9 julio, 2024

Dirac Jordan atomic model: characteristics and postulates

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

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

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

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 they depend on certain discrete quantum numbers, which define the energy and angular momentum of the electron.

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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. And furthermore, specifying one of the variables leads to a 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 a total energy E that depends on its linear momentum. p thus:

E = (p2)/ 2m

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

p = -i ħ ∂ /∂r

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

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

H = (ħ2/2m) ∂2 /∂r2

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

H = (ħ2/2m) ∂2 /∂r2 – 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 = (m2 c4 + p2 c2)1/2

In the above relationship, if the particle has momentum p = 0 then we have the famous equation E = mc2 which relates the rest energy of any particle of mass m to 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 his fourth postulate emerged.

fourth postulate: to get rid of the square root in the relativistic energy formula, Dirac proposed the following structure for E2:

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

Dirac’s equation

Dirac’s equation was first stated for the free electron, using the structure proposed in the fourth postulate. It remains as follows:

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

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

It is immediately observed that the wave function ceases to be a scalar function and becomes a vector of four components called 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 TO in the hamiltonian:

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

– Complies with 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 the 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 TO it is zero if the stationary central proton is assumed and that is why it does not appear.

The reason why Dirac’s corrections to Schrodinger’s 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, making these terms numerically small.

Relativistic corrections to the energy spectrum

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

In the same way, the Dirac model allows finding the fine structure correction in the energy levels of hydrogen.

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

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

For example, the solutions to Dirac’s equation correctly predict an observed shift at the 2s level. 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 splitting of the emission spectrum lines of atoms, which is a direct consequence of electronic spin.

Articles of interest

Broglie’s atomic model.

Chadwick’s atomic model.

Heisenberg’s atomic model.

Perrin’s atomic model.

Thomson’s atomic model.

Dalton’s atomic model.

Schrödinger’s atomic model.

Democritus’s atomic model.

Leucippus’ atomic model.

Bohr’s atomic model.

Current atomic model.

References

Atomic theory. Retrieved from wikipedia.org.
Electron Magnetic Moment. Retrieved from wikipedia.org.
Quanta: A handbook of concepts. (1974). Oxford University Press. Retrieved from Wikipedia.org.
Dirac Jordan’s atomic model. Retrieved from prezi.com.
The New Quantum Universe. Cambridge University Press. Retrieved from Wikipedia.org.

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