Comparing Dirac and Schrodinger Equations for the Hydrogen Atom

[foobar]

The Schrodinger equation solved for the hydrogen atom gave good agreement with spectral lines, except for line doublets.

To account for these electron spin theory was grafted onto the theory, despite the problem of electron being a point particle.

In 1928 Dirac gives his different answer to this doublet problem with the Dirac equation giving the correct spin for the electron. The spin being connected to relativity.

My question is does the Dirac equation when solved for the Hydrogen atom give all the other correct predictions for spectral lines as the Schrodinger solution does ?

Or are there many other problems/difficulties/unknowns introduced by the Dirac solution ?

 

[M Quack]

The Pauli equation (Schroedinger + spin and a few other relativistic corrections) can be derived as the non-relativistic limit of the Dirac equation. As such it correctly gives the same results as the Schroedinger/Pauli equation for the spectral lines of the Hydrogen atom.

The Dirac equation gives 4-component spinors which are usually interpreted as spin-up particle, spin-down particle, spin-up antiparticle and spin-down antiparticle. It kind of allows the creation of particle/antiparticle pairs out of pure energy. In the hydrogen atom, these effect can be neglected., but in a more general context, it requires the introduction of the "second quantization" that systematically treats the creation and annihilation of particles. That is a big can of worms with infinities creeping up that require "renormalization" and other similar candies.

We all know by now that antiparticles exist, so having predicted that is a huge success for the Dirac equation. But mathematically, renormalization theory still seems to be a mess even if many hugely successful predictions have come out.

I am sure someone with better knowledge will have a more qualified opinion on this :-)

 

[Bill_K]

No need to make a nonrelativisitic approximation, the full Dirac equation for a central Coulomb potential can be solved exactly. (See one of the standard QM references, such as Messiah vol II.) The resulting energy levels include most of the known corrections such as hyperfine splitting (spin-orbit coupling) and the relativistically correct kinetic energy, which becomes important for inner electrons in high-Z atoms. What it does not include are QFT corrections such as the Lamb shift.

More generally, treating the Dirac equation as an equation for a single particle in an external potential is a valid approximation as long as V(x) << mc². At that point pair production becomes important.

 

[foobar]

OK thanks for that.

pretty impressive then.

 

[dextercioby]

The hyperfine splitting is not a consequence of the Dirac equation. The fine splitting is.

 

[Jano L.]

Some time ago on this forum we had a discussion about the stability of the atom according to the quantum theory, and one contributor made an interesting remark that due to the fact that in Dirac's eq. the energy is linear in momenta, according to this eq. models of many-electron atoms become unstable, in contrary to models based on Schr. eq, where energy is quadratic in momenta. Maybe in quantum field theory they become stable again, but I think the consequences of the equations are so hard to infer that we do not know for sure.

No need to make a nonrelativisitic approximation, the full Dirac equation for a central Coulomb potential can be solved exactly.

Bill, this makes an impression that the model is fully relativistic and can be solved exactly. But "exactly" above means only exactly within the approximate model. The approximation is that the electron moves in central electrostatic field of the proton. In relativistic theory, this is necessarily an approximation to field worse than 1/c^2 (accompanied by total neglect of readiation from the proton and retardation of its field). Better approximation has to be achieved in other way, maybe somehow in quantum field theory or otherwise.

More generally, treating the Dirac equation as an equation for a single particle in an external potential is a valid approximation as long as V(x) << mc2.

Can you please explain this? What is V(x)?

 

[nonequilibrium]

What it does not include are QFT corrections such as the Lamb shift.

So the dirac equation isn't a form of QFT? I've been told different things, but that being said, I haven't had a course in QFT yet (I'm now having a course in elementary particles using Griffiths' book, which does discuss the Dirac equation and such)

 

[dextercioby]

[...]The approximation is that the electron moves in central electrostatic field of the proton. [...]

Assumed of infinite mass, which is an approximation done in this case specifically, but is unjustified in the non-specially-relativistic Schroedinger equation. The electrostatic approximation for the e-m interaction electron/proton in the Schroedinger equation is justified, of course, because the particles' dynamics is non-specially-relativistic as well.

