Questions about Schrödinger's Equation: Help Needed!

[Lindsayyyy]

Hi everyone

I have to questions which I don't have an answer to:

1. the solution for the Schrödinger's equation are continuous (for time as well as for the location). But why do I get discrete values for the energies for example (let's say in hydrogen) ?

2. Is there a spherical harmonic where the probability is unequal to zero when |r|=0 ? (probability to be in the core of the atom)

to
1. I'm not sure but my guess would be it has to do with Bohrs postulates. The electron is a standing wave in order not to crash into the core, so that's why we have discrete values.

2. I'd say no, but I can't argue against it. I mean maybe there's a difference between the maths and the intuition.

Thanks for your help.

 

[DeShark]

1. The hydrogen atom (in the absense of an external potential) has a potential which doesn't depend on time. It's just the coulomb attraction between the electron and the proton (and you can make things easy by transforming your viewpoint to the centre of mass reference frame).

As it is time-independent, the schroedinger equation can be separated (à la separation of variables - mathematic technique that you should know) into time dependent parts and time independent parts.

The time-independent SE just reads Ĥψ(x,y,z) = Eψ(x,y,z) i.e. it asks you to find the eigenstates and eigenvalues of the hamiltonian operator.

This is "just" mathematics and the solutions can be found in pretty much any book on QM. For negative values of E (i.e. bound states), the eigenvalues are discrete (i.e. non-continuous). It is essentially your typical "particle in a box" type mathematics.

2. The "first" (if I may call it so) solution, with n=1, l=0, m=0, has a non-zero value at r=0. It exponentially decays away from the centre.

 

[waveandmatter]

To question 1: the solutons of the schrödinger equation corresponding to the energy levels of an atom are supposed to be solutions that are constant (except a phase factor) over time (== eigenfunctions of the hamiltonian). That's intuitive i guess, because they are supposed to be "stable orbits".
For electrons bound in a potential, the eigenfunctions are always some kind of standing wave and those have discrete eigenvalues. In this case it is basically the standing wave around the atom. For large eigenvalues however, the spectrum should turn continous, because then we have free electrons (ie plane waves, which can take on any frequency and thus energy).
I think the Bohr postulates actually can be explained by this, not the other way round.

(Continous equations often have discrete eigenvalues, take for example waveguide or cavity modes in optics.)

 

[Lindsayyyy]

thanks for your help thus far.
I've seen the solution for the Schrödinger equation for the hydrogen atom. And I see that I get discrete solutions, but it's still not physically clear to me why I have a continious wave function but discrete eigenvalues. How can I describe that more on the physical side rather than on the math side, if you know what I mean.

 

[waveandmatter]

And I see that I get discrete solutions, but it's still not physically clear to me why I have a continious wave function but discrete eigenvalues.

The discrete eigenvalue spectrum basically counts how many (eigen-)solutions there are (finite, discretely infinite, continously infinite).
It does not say anything about wether those solutions themselves are continuous or discrete!
In qm i actually can only think of examples where the wavefunctions themselves are continous...

Think of a wave in a box: The standing wave patterns are all continuous functions of space and time. But there are only discrete wavelengths and thus energy levels allowed because an integer wavelength has to fit into the box to get a wavefunction that is quasi-static.
For the atom it is the same reasons in a more complicated geometry.