV. I would be able to get from E to n though.

 

[bradx888]

I've had massive trouble trying to do powers and subscripts. Hopefully it's clear enough.

 

[BvU]

Hello Brad, welcome to PF :)

I have subbed the solution into the equation

Good. You are to find that your left hand side yields a constant times $\Psi$. The constant is E.

Did you find that ? If not, show what you did find.

 

[bradx888]

Hello Brad, welcome to PF :)Good. You are to find that your left hand side yields a constant times $\Psi$. The constant is E.

Did you find that ? If not, show what you did find.

Ok I'm in the process of trying to get it like that. I'm struggling to factorise out the 'r's so may be a little while :P

 

[bradx888]

Hello Brad, welcome to PF :)Good. You are to find that your left hand side yields a constant times $\Psi$. The constant is E.

Did you find that ? If not, show what you did find.

Hi,

After working on it all day, I haven't got what is needed.
A lot of the terms have canceled out. I may have done it wrong but have checked thoroughly.

On the left hand side I am left with,

Ae^-r/3ae^iφsinθ(-r²/9a³ + ((3a + 8)/3a²)r - (12ħ² + 6m)/aħ²

Please please help :(

 

[vela]

ψ =A(6r –r2/a0) exp[-^r3/a0] sinθ exp[+iφ],

-ħ2/2m[1/r2 δ/δr(r2 δ/δr) + 1/(r2sinθ) ∂/∂θ(sinθδ/δθ) + 1/(r2sinθ) δ2/δφ2]ψ - Vψ = Eψ

You should make the effort to learn LaTeX so you can write mathematics. There's a good tutorial here:
https://www.physicsforums.com/threads/physics-forums-faq-and-howto.617567/#post-3977517
LaTeX is pretty straightforward. If you quote this post, you can see the markup used to generate the equations below:
$$\psi = A \left(6r-\frac{r^2}{a_0}\right) e^{-r^3/a_0}\sin\theta e^{i\phi} \\
-\frac{\hbar^2}{2m}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) +
\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial}{\partial \theta}\right) +
\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right]\psi - V\psi = E\psi.$$ I don't know if it was just a typo, but in the last term of $\nabla^2$, $\sin\theta$ should be squared.

You need to show your work. Just posting what you end up with isn't very helpful since your mistakes, if any, occurred in earlier steps. Unless we can see them, we don't know where you went wrong.