 

[dextercioby]

So the dirac equation isn't a form of QFT? I've been told different things, but that being said, I haven't had a course in QFT yet (I'm now having a course in elementary particles using Griffiths' book, which does discuss the Dirac equation and such)

If the Dirac field were a true classical field (like the e-m field or the gravitational field), then it would have a Lagrangian whose Euler-Lagrange equations would be the Dirac equations.

 

[nonequilibrium]

So it is a form of QFT? (I'm sorry if I'm completely missing your point, I'm not familiar enough with these topics)

 

[dextercioby]

By my version of QFT, it's not, it's a form of relativistic quantum mechanics, that is quantum mechanics (like the one of Schroedinger or Heisenberg) + special relativity instead of Galilean relativity.

 

[nonequilibrium]

Interesting, although I'm slightly confused, since Griffiths seems to mention a Lagrangian for the Dirac equation, while you seem to say it doesn't exist?

 

[dextercioby]

What Griffiths and many others are using is a <fabrication> of a CLASSICAL Lagrangian for a theory which is purely quantum mechanical. It has no physical meaning, for classical physics cannot account for the existence of the Dirac field.

Surely, one can build classical theories of electrons, but not with spinors.

 

[nonequilibrium]

What is a good source for enlightenment on these matters?

 

[dextercioby]

I would reccomend starting with a course on formal QFT, then you can ask yourself questions like I did. A solid background in mathematics is a must, some answers to your present & future questions are impossible to grasp without it.

 

[Bill_K]

Assumed of infinite mass,

dextercioby, No one will deny you the right to make the assumption that the proton has infinite mass, but it is pointless and counterproductive. The idea in replacing the Schrodinger equation with the Dirac equation is to make a better approximation, not a worse one. Failure to use the reduced mass means you won't even be able to reproduce the Bohr-Sommerfeld energy levels.

 

[dextercioby]

Replacing m_e by μ is indeed a useful trick and is reccomended by Messiah (the only one to my knowledge) to account for the error introduced by the assumption that the nucleus mass is infinite, however, due to the difficuly of separating the 2 particle motion into COM motion and relative "dummy" particle motion in the context of SR*, it remains as a trick, not as a result of specially relativistic dynamics.

*Here again I refer to Greiner's text on field equations as in the other recent thread on this subject (H atom and Dirac's eqn.)

 

[Dickfore]

Replacing m_e by μ is indeed a useful trick and is reccomended by Messiah (the only one to my knowledge)

Lol, this is a standard procedure in every textbook on Classical Mechanics for a two-body problem interacting with a central force.

 

[dextercioby]

Yes, because in Newtonian mechanics it is well justified. In special relativity, it's not.

 

[Dickfore]

Yes, because in Newtonian mechanics it is well justified. In special relativity, it's not.

Of course. But, then again, there is no concept of potential energy in SR.

 

[dextercioby]

True, but there's the notion of electromagnetic field potential to replace it.

 

[Dickfore]

True, but there's the notion of electromagnetic field potential to replace it.

If by electromagnetic field potential, you mean the 4-potential $(\phi, \vec{A})$, then yes, which is time dependent and depends on all previous locations and velocities of the proton and the electron! The dynamics of these particles, in turn, depends on the field. This problem, to my knowledge has not been solved exactly in non-quantum mechanics, nor in quantum.

 

[lugita15]

If by electromagnetic field potential, you mean the 4-potential $(\phi, \vec{A})$, then yes, which is time dependent and depends on all previous locations and velocities of the proton and the electron!

Don't Maxwell's equations relate electromagnetic field and potentials to charge and current density in the present (or rather the past a lightspeed-delay ago)? What does the entire history of the particles matter?

This problem, to my knowledge has not been solved exactly in non-quantum mechanics, nor in quantum.

Could you elaborate on this problem? What is it called?

 

[Dickfore]

Don't Maxwell's equations relate electromagnetic field and potentials to charge and current density in the present (or rather the past a lightspeed-delay ago)? What does the entire history of the particles matter?

No. ME relate the second derivatives (D'Alembertians) to the instantaneous charge-current density at that point. Due to retardation effects, the solution depends on all densities lying on the light cone passing through the space-time point we are considering. You need to find where the world line of each particle intersects this cone to calculate its contribution to the potential at that point.

