Relativistic Mechanics of Hydrogen Like S Orbials at high Z

[ChmDudeCB]

In the middle of studying for my matrix mechanics/atomic orbial/diatomic bonding final and something has been itching at my buttcrack for a while.

Above Z=20 or so the lower orbitals (that don't actually exist so I've come to find out) have to be calculated with relativistic QM because the classical picture of their orbital speed becomes so great that the electron would have to be traveling faster than c (again, electrons may as well not even exist as far as I'm actually concerned, but cut me slack since its only been 6mo or so since I first started studying QM).

My question is, what sort of time dilation are we talking about for those inner electrons? Does it have any quantifiable effect?

Is there gravitational time dilation at those scales?

Given the Pauli exclusion principle, do we actually have to worry about relativistic effects on electrons trapped that deeply in the atom and if so what are the relativistic effects on real world calculations?

I should be able to understand any QM speak up to Dirac Notation which I have only grazed over (not required until graduate level courses). I realize this is a question best fitted for my professor but between now and the end of the semester I need to spend my time around him picking his brain for my final.

Thanks.

 

[alxm]

Above Z=20 or so the lower orbitals [..] have to be calculated with relativistic QM because the classical picture of their orbital speed becomes so great that the electron would have to be traveling faster than c

I haven't heard that; I'm not sure it's true. They do certainly pick up significant relativistic momentum though. Nor do I know what the time dilation is offhand. (although you could get an idea if you just assumed the excitation energy = the electron's kinetic energy). I don't know how you could possibly quantify the time dilation. Electrons don't decay or anything.

Gravitational effects aren't significant. They're taken into consideration even in the most exact calculations.

Given the Pauli exclusion principle, do we actually have to worry about relativistic effects on electrons trapped that deeply in the atom and if so what are the relativistic effects on real world calculations?

Well, one example everyone can see for themselves is https://www.physicsforums.com/showpost.php?p=2334217&postcount=8", which is due to a relativistic shift in levels. All the other electrons are affected in turn by the core electrons, so in heavy elements, even chemical bonding (which is strictly a valence-electron phenomenon) is affected to a significant degree.

There's a whole bunch of SR effects, but most are not very significant for light elements. The main two are: The relativistic momentum affecting the core orbitals and spin-orbit coupling effects. On top of that, there are some others; (see e.g. the Breit-Pauli Hamiltonian)

For chemical calculations you need to start taking relativistic effects into account for third-period elements, at which point a simple pseudopotential (ECP) is often enough, and increasingly so the farther down you get. (Forget about the Schrödinger equation for Actinides!)

 

[Vanadium 50]

In the middle of studying for my matrix mechanics/atomic orbial/diatomic bonding final and something has been itching at my buttcrack for a while.

I think that's better treated with diet, but let's not go there at the moment.

Above Z=20 or so the lower orbitals (that don't actually exist so I've come to find out) have to be calculated with relativistic QM because the classical picture of their orbital speed becomes so great that the electron would have to be traveling faster than c

Not so. The expectation velocity for the lowest orbital is (Z/137)c.

 

[kanato]

The main effect of relativistic quantum mechanics for electrons in atoms is spin-orbit coupling. This mixes the states so that instead of e.g. three 2p orbitals with two spins each, you have two 2p(1/2) and four 2p(3/2) orbitals. The spin-orbit interaction is the largest for these orbitals, so the 2p1/2 and 2p3/2 orbitals will be well split, and then there may be very minor hyperfine or crystal field (in a solid) effects that split the degeneracies further. For most real world calculations relativistic effects of the deep core electrons could be ignored, but they may however be important for valence or semi-core electrons.

I'm not sure what you mean by "the lower orbitals don't actually exist." There are several x-ray experiments that probe excitations from deep core orbitals to valence orbitals (ie. x-ray absorption edge may probe the 2p3/2 to 5d excitation in Yb metal).

 

[ChmDudeCB]

When I say the lower orbitals don't exist, I mean they are beginning to stray from being hydrogenic to a significant degree.