ATOMIC MODEL OF DIRAC JORDAN: CHARACTERISTICS AND POSTULATES - PHYSICAL - 2024

The Dirac-Jordan atomic model is the relativistic generalization of the Hamiltonian operator in the equation that describes the quantum wave function of the electron. Unlike the previous model, that of Schrodinger, it is not necessary to impose the spin by means of the Pauli exclusion principle, since it appears naturally.

In addition, the Dirac-Jordan model incorporates relativistic corrections, the spin-orbit interaction and the Darwin term, which account for the fine structure of the electronic levels of the atom.

Figure 1. Electronic orbitals in the hydrogen atom for the first three energy levels. Source: Wikimedia Commons.

Starting in 1928, the scientists Paul AM Dirac (1902-1984) and Pascual Jordan (1902-1980), set out to generalize the quantum mechanics developed by Schrodinger, to include Einstein's special relativity corrections.

Dirac starts from the Schrodinger equation, which consists of a differential operator, called the Hamiltonian, which operates on a function known as the electron wave function. However, Schrodinger did not take into account relativistic effects.

The solutions of the wave function allow us to calculate the regions where with a certain degree of probability the electron will be found around the nucleus. These regions or zones are called orbitals and depend on certain discrete quantum numbers, which define the energy and angular momentum of the electron.

Postulates

In quantum mechanical theories, whether relativistic or not, there is no concept of orbits, since neither the position nor the speed of the electron can be specified simultaneously. Furthermore, specifying one of the variables leads to total imprecision in the other.

For its part, the Hamiltonian is a mathematical operator that acts on the quantum wave function and is built from the energy of the electron. For example, a free electron has total energy E that depends on its linear momentum p like this:

E = (p ²) / 2m

To construct the Hamiltonian, we start from this expression and substitute p for the quantum operator for momentum:

p = -i ħ ∂ / ∂ r

It is important to note that the p and p terms are different, since the first one is the momentum and the other is the differential operator associated with the momentum.

Additionally, i is the imaginary unit and ħ the Planck constant divided by 2π, in this way the Hamiltonian operator H of the free electron is obtained:

H = (ħ ² / 2m) ∂ ² / ∂ r ²

To find the hamiltonian of the electron in the atom, add the interaction of the electron with the nucleus:

H = (ħ2 / 2m) ∂ ² / ∂ r ² - eΦ (r)

In the previous expression -e is the electric charge of the electron and Φ (r) the electrostatic potential produced by the central nucleus.

Now, the operator H acts on the wave function ψ according to the Schrodinger equation, which is written like this:

H ψ = (i ħ ∂ / ∂t) ψ

Dirac's four postulates

First postulate: the relativistic wave equation has the same structure as the Schrodinger wave equation, what changes is the H:

H ψ = (i ħ ∂ / ∂t) ψ

Second postulate: the Hamiltonian operator is constructed starting from Einstein's energy-momentum relationship, which is written as follows:

E = (m ² c ⁴ + p ² c ²) ^1/2

In the previous relation, if the particle has momentum p = 0 then we have the famous equation E = mc ² that relates the energy at rest of any particle of mass m with the speed of light c.

Third postulate: to obtain the Hamiltonian operator, the same quantization rule used in the Schrodinger equation is used:

p = -i ħ ∂ / ∂ r

At the beginning, it was not clear how to handle this differential operator acting within a square root, so Dirac set out to obtain a linear Hamiltonian operator on the momentum operator and from there arose his fourth postulate.

Fourth postulate: to get rid of the square root in the relativistic energy formula, Dirac proposed the following structure for E ²:

Of course, it is necessary to determine the alpha coefficients (α0, α1, α2, α3) for this to be true.

Dirac's equation

In its compact form, the Dirac equation is considered one of the most beautiful mathematical equations in the world:

Figure 2. Dirac equation in compact form. Source: F. Zapata.

And that's when it becomes clear that the constant alphas cannot be scalar quantities. The only way that the equality of the fourth postulate is fulfilled is that they are constant 4 × 4 matrices, which are known as Dirac matrices:

We immediately observe that the wave function ceases to be a scalar function and becomes a vector with four components called a spinor:

The Dirac-Jordan atom

To obtain the atomic model it is necessary to go from the equation of the free electron to that of the electron in the electromagnetic field produced by the atomic nucleus. This interaction is taken into account by incorporating the scalar potential Φ and the vector potential A in the Hamiltonian:

The wave function (spinor) that results from incorporating this Hamiltonian has the following characteristics:

- Fulfills special relativity, since it takes into account the intrinsic energy of the electron (first term of the relativistic Hamiltonian)

- It has four solutions corresponding to the four components of spinor

- The first two solutions correspond one to spin + ½ and the other to spin - ½

- Finally, the other two solutions predict the existence of antimatter, since they correspond to that of positrons with opposite spins.

The great advantage of the Dirac equation is that the corrections to the basic Schrodinger Hamiltonian H (o) can be broken down into several terms that we will show below:

In the previous expression V is the scalar potential, since the vector potential A is zero if the central proton is assumed to be stationary and therefore does not appear.

The reason that the Dirac corrections to the Schrodinger solutions in the wave function are subtle. They arise from the fact that the last three terms of the corrected Hamiltonian are all divided by the speed c of light squared, a huge number, which makes these terms numerically small.

Relativistic corrections to the energy spectrum

Using the Dirac-Jordan equation we find corrections to the energy spectrum of the electron in the hydrogen atom. Corrections for energy in atoms with more than one electron in approximate form are also found through a methodology known as perturbation theory.

Similarly, the Dirac model allows us to find the fine structure correction in the hydrogen energy levels.

However, even more subtle corrections such as the hyperfine structure and the Lamb shift are obtained from more advanced models such as quantum field theory, which was born precisely from the contributions of the Dirac model.

The following figure shows what Dirac's relativistic corrections to energy levels look like:

Figure 3. Corrections of the Dirac model to the levels of the hydrogen atom. Source: Wikimedia Commons.

For example, the solutions to the Dirac equation correctly predict an observed shift at level 2s. It is the well-known fine structure correction in the Lyman-alpha line of the hydrogen spectrum (see figure 3).

By the way, the fine structure is the name given in atomic physics to the doubling of the lines of the emission spectrum of atoms, which is a direct consequence of electronic spin.

Figure 4. Fine structure splitting for the ground state n = 1 and the first excited state n = 2 in the hydrogen atom. Source: R Wirnata. Relativistic corrections to hydrogen-like atoms. Researchgate.net

Articles of interest

De Broglie atomic model.

Chadwick's atomic model.

Heisenberg atomic model.

Perrin's atomic model.

Thomson's atomic model.

Dalton's atomic model.

Schrödinger's atomic model.

Atomic model of Democritus.

Bohr's atomic model.

References

1. Atomic theory. Recovered from wikipedia.org.
2. Electron Magnetic Moment. Recovered from wikipedia.org.
3. Quanta: A handbook of concepts. (1974). Oxford University Press. Recovered from Wikipedia.org.
4. Dirac Jordan atomic model. Recovered from prezi.com.
5. The New Quantum Universe. Cambridge University Press. Recovered from Wikipedia.org.