Fine Structure - Kinetic Energy Relativistic Correction

Kinetic Energy Relativistic Correction

Classically, the kinetic energy term of the Hamiltonian is

However, when considering special relativity, we must use a relativistic form of the kinetic energy,

where the first term is the total relativistic energy, and the second term is the rest energy of the electron. Expanding this in a Taylor series, we find

Then, the first order correction to the Hamiltonian is

Using this as a perturbation, we can calculate the first order energy corrections due to relativistic effects.

where is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see

We can use this result to further calculate the relativistic correction:

For the hydrogen atom, and where is the Bohr Radius, is the principal quantum number and is the azimuthal quantum number. Therefore the relativistic correction for the hydrogen atom is

where we have used:

On final calculation, the order of magnitude for the relativistic correction to the ground state is .

Note: In reality, is not a Hermitian operator for hydrogen-like s-orbitals . The use of first order quantum perturbation theory requires that the perturbing Hamiltonian be Hermitian. Thus, the proof shown above is not entirely rigorous when . Despite this shortcoming, comparison with the exact answer (derived from the Dirac equation) shows that the result shown above is correct to the first order, even when .