Yes, thank you for explaining it to me. I have a much better understanding now.

[papanatas]

Hi, I am a physics student and i was asked to answer some questions about Hydrogen atom wavefunctions. I hope you can help me (sorry for my english, is not my motherlanguage, i will try to explain myself properly)

1. In order to find hamiltonian eigenfunctions of Hydrogen atom, we make then be simultaneously eigenfunctions of the operators H (hamiltonian) L² and L_z. What does this condition mean? Why do we do this?

2. Schrödinger ecuation for an electron moving under the action of an electrial (central, spherical) potential, without magnetic interaction or relativistic corrections, can be solved exactly, giving eigenfunctions:
Ψ_n,l,m(r,θ,φ)=R(r)⋅P(θ)⋅G(φ)

What do quantum numbers n,l and m mean?
Why are variables separable? (I am mostly interested in this one)
What do R, P and G mean?

I don't know if i made myself clear, I am looking forward your answers. Thanks.

 

[PeroK]

Hi, I am a physics student and i was asked to answer some questions about Hydrogen atom wavefunctions. I hope you can help me (sorry for my english, is not my motherlanguage, i will try to explain myself properly)

1. In order to find hamiltonian eigenfunctions of Hydrogen atom, we make then be simultaneously eigenfunctions of the operators H (hamiltonian) L² and L_z. What does this condition mean? Why do we do this?

2. Schrödinger ecuation for an electron moving under the action of an electrial (central, spherical) potential, without magnetic interaction or relativistic corrections, can be solved exactly, giving eigenfunctions:
Ψ_n,l,m(r,θ,φ)=R(r)⋅P(θ)⋅G(φ)

What do quantum numbers n,l and m mean?
Why are variables separable? (I am mostly interested in this one)
What do R, P and G mean?

I don't know if i made myself clear, I am looking forward your answers. Thanks.

When you try to solve the Schroedinger equation, you tackle it using the separation of variables technique. Note that for an arbitrary PDE (partial differential equation) there is no guarantee that there will be separable solutions. Moreover, separation of variables is just about the only analytic technique at your disposal for solving PDE's. The hydrogen atom, therefore, is a special case, where things are simple enough to yield an analytic, separable solution.

This gives you a set of solutions $\lbrace \psi_{nlm} \rbrace$ with some good mathematical and physical properties:

Mathematically, they are mutually orthogonal and complete, in the sense that any function can be expressed as a linear combination of these functions. This allows you to express every general solution to the Schroedinger equation as a linear combination of these functions. Note that it is important not to forget the separable time component:

$$\Psi = \sum_{n, l, m} \psi_{nlm} exp(-iE_nt/ \hbar)$$

So, mathematically these functions are important as they represent the building blocks of any solution to the Schroedinger equation.

They are also physically important as they represent eigenstates of Energy $H$, Orbital Angular Momentum $L^2$ and z-component of angular momentum $L_z$, represented by the quantum numbers $n, l, m$.

If you measure the energy of the Hydrogen atom, you must get one of the values $E_n$; if you measure the total orbital angular momentum squared, you must get $\hbar^2 l(l+1)$, for some $l < n$; and, if you measure the z-component of angular momentum, you must get $\hbar m$, for some $m$ with $|m| \le l$.

In other words, the mathematical building blocks you found fit in precisely with the possible measurements of these observables.

Does that make sense?