Relativistic Quantum Mechanics

[ziad1985]

I got an idea today , what different thing will we get if we applied the Klein-Gordon Equations to The Simple harmonic oscillator?
then take it to the hydrogen atome and see how much corrections will we get??
wouldn't we get more precise results?or it just can't be done?did someone already did this work or what?

 

[inha]

How on Earth would doing a KG harmonic oscillator and then "taking it" (I have no idea what you mean by this) to hydrogen atom give anything more accurate than what's already gotten via solving the Dirac eq?

 

[ziad1985]

I don't remember how exactly to do it , but you can derive hydrogen energy and other stuff from The simple harmonic oscillator.
anyway forget that for a second,I'm still learning Non-relativistic Quantum mechanics , but it was an idea crossed my mind.
so dirac eq involves solving the relativistic wave eq of the hydrogen atom?

 

[Perturbation]

You mean using a relativistic field equation to predict the bound states etc. of hydrogen, instead of the Schrödinger equation? You'd have to use QED, the Klein-Gordon and free-Dirac fields don't account for electromagnetic interactions, well the latter can if you apply minimal coupling but without quantum field theory one can only account for first order interactions. If you do apply QED to hydrogen you get things like the lamb-shift: very slight shifts in the emission spectra of hydrogen from the non-relativistic prediction (i.e. Schrödinger).

 

[ziad1985]

hmmm , but what kind of corrections will i get solving the Relativistic eq ?

 

[ziad1985]

i mean in comparision to non-relativistic , and what is the order of precision?

 

[Physics Monkey]

ziad,

Historically, Schrödinger actually wrote the Klein-Gordon equation down first as a putative wave equation for the electron. As the story goes, he solved the Klein-Gordon equation for hydrogen and found that the results did not agree with the observed spectra. His second attempt was the Schrödinger equation we all know. Furthermore, as has been mentioned above, it is the Dirac equation that describes the electron, and in the context of relativistic wave mechanics, it is this equation that should be used to compute corrections. There are well known difficulties with this approach that must eventually be dealt with by using the powerful methods of quantum field theory. In the end, the full radiative corrections of QED are evident as in the Lamb shift, the anomalous magnetic moment, and so on.

 

[ZapperZ]

I don't remember how exactly to do it , but you can derive hydrogen energy and other stuff from The simple harmonic oscillator.
anyway forget that for a second,

I can't, because this would be VERY strange.

Look at the solution to the SHO - you get Hermite polynomials. Now look at the solution to the H atom. Do you see any Hermite polynomials anywhere? What about the angular momentum part in SHO? Any sign of it?

And what about the fact that when you do SHO problems, you put the center of the potential at the origin, and you have a symmetric potential around that point. You have both - and + displacement of the oscillator. Do you have such a solution in H atom? is "-r" (not a vector) for the radial part a physical solution in spherical coord?

Zz.

 

[ziad1985]

Physics Monkey: i see thank you a lot.
but this bring me to another question what are the real uses of the klein-gordon equations , where do they benefit the most,other then using them for relativistic moving particles?

ZapperZ:I honestly don't remember where i saw that , I think maybe it was in a book i was reading, it was an approch to get the energy of the atome , but maybe I'm mixing some other thing up or the writer was a whacko , which the latter seem more convincing because i still don't understand how did he do it.

 

[akhmeteli]

I can't, because this would be VERY strange.

Look at the solution to the SHO - you get Hermite polynomials. Now look at the solution to the H atom. Do you see any Hermite polynomials anywhere? What about the angular momentum part in SHO? Any sign of it?

And what about the fact that when you do SHO problems, you put the center of the potential at the origin, and you have a symmetric potential around that point. You have both - and + displacement of the oscillator. Do you have such a solution in H atom? is "-r" (not a vector) for the radial part a physical solution in spherical coord?

Zz.

Actually, as far as I remember, there is an equivalency between the Schroedinger 3d Coulomb and 4d oscillator problems, it arises from the dynamical symmetry of 3d Coulomb problem discovered by Fock. To demonstrate this equivalency one writes 3d Coulomb in momentum space and introduces stereographic coordinates, or something like that. Sorry, I am very vague here, and I cannot give a reference to online resources, but this reference seems to be relevant, although I haven't read the article itself: http://prola.aps.org/abstract/PRA/v22/i2/p333_1
I agree, however, that it would be very strange to see this in an entry level course

 

[dextercioby]

If you solve the minimally coupled KG equation for the H atom, you get a formula containing a sq.root (just like in the Dirac case, see Greiner's book on "Relativistic Wave equations") and if you do power series expansion of that sqrt, you'll get some terms which can be found in the Dirac case (like the original Balmer formula, the relativistic correction and the B coupled to L part), but some will not occur: namely B coupled to S and L coupled to S. WHY ? Well, in CFT KG is the equation of motion for a spin 0 particle, while in the Dirac case: C,P,T and all possible combinations invariance + spin 1/2 determine the form of the equation.

As for H atom vs 3D harmonic oscillator, well the wave function contains the same angular part (L_z L^2 and H are a c.s.c.o. for both systems) and the radial one is a confluent hypergeometric function which degenerates to two different types of orthogonal polynomials: Laguerre & Hermite.

Daniel.