The relativistic electron
In this chapter, we’ll be looking at the relativistic quantum mechanics for a single, free electron. We will merely introduce the important concepts and implications, instead of a fully formal relativistic treatment of the Dirac electron/field.
The Klein-Gordon equation
There is a fundamental flaw with the time-dependent Schrödinger equation \[\begin{equation} \require{physics} i \hbar \pdv{t} \Psi(\vb{r}, t) = h^c \Psi(\vb{r},t) % \thinspace , \end{equation}\] and that is that that it is not Lorentz covariant: time and space coordinates are not treated in the same way. We could however, use the squared relativistic energy for the electron: \[\begin{equation} E^2 = c^2 \norm{\vb{p}}^2 + m_e^2 c^4 \end{equation}\] and subsequently use the quantum mechanical correspondence principle \[\begin{align} & E \rightarrow i \hbar \pdv{t} \\ & \vb{p} \rightarrow -i \hbar \nabla % \label{eq:quantization_momentum} \thinspace , \end{align}\] we do end up with a Lorentz covariant equation: \[\begin{equation} \qty[ % \frac{1}{c^2} \pdv[2]{t} % - \laplacian % + \qty(\frac{m_e c}{\hbar})^2 % ] \Psi(\vb{r},t) = 0 % \thinspace , \end{equation}\] called the Klein-Gordon equation. In covariant form, it would read: \[\begin{equation} \qty[ % \partial_\mu \partial^\mu % + \qty(\frac{m_e c}{\hbar})^2 % ] \Psi(\vb{r},t) = 0 % \thinspace , \end{equation}\] from which it is evident (because only Lorentz scalars appear) that the Klein-Gordon equation is Lorentz covariant. The solutions to the Klein-Gording equation are the plane waves: \[\begin{equation} \Psi(\vb{r}, t) % = \exp( % \frac{i}{\hbar} (Et - \vb{p} \vdot \vb{r}) % ) \thinspace , \end{equation}\] characterized by the energy \[\begin{equation} E = \pm \sqrt{c^2 \norm{\vb{p}}^2 + m_e^2 c^4} \thinspace . \end{equation}\]
However, for electrons, there are a few problems with this Klein-Gordon equation. First of all, the occurrence of negative energies are hard to explain, second, it cannot explain the occurrence of the spin of an electron and third, the Klein-Gordon probability density (derived from the Klein-Gordon continuity equation) is not positive definite. This means that the Klein-Gordon was a nice first step to describe the relativistic electron, but there is better equation of motion for the free, relativistic electron.
The Dirac equation
The Dirac equation for a freely moving electron \[\begin{equation} i \hbar \pdv{t} \Psi(\vb{r},t) % = h^{c,\text{D}} \Psi(\vb{r},t) \end{equation}\] is based on the following form for the Hamiltonian: \[\begin{equation} \label{eq:Dirac_equation} h^{c,\text{D}} % = -i \hbar c \alpha^k \partial_k + \beta m_e c^2 % \thinspace , \end{equation}\] in which \(\alpha^k\) and \(\beta\) are yet unspecified objects. The components \(\alpha^i\) are not components of a contravariant vector, but they are introduced to employ Einstein’s summation convention. If we now require that \[\begin{equation} \label{eq:requirements_Dirac_matrices} \begin{cases} & \comm{\alpha^i}{\alpha^j}_+ = 2 \delta_{ij} \\ & \comm{\alpha^i}{\beta}_+ = 0 \\ & \beta^2 = 1 \end{cases} \thinspace , \end{equation}\] from which, by the way, follows that \[\begin{equation} (\alpha^i)^2 = 1 % \thinspace , \end{equation}\] we can verify that the Klein-Gordon equation is in fact embedded in the Dirac equation. This means that, given that the objects \(\alpha^k\) and \(\beta\) behave like these requirements, the Dirac equation is Lorentz covariant. Gauge invariance of the Dirac equation is discussed in subsection \(\eqref{sec:Dirac_electron_external_fields}\).
The standard representation
There are different ways of choosing the objects \(\boldsymbol{\alpha} = (\alpha_x, \alpha_y, \alpha_z)\) and \(\beta\) such that the requirements in equation \(\eqref{eq:requirements_Dirac_matrices}\) are fulfilled. One particular choice of representation is called the standard representation, in which the \(\alpha\)- and \(\beta\)-matrices are both \((4 \times 4)\)-matrices. In this representation, the \(\alpha\)-matrices are given by \[\begin{equation} \label{eq:Dirac_alpha_matrices} \alpha^k = % \begin{pmatrix} % 0 & \sigma_k \\ % \sigma_k & 0 % \end{pmatrix} \thinspace , \end{equation}\] in which \(\sigma_k\) are the Pauli spin matrices (cfr. equation \(\eqref{eq:Pauli_spin_matrices}\)). In the standard representation, the \((4 \times 4)\)-\(\beta\) matrix is given by: \[\begin{equation} \label{eq:Dirac_beta_matrix} \beta = % \begin{pmatrix} % I_2 & 0 \\ % 0 & -I_2 % \end{pmatrix} \thinspace . \end{equation}\] From the previous definitions, it automatically follows that all \(\alpha^k\) and \(\beta\)-matrices are Hermitian, which should be the case since the Dirac Hamiltonian must be Hermitian.
