Exploring the Hydrogen Atom - Molu's Questions

[loom91]

Hi,

I had a few questions about the most well behaved of atoms.

1) Solving the Schroedinger equation gives us a basis (say the energy eigenstates) such that all possible wavefunctions can be written as a linear combination of these stationery solutions. This means that the system is not confined to these stationery solutions but to their combinations, which are not necessarily eigenstates of the (time-independent) Hamiltonian.

Then why are the lines observed in the spectra are only those corresponding to transitions between the eigenstates labeled with (n,l,m,s)? Does the electron in a H atom only occupy these states as Bohr would have believed?

2)In a H atom, the energy depends only on n, not on l. Then why do we observe distinct lines for different l values of initial and final states?

3)The usual solution for the H atom is obtained by treating the nucleus as a classical point charge and utilising the classical Coulomb potential. Is it possible to obtain a completely quantum solution of the H atom? If so, what equations would be used?

Thanks for your help.

Molu

 

[OlderDan]

Hi,

I had a few questions about the most well behaved of atoms.

1) Solving the Schroedinger equation gives us a basis (say the energy eigenstates) such that all possible wavefunctions can be written as a linear combination of these stationery solutions. This means that the system is not confined to these stationery solutions but to their combinations, which are not necessarily eigenstates of the (time-independent) Hamiltonian.

Then why are the lines observed in the spectra are only those corresponding to transitions between the eigenstates labeled with (n,l,m,s)? Does the electron in a H atom only occupy these states as Bohr would have believed?

This is all tied up in the philosophical interpretation of QM. If you had an electron in a superposition of states as you describe, and you measured the energy of the electron, what might you find? Any one of many possible eigenenergies with probability related to the linear coefficients in the superposition. Absorption or emission of a photon is effecively a mesaurement of the energy of the electron befoe and after the transition. The electron transitions from one eigenenergy to another in this process.
2)In a H atom, the energy depends only on n, not on l. Then why do we observe distinct lines for different l values of initial and final states?

Take a look here http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydfin.html

3)The usual solution for the H atom is obtained by treating the nucleus as a classical point charge and utilising the classical Coulomb potential. Is it possible to obtain a completely quantum solution of the H atom? If so, what equations would be used?

For points outside the nucleus, it does not have to be a point particle to give the Coulomb potentail. It just has to be a spherically symmetric charge distribution. One could assume any other spherically symmetric charge distribution for the nucleus and at least in principle solve the equation. The radial wave functions would have to be modified to be consistent with the radial variation in the charge density. Beyond that, one would have to start delving into the charge distribution (potentially time dependent) of the proton for the simple hydrogen atom, or the more complex structure of the nucleus for heavier hydrogen-like atoms. In fact, the nucleus is so dense compared to the size of an atom that one expects at most only minor perturbations of the electric potential at distances from the nucleus where the electron is most likely to be found. If this were not true, the simpler calculation using the classical point nucleus would not have been so successful in explaining the behavior of the hydrogen atom.

http://dnp.nscl.msu.edu/current/proton.html

Thanks for your help.

Molu

Some comments included in the quote area

 

[loom91]

2)That link is about spin splitting. I'm talking about different lines for different value of l. Like 2s and 2p.

3)But the coulomb potential is from CED. Wouldn't it be changed in QED?

Thanks for the help.

Molu

 

[OlderDan]

2)That link is about spin splitting. I'm talking about different lines for different value of l. Like 2s and 2p.

3)But the coulomb potential is from CED. Wouldn't it be changed in QED?

Thanks for the help.

Molu

Is this what you are looking for?

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/lamb.html#c3

http://en.wikipedia.org/wiki/Lamb_shift

I know very little about QED, but it is related to this effect.

 

[dextercioby]

3)The usual solution for the H atom is obtained by treating the nucleus as a classical point charge and utilising the classical Coulomb potential. Is it possible to obtain a completely quantum solution of the H atom? If so, what equations would be used?

The Dirac equation has a full exact solution is one treats the electrostatic field b/w the pointlike electron & proton classically. Corrections to account for the finite (i.e. nonzero) size of the nucleus (in H-like ions) have been made. Also, there's a perturbative resolution of this problem in QED, where, of course, the em interaction is treated QM-ally. See the textbooks of Lifschitz, Pitayevski and Berestetskii, Jauch and Rohrlich or Akhiezer and Berstetskii.

Daniel.

Da

 

[loom91]

Is this what you are looking for?

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/lamb.html#c3

http://en.wikipedia.org/wiki/Lamb_shift

I know very little about QED, but it is related to this effect.

That is more-or-less what I was looking for, thank you.

I know that the H atom admits closed-form solutions in both the non-relativistic (Schroedinger) model and the semi-relativistic (Dirac) model. In QED is it again exactly solvable or do we use perturbation theory or some other approximating technique? Thanks.

Molu

 

[dextercioby]

Perturbation theory and see Akhiezer's book for the formalism for tackling bound states in QED.

Daniel.

 

[loom91]

So the H atom is not exactly solvable in QED?

 

[dextercioby]

No. See for instance the first volume of Weinberg, if you don't have Akhiezer.

Daniel.

 

[loom91]

I doubt I could make head or tail of a QFT textbook (even if I managed to find one somehow) without first properly learning non-relativistic quantum mechanics.

 

[dextercioby]

That's true. Mumbles to himself: <<I doubt he could make head or tail of a non-relativistic quantum mechanics textbook (even if he managed to find one somehow) without first properly learning mathematics>>.

Daniel.

 

[loom91]

That's true. Mumbles to himself: <<I doubt he could make head or tail of a non-relativistic quantum mechanics textbook (even if he managed to find one somehow) without first properly learning mathematics>>.

Daniel.

I protest. I can make both head and tail of a non-relativistic quantum mechanics textbook (Griffiths, to be specific), though I may not be able to savour the unifying insights provided by a deeper knowledge of the underlying mathematics like Fourier analysis.

Molu