Then, the acceleration of each particle depends on the derivatives of the potentials (the fields) at a particular space-time point. Finding a derivative of a function is a "nonlocal" operation in the following sense:

Expand the function $f(x) = \int{\frac{dk}{2\pi} \, e^{i k \, x} \tilde{f}(k)}$, where $\tilde{f}(k) = \int{dx' \, f(x') \, e^{-i k \, x'}}$ is the Fourier transform. The derivative is:
$$f'(x) = \int{\frac{dk}{2 \pi} \, i \, k \, e^{i k \, x} \, \int{dx' \, e^{-i k \, x'} \, f(x')}}$$
$$f'(x) = \int{dx' \, \left( -\int{\frac{d k}{2 \pi \, i} \, k \, e^{i k \, (x - x')}} \right) \, f(x')}$$
It is expressed as an integral operator that needs the values of the function for all possible values of x'.

From the previous point, we need the intersection of the past world line of the particle with all the world cones starting from every point in space. This basically amounts to knowing the whole past world line of the particle.

Could you elaborate on this problem? What is it called?

I don't know if it even has a special name. Relativistic two-body problem?! (when I type that, it refers to two point particles interacting only gravitationally http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity. When I wrote electrodynamics relativistic two body problem, I got the following article "10.1103/PhysRevD.4.2956" )

 

[lugita15]

No. ME relate the second derivatives (D'Alembertians) to the instantaneous charge-current density at that point. Due to retardation effects, the solution depends on all densities lying on the light cone passing through the space-time point we are considering. You need to find where the world line of each particle intersects this cone to calculate its contribution to the potential at that point.

OK, that's true, but you still don't need to know the entire history of the particles. You just need to know about at the time(s) at which their worldlines intersect the light cones of the point.

 

[Dickfore]

OK, that's true, but you still don't need to know the entire history of the particles. You just need to know about at the time(s) at which their worldlines intersect the light cones of the point.

You didn't read the remainder of my post.

 

[PhilDSP]

Or are there many other problems/difficulties/unknowns introduced by the Dirac solution ?

Because of the very high importance of the Dirac equation, I think you're certainly justified in asking to be presented with a higher level of rigor and views from various perspectives. Here are some of the less subtle difficulties presented by several authors:

1. "The Lorentz group, being 'non-compact', has no faithful, finite-dimensional representations that are unitary" Peskin and Schroeder p. 41. The Dirac equation is Lorentz invariant under boosts (expect for the mass term) and so lacks a unitary representation. Thaller (below) adds further detail on p. 386 that that applies to 4 dimensional complex number space $\mathbb C^4$ but not Hilbert space $\mathbb L^2(\mathbb R^3)^4$.

2. "In the presence of external fields, the Dirac equation cannot be invariant under Lorentz transformations" B. Thaller "Advanced Visual Quantum Mechanics" p. 377.

3. "The Klein paradox occurs for a high one-dimensional electrostatic potential step" Thaller p. 395.

4. The rotation symmetry of the Dirac equation is not complete or correct. Calculations for the magnetic field for directions not orthogonal to the plane of the trajectory fail to match experimental results. L. de Broglie "L 'electron magnetique" 1934 p. 138.

5. "The most important failure of the model seems to be that the magnitude of the resultant orbital angular momentum of an electron moving in an orbit in a central field of force is not a constant, as the model leads one to expect." P. A. M. Dirac "The Quantum Theory of the Electron" 1928 p. 610.

 

[akhmeteli]

Surely, one can build classical theories of electrons, but not with spinors.

Could you explain your statement or give a reference, as many people call the Dirac equation "classical" (see, e.g., http://catdir.loc.gov/catdir/samples/cam031/00268314.pdf , pp. 11-12)?

 

[dextercioby]

I'm sure I've adressed this issue before. The so-called Dirac Lagrangian which produces the Dirac eqns. for the spinor and its adjoint is an artefact, because the "objects" (mathematical items) Psi and Psibar are purely quantum theoretical without any classical counterparts, while for example the g_{ab} from the Hilbert-Einstein action is a purely classical object. This artefact is known to be useful if one is using the path integral approach to the quantum Dirac field, because this method requires the existence of a <classical> action/Lagrangian. So we simply 'manufacture' this Lagrangian and make the "objects" in it anticommutative, instead of commutative, anticommutativity being a feature without any connection to classical physics.

