In summary, the difference between the hydrogen atom Dirac equation and the spin symmetry Dirac equation lies in the presence or absence of a scalar potential. In the spin symmetry case, where the scalar and vector potentials are equal, the electron experiences the same force in all directions and the potential is spherically symmetric. This makes it easier to solve for the energy compared to the hydrogen atom case, where the scalar potential is zero and the potential is not spherically symmetric. The absence of spherical symmetry in the potential results in a more complex solution for the energy.
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Homework Statement



Exact spin symmetry in the Dirac equation occurs when there is both a scalar and a vector potential, and they are equal to each other. What physical effect is absent in this case, that does exist in the Dirac solution for the hydrogen atom (vector potential = Coulomb and scalar potential = 0)?

Homework Equations



Hydrogen atom Dirac equation:
##(\vec{\alpha} \cdot \vec{p}c+\beta mc^2)\psi = (E-V^v_0(r))\psi ##
##V^v_0(r) = -e^2/r##

Spin symmetry Dirac equation:
##(\vec{\alpha} \cdot \vec{p}c+\beta (mc^2+V_s(r)))\psi = (E-V^v_0(r))\psi ##
##V^v_0(r) = V_s(r)##

The Attempt at a Solution



Taking the non relativistic limit of both equations I found that it was hard to solve for the non relativistic energy in the hydrogen atom case, but the spin-symmetry case immediately led to a Schrodinger-type equation with ##E=E_{NR}##. However I don't know what this says about the hydrogen atom case, more specifically what physical effect is missing in the spin symmetry case that makes it easy to solve for the energy.
 
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In the spin symmetry case, the scalar potential and vector potential are equal to each other, meaning that the electron experiences the same force in all directions. This results in a spherically symmetric potential, which makes it easier to solve for the energy.

In the hydrogen atom case, the vector potential is the Coulomb potential, which is spherically symmetric. However, the scalar potential is zero, meaning that the electron experiences a different force in the radial direction compared to the transverse directions. This breaks the spherical symmetry of the potential and makes it more difficult to solve for the energy.

Therefore, the physical effect that is absent in the spin symmetry case is the breaking of spherical symmetry in the potential. This has a significant effect on the behavior of the electron and makes the solution more complex.
 

1. What is the Dirac hydrogen atom?

The Dirac hydrogen atom is a theoretical model used to describe the behavior of a single electron in the hydrogen atom. It incorporates both the principles of quantum mechanics and special relativity to provide a more accurate description of the atom's energy levels and electron behavior.

2. What is spin symmetry in the context of the Dirac hydrogen atom?

Spin symmetry refers to the property of the Dirac equation that allows for the separation of spin and orbital angular momentum. This means that the spin of the electron can be described separately from its motion around the nucleus, simplifying the calculation of energy levels in the atom.

3. How does the Dirac hydrogen atom differ from the classical model?

The classical model of the hydrogen atom, proposed by Niels Bohr, does not take into account the principles of quantum mechanics and special relativity. The Dirac hydrogen atom, on the other hand, incorporates these principles and provides a more accurate description of the atom's energy levels and electron behavior.

4. What are the benefits of using the Dirac hydrogen atom model?

The Dirac hydrogen atom model allows for a more accurate prediction of energy levels and electron behavior in the hydrogen atom. It also provides a better understanding of the underlying principles of quantum mechanics and special relativity. Additionally, it has been used to successfully describe the behavior of other atoms and molecules.

5. Are there any limitations to the Dirac hydrogen atom model?

While the Dirac hydrogen atom model is more accurate than the classical model, it is not a perfect representation of the atom. It does not take into account the effects of quantum electrodynamics, which are necessary for a complete understanding of the atom. Additionally, it does not account for the presence of multiple electrons in an atom or the interactions between them.

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