Normalizaed wave function of the hydrogen atom

[asdf1]

how do you find the normalized wave functions of the hydrogen atom for n=1, l=0 and ml=0?
in my textbook, it's a table, but i have no idea where the figures come from...

 

[jtbell]

how do you find the normalized wave functions of the hydrogen atom for n=1, l=0 and ml=0?
in my textbook, it's a table, but i have no idea where the figures come from...

You do it by finding the solution for the Schrödinger equation for the hydrogen atom. This means finding the solutions of the individual differential equations for the functions $R(r)$, $\Theta(\theta)$ and $\Phi(\phi)$.

Many introductory textbooks work through the solution for $\Phi(\phi)$ because its equation is rather easy. But I didn't see the complete solution for $\Theta(\theta)$ and $\Phi(\phi)$ until my graduate school QM courses. They're that messy!

Oh wait, I was thinking of the general solution for any n, l, m. You want specifically n=1, l=0, m=0. That case might not be too bad, after you substiute those values of n, l, m into the individual differential equations for the three variables. $\Theta(\theta)$ and $\Phi(\phi)$ both turn out to be constant in this case, so their differential equations must be very simple!

 

[asdf1]

$\Theta(\theta)$
but the original function isn't known, right? so how do you solve it?

 

[jtbell]

That's what the Schrödinger equation is for, or rather the individual ordinary differential equations that you get after you separate the S.E.

Solving an algebraic equation like $x^2 - 5x + 6 = 0$ gives you a number for $x$, or a set of numbers. Solving a differential equation like $d^2 \Phi(\phi) / d \phi^2 = -m_l^2 \Phi(\phi)$ gives you a function for $\Phi(\phi)$, or a set of functions.

 

[asdf1]

thank you very much! :)