The Dirac equation and the spectrum of the hydrogen atom

[Sophrosyne]

I was reading that one of the successes of the Dirac equation was that it was able to account for the fine structure of some of the differences in the spectrum of the hydrogen atom.

But the Dirac equation is about subatomic particles moving at relativistic velocities. But an electron around the hydrogen atom is in a superposition state. It's not supposed to have a velocity, does it? And if it does, is it really at relativistic velocities? What kind of numbers are we talking about here?

 

[PeterDonis]

I was reading

Where? Can you give a reference?

 

[PeterDonis]

the Dirac equation is about subatomic particles moving at relativistic velocities

It describes particles moving at relativistic velocities, but it is certainly not limited to doing so. The reason it worked better in predicting the fine structure of hydrogen was that it properly incorporated the effects of electron spin, not that electrons in atoms are moving at relativistic velocities.

 

[Sophrosyne]

Where? Can you give a reference?

I am embarrassed to say, but it was some YouTube videos. I can track them down if you are really interested.

But do we know if velocities of the electron can be measured around the hydrogen atom? Of course, this would sacrifice position measurements based on Heisenberg uncertainty, but what kind of numbers do we get when attempts ARE made to measure it?

 

[Sophrosyne]

It describes particles moving at relativistic velocities, but it is certainly not limited to doing so. The reason it worked better in predicting the fine structure of hydrogen was that it properly incorporated the effects of electron spin, not that electrons in atoms are moving at relativistic velocities.

It incorporates spin of particles moving at relativistic speeds. Otherwise, those effects can be calculated using classical treatments.

What value would have to be put into the velocity value in the Lorentz transform of the equation if you are using the Dirac equation?

 

[PeterDonis]

It incorporates spin of particles moving at relativistic speeds.

No, it incorporates spin of spin-1/2 particles (fermions), period. There is no restriction that they must be moving at relativistic speeds. See below.

Otherwise, those effects can be calculated using classical treatments.

Yes, if you don't mind getting wrong answers. Physicists in the 1920s did mind that, which is one reason why Dirac's equation was such a breakthrough.

One way of interpreting this is that Dirac discovered that spin, in and of itself, requires a relativistic treatment to be modeled correctly--i.e., that spin, even in particles moving much more slowly than light, is a manifestation of relativistic effects. Nowadays we understand this as an aspect of group theory: the group SU(2), which describes spin, is a subgroup of SO(3, 1), which describes Lorentz invariance in general.

 

[PeterDonis]

What value would have to be put into the velocity value in the Lorentz transform of the equation if you are using the Dirac equation?

Um, whatever the relative velocity of the frames is? The Lorentz transform is the same for the Dirac equation as for any other relativistically covariant equation.

 

[PeterDonis]

do we know if velocities of the electron can be measured around the hydrogen atom?

AFAIK nobody has attempted to do this, probably because it would be very difficult technically and would not provide any real value to us, since we already have models that accurately predict the quantities we care about, such as the energy levels of electrons in atoms.

 

[PeterDonis]

I am embarrassed to say, but it was some YouTube videos.

Your embarrassment indicates that you are aware that this is not a good source of information. Certainly not if you want to have an "I" level discussion. I have adjusted the level of this thread to "B".

For better information, you should consult textbooks or peer-reviewed papers.

 

[Sophrosyne]

Your embarrassment indicates that you are aware that this is not a good source of information. Certainly not if you want to have an "I" level discussion. I have adjusted the level of this thread to "B".

For better information, you should consult textbooks or peer-reviewed papers.

I see. This was helpful. Thanks!

 

[DrDu]

V is an observable like any other. You can obtain it as $v= i/\hbar [H,x]=c\alpha$.
I think you can measure the velocity distribution e.g. from the Doppler shift of photons scattered off the electrons. Also note that the Dirac equation not only describes hydrogen but also hydrogen like ions.
While the relativistic effects in hydrogen are small (but have nevertheless been measured with high accuracy) they are immense in hydrogen like ions like U$^{91+}$.