Hydrogen-like atom

The one-electron wave function solutions are referred to as hydrogen-like atomic orbitals.

In the solution to the Schrödinger equation, which is non-relativistic, hydrogen-like atomic orbitals are eigenfunctions of the one-electron angular momentum operator L and its z component Lz.

This principle restricts the allowed values of the four quantum numbers in electron configurations of more-electron atoms.

Numerical methods must be applied in order to obtain (approximate) wavefunctions or other properties from quantum mechanical calculations.

Due to the spherical symmetry (of the Hamiltonian), the total angular momentum J of an atom is a conserved quantity.

Many numerical procedures start from products of atomic orbitals that are eigenfunctions of the one-electron operators L and Lz.

In quantum chemical calculations hydrogen-like atomic orbitals cannot serve as an expansion basis, because they are not complete.

The non-square-integrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space.

[1] In the simplest model, the atomic orbitals of hydrogen-like atoms/ions are solutions to the Schrödinger equation in a spherically symmetric potential.

It is customary to subscript the wave functions with the values of the quantum numbers they depend on.

Note that L2 and Lz commute and have a common eigenstate, which is in accordance with Heisenberg's uncertainty principle.

Since Lx and Ly do not commute with Lz, it is not possible to find a state that is an eigenstate of all three components simultaneously.

When one takes this coupling into account, the spin and the orbital angular momentum are no longer conserved, which can be pictured by the electron precessing.

In 1928 in England Paul Dirac found an equation that was fully compatible with special relativity.

The equation was solved for hydrogen-like atoms the same year (assuming a simple Coulomb potential around a point charge) by the German Walter Gordon.

The quantum number j determines the sum of the squares of the three angular momenta to be j(j+1) (times ħ2, see Planck constant).

The splitting of the energies of states of the same principal quantum number n due to differences in j is called fine structure.

Note that if Z were able to be more than 137 (higher than any known element) then we would have a negative value inside the square root for the S1/2 and P1/2 orbitals, which means they would not exist.

The accuracy of the energy difference between the lowest two hydrogen states calculated from the Schrödinger solution is about 9 ppm (90 μeV too low, out of around 10 eV), whereas the accuracy of the Dirac equation for the same energy difference is about 3 ppm (too high).

The Schrödinger solution always puts the states at slightly higher energies than the more accurate Dirac equation.

[2] The modifications of the energy due to using the Dirac equation rather than the Schrödinger solution is of the order of α2, and for this reason α is called the fine-structure constant.

are: where A is a normalization constant involving the gamma function: Notice that because of the factor Zα, f(r) is small compared to g(r).

Note that the dominant term is quite similar to corresponding the Schrödinger solution – the upper index on the Laguerre polynomial is slightly less (2γ+1 or 2γ−1 rather than 2ℓ+1, which is the nearest integer), as is the power of ρ (γ or γ−1 instead of ℓ, the nearest integer).

The normalization factor makes the integral over all space of the square of the absolute value equal to 1.

For the 2S1/2 spin up orbital, we have: Now the first component is S-like and there is a radius near ρ = 2 where it goes to zero, whereas the bottom two-component part is P-like.

Negative-energy solutions to Dirac's equation exist even in the absence of a Coulomb force exerted by a nucleus.

If one of these negative-energy states is not filled, this manifests itself as though there is an electron which is repelled by a positively-charged nucleus.

This prompted Dirac to hypothesize the existence of positively-charged electrons, and his prediction was confirmed with the discovery of the positron.

The Dirac equation with a simple Coulomb potential generated by a point-like non-magnetic nucleus was not the last word, and its predictions differ from experimental results as mentioned earlier.

More accurate results include the Lamb shift (radiative corrections arising from quantum electrodynamics)[3] and hyperfine structure.