12.4: Dirac Theory of the Hydrogen Atom

The theory of Paul Dirac represents an attempt to unify the theories of quantum mechanics and special relativity. That is, one seeks a formulation of quantum mechanics which is Lorentz invariant, and hence consistent with special relativity. For a free particle, relativity states that the energy is given by

\[E + p^2c^2 + m^2c^4\]

Associating \(E\) with a Hamiltonian in quantum mechanics, one has

\[H^2 = p^2c^2 + m^2c^4 \label{59}\]

If \(H\) and \(p\) are associated with the same operators as in Schrödinger theory, then one expects the wave equation

\[ - \hbar \dfrac{\partial^2}{\partial t^2} \Psi = ( -\hbar^2 \nabla^2 c^2 + m^2 c^4) \Psi \label{60}\]

This is known as the Klein-Gordan Equation. Unfortunately, attempts to utilize this equation are not successful, since that which one would wish to interpret as a probability distribution turns out to be not positive definite. To alleviate this problem, the square root may be taken to get

\[ H = \sqrt{ p^2c^2 + m^2c^4} \label{61}\]

However, this creates a new problem. What is meant by the square root of an operator? The approach is to guess the form of the answer, and the correct guess turns out to be

\[H = c \alpha \cdot p + \beta mc^2 \label{62}\]

With this form of the Hamiltonian, the wave equation can be written

\[ i \hbar \dfrac{\partial \chi}{\partial t} = (c \alpha \cdot p + \beta mc^2) \chi \label{63}\]

In order for this to be valid, one hopes that when it is squared the Klein-Gordan equation is recovered. For this to be true, Equation \(\ref{63}\) must be interpreted as a matrix equation, where \(\alpha\) and \(\beta\) are at least \(4 \times 4\) matrices and the wavefunction \(\chi\) is a four-component column matrix.

It turns out that Equation \(\ref{63}\) describes only a particle with spin 1/2. This is fine for application to the hydrogen atom, since the electron has spin 1/2, but why should it be so? The answer is that the linearization of the Klein-Gordan equation is not unique. The particular linearization used here is the simplest one, and happens to describe a particle of spin 1/2, but other more complicated Hamiltonians may be constructed to describe particles of spin 0,1,5/2 and so on. The fact that the relativistic Dirac theory automatically includes the effects of spin leads to an interesting conclusion - spin is a relativistic effect. It can be added by hand to the non-relativistic Schödinger theory with satisfactory results, but spin is a natural consequence of treating quantum mechanics in a completely relativistic fashion.

Spin is a relativistic effect

Including the potential now in the Hamiltonian, Equation \(\ref{63}\) becomes

\[i \hbar \frac{\partial \chi}{\partial t}=\left(c \alpha \cdot \mathbf{p}+\beta m c^2-\frac{Z e^2}{r}\right) \chi .\]

When the square root was taken to linearize the Klein-Gordan equation, both a positive and a negative energy solution was introduced. One can write the wavefunction

\[\chi=\binom{\Psi_{\text {+}}}{\Psi_{-}},\]

where \(\Psi_+\) represents the two components of \(\chi\) associated with the positive energy solution and \(\Psi_-\) represents the components associated with the negative energy solution. The physical interpretation is that \(\Psi_+\) is the particle solution, and \(\Psi_-\) represents an anti-particle. Anti-particles are thus predicted by Dirac theory, and the discovery of anti-particles obviously represents a huge triumph for the theory. In hydrogen, however, the contribution of \(\Psi_-\) is small compared to \(\Psi_+\). With enough effort, the equations for \(\Psi_+\) and \(\Psi_-\) can be decoupled to whatever order is desired. When this is done, the Hamiltonian to order \(v^2/c^2\) can be written

\[ H + H_s + \Delta H _{rel} + \Delta H_{so} + \Delta H_d \label{66}\]

where

• \(H_s\) is the original Schrödinger Hamiltonian,
• \(\Delta H _{rel}\) is the relativistic correction to the kinetic energy,
• \(\Delta H_{so}\) is the spin-orbit term, and
• \(\Delta H_d\) is the previously mentioned Darwin term.

The Darwin Term

The physical origin of the Darwin term is a phenomenon in Dirac theory called zitterbewegung, whereby the electron does not move smoothly, but instead undergoes extremely rapid small-scale fluctuations, causing the electron to see a smeared-out Coulomb potential of the nucleus.

The Darwin term may be written

\[ \Delta H_d = -\dfrac{e\hbar^2}{8m^2c^2} \nabla^2 \Psi \label{67}\]

For the hydrogenic-atom potential \( V = \dfrac{Ze}{r}\), Equation \(\ref{67}\) is

\[ H_d = -\dfrac{Ze^2\pi \hbar^2}{2 m^2c^2} \delta^3(r) \label{68}\]

When first-order perturbation theory is applied, the energy correction depends on \(| \Psi(0)|^2\). This term will only contribute for s states (i.e., (l=0)), since only these wavefunctions have non-zero probability for finding the electron at the origin. The energy correction for \(l=0\) can be calculated to be

\[ \Delta E _d = (Z \alpha)^4 mc^2 \dfrac{1}{2n^3} \label{69}\]

Including this term, the fine-structure splitting can be reproduced for all \(l\). All the effects that go into fine structure are thus a natural consequence of the Dirac theory.

The hydrogen atom can be solved exactly in Dirac theory, where the states found are simultaneous eigenstates of \(H, J^2\), and \(J_z\), since these operators can be shown to mutually commute. The exact energy levels in Dirac theory are

\[E_{n j}=m c^2\left[1+\left(\frac{Z \alpha}{n-\left(j+\frac{1}{2}\right)+\sqrt{\left(j+\frac{1}{2}\right)^2-(Z \alpha)^2}}\right)^2\right]^{-1 / 2} .\]

This can be expanded in powers of \(Z \alpha\), yielding

\[E_{n j}=m c^2\left\{1-\frac{1}{2} \frac{(Z \alpha)^2}{n^2}\left[1+\frac{(Z \alpha)^2}{n}\left(\frac{1}{j+\frac{1}{2}}-\frac{3}{4 n}\right)\right]+\ldots\right\}\]

This includes an amount \(m c^2\) due to the relativistic energy associated with the rest mass of the electron, along with the principle energy levels and fine structure, in exact agreement to order \((Z \alpha)^4\) with what was previously calculated. However, even this exact solution in Dirac theory is not a complete description of the hydrogen atom, and so the the next section describes further effects not yet discussed.

Footnote

See Bjorken and Drell Chapter 4 for a thorough discussion of the transfer

Contributors and Attributions

• Randal Telfer (JWST Astronomical Optics Scientist, Space Telescope Science Institute)