 

[PhilDSP]

Lagrangian and make the "objects" in it anticommutative, instead of commutative, anticommutativity being a feature without any connection to classical physics.

I'm not sure I quite agree with that. I recently found that the 3 x 3 matrix representation of the curl operation (harkening all the way back to J. C. Maxwell) anti-commutes with the vector or diagonal vector-matrix it's applied to.

 

[akhmeteli]

I'm sure I've adressed this issue before. The so-called Dirac Lagrangian which produces the Dirac eqns. for the spinor and its adjoint is an artefact, because the "objects" (mathematical items) Psi and Psibar are purely quantum theoretical without any classical counterparts, while for example the g_{ab} from the Hilbert-Einstein action is a purely classical object. This artefact is known to be useful if one is using the path integral approach to the quantum Dirac field, because this method requires the existence of a <classical> action/Lagrangian. So we simply 'manufacture' this Lagrangian and make the "objects" in it anticommutative, instead of commutative, anticommutativity being a feature without any connection to classical physics.

With all due respect, I did not see any solid arguments supporting your point of view in your previous posts, and I don't see such arguments in your latest post. You just say without any proof (or maybe a reference to such proof) that "Psi and Psibar are purely quantum theoretical without any classical counterparts", although, as I said, one can treat the Dirac equation as an equation describing a classical field. The (non-second-quantized) Dirac equation is not an ideal theory, but it is a damn good theory. Your opinion is just your opinion, not a fact.

 

[Jano L.]

...one can treat the Dirac equation as an equation describing a classical field.

But what do you mean by " classical field " ? Psi is a complex-valued four-component function. There is no such field in classical physics. Why call it classical?

Perhaps it would be better to agree on some less ear-earing expression, say "Psi is a c - number field quantity."

 

[akhmeteli]

But what do you mean by " classical field " ?

The same as the author of the book I quoted in post 28 in this thread (and I can assure you , this is a pretty standard parlance): "This chapter is devoted to a brief summary of classical field theory. The reader should not be surprised by such formulations as, for example, the ‘classical’ Dirac field which describes the spin ½ particle. Our ultimate goal is quantum field theory and our classical fields in this chapter are not necessarily only the fields which describe the classical forces observed in Nature."

Psi is a complex-valued four-component function. There is no such field in classical physics. Why call it classical?

So why the real four-component potential of electromagnetic field can be regarded as classical, but the complex-valued four-component function of the Dirac equation cannot?

Furthermore, in a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Moreover, this remaining component can be made real by a gauge transform. (Journal of Mathematical Physics, 52, 082303 (2011) (http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 or http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )). So you may replace Psi by just one real function. As for "no such field in classical physics"... My feeling is this argument has only historical significance nowadays.

Perhaps it would be better to agree on some less ear-earing expression, say "Psi is a c - number field quantity."

Your expression is certainly OK, but that does not mean any other expression is wrong. However, the phrase that I questioned - "one can build classical theories of electrons, but not with spinors", does not seem correct to me.

 

[Jano L.]

So why the real four-component potential of electromagnetic field can be regarded as classical, but the complex-valued four-component function of the Dirac equation cannot?

The em potential is a part of classical theory, and can be defined in terms of classical physics, say physical quantities rho and j, or E and B. Then it makes no harm to call it classical.

However, as far as I know, Dirac's Psi does not have a definition in terms of classical physics. Can you give one? If so, we could say Psi is a classical field. If not, the use of the word classical in my opinion is seriously hampering correct understanding of both classical and quantum theory.

 

[PhilDSP]

I recently found that the 3 x 3 matrix representation of the curl operation (harkening all the way back to J. C. Maxwell) anti-commutes with the vector or diagonal vector-matrix it's applied to.

I have to correct myself on this. The relationship is actually $\mathsf A \mathsf B + \mathsf B^t \mathsf A = 0$