Since, in the standard representation, the objects \(\alpha^k\) and \(\beta\) are \(4 \times 4\)-matrices, the state vector \(\Psi(\vb{r}, t)\) must also be \(4\)-dimensional and is called a \(4\)-spinor or bispinor. Because of the block structure of the \(\alpha\)- and \(\beta\)-matrices, this 4-spinor is often split up into two 2-spinors: \[\begin{equation} \Psi(\vb{r}, t) = % \begin{pmatrix} % \Psi^{(1)}(\vb{r}, t) \\ \Psi^{(2)}(\vb{r}, t) \\ \Psi^{(3)}(\vb{r}, t) \\ \Psi^{(4)}(\vb{r}, t) % \end{pmatrix} % = % \begin{pmatrix} % \Psi^\text{L}(\vb{r}, t) \\ \Psi^\text{S}(\vb{r}, t) % \end{pmatrix} \thinspace , \end{equation}\] which are called the large and the small components, but more on that later.
If we introduce the \(k\)-th component of the momentum operator in coordinate representation as (cfr. the quantization of the momentum in equation \(\eqref{eq:quantization_momentum}\)): \[\begin{equation} p^c_k = -i \hbar \partial_k \end{equation}\] then we can write the Dirac equation in a compact form as \[\begin{equation} i \hbar \pdv{t} \Psi(\vb{r}, t) % = \qty( % c \boldsymbol{\alpha} \vdot \vb{p}^c + \beta m_e c^2 % ) \Psi(\vb{r}, t) % \thinspace . \end{equation}\] By subsequently introducing the small and large components, we can see that the Dirac equation admits a \((2 \times 2)\)-superstructure: \[\begin{equation} i \hbar \pdv{t} % \begin{pmatrix} % \Psi^\text{L}(\vb{r}, t) \\ \Psi^\text{S}(\vb{r}, t) % \end{pmatrix} = \begin{pmatrix} m_e c^2 & c \boldsymbol{\sigma} \vdot \vb{p}^c \\ c \boldsymbol{\sigma} \vdot \vb{p}^c & - m_e c^2 \end{pmatrix} % \begin{pmatrix} % \Psi^\text{L}(\vb{r}, t) \\ \Psi^\text{S}(\vb{r}, t) % \end{pmatrix} \thinspace , \end{equation}\] or in full \(4\)-component form: \[\begin{equation} i \hbar \pdv{t} % \begin{pmatrix} % \Psi^{(1)}(\vb{r}, t) \\ \Psi^{(2)}(\vb{r}, t) \\ \Psi^{(3)}(\vb{r}, t) \\ \Psi^{(4)}(\vb{r}, t) % \end{pmatrix} = \begin{pmatrix} m_e c^2 % & 0 % & p^c_z % & p^c_x - i p^c_y \\ 0 % & m_e c^2 % & p^c_x + i p^c_y % & -p_z \\ p^c_z % & p^c_x - i p^c_y % & -m_e c^2 % & 0 \\ p^c_x + i p^c_y % & -p_z % & 0 % & - m_e c^2 \end{pmatrix} \begin{pmatrix} % \Psi^{(1)}(\vb{r}, t) \\ \Psi^{(2)}(\vb{r}, t) \\ \Psi^{(3)}(\vb{r}, t) \\ \Psi^{(4)}(\vb{r}, t) % \end{pmatrix} \thinspace . \end{equation}\] in which we have used the at first sight strange ‘inner product’ of the Pauli sigma matrices with the momentum operator: \[\begin{equation} \boldsymbol{\sigma} \vdot \vb{p}^c % = \begin{pmatrix} % p^c_z & p^c_x -i p^c_y \\ p^c_x + i p^c_y & -p_z % \end{pmatrix} \thinspace . \end{equation}\] By the way, this operator can be equivalently written as: \[\begin{equation} \boldsymbol{\sigma} \vdot \vb{p}^c % = \frac{1}{r^2} ( % \boldsymbol{\sigma} \vdot \vb{r} % ) [ % \vb{r} \vdot \vb{p}^c % + i (\boldsymbol{\sigma} \vdot \vb{l}) % ] % \thinspace , \end{equation}\] in which the so-called spin-orbit coupling term is made explicit in the last term.
Other representations
We can write Dirac’s equation in many different forms. Let us start by introducing the Dirac matrices \(\gamma^\mu = (\gamma^0, \gamma^i)\): \[\begin{equation} \gamma^0 = \beta = % \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 % \end{pmatrix} \thinspace , \end{equation}\] which is a Hermitian matrix, and \[\begin{equation} \gamma^i = \beta \alpha^i % = \begin{pmatrix} 0 & \sigma_k \\ -\sigma_k & 0 \end{pmatrix} \thinspace , \end{equation}\] which are anti-Hermitian matrices. They furthermore fulfill the anticommutation relations \[\begin{equation} \comm{\gamma^\mu}{\gamma^\nu}_+ = 2 \eta^{\mu \nu} % \label{eq:anticommutator_gamma_matrices} % \thinspace , \end{equation}\] which is said to define a Clifford algebra. It is then possible to obtain different representations of the Dirac equations by doing similarity transformations of the form \[\begin{equation} \tilde{\gamma}^\mu = M \gamma^\mu M^{-1} \end{equation}\] with \(M\) an arbitrary invertible matrix. The Dirac matrices \(\gamma^\mu\) are used in the Einstein summation convention, but are not strictly Lorentz 4-vectors. In fact, they are Lorentz scalars, as they have the same values in every frame of reference. Just by an abuse of notation, we can calculate the seemingly covariant components as \[\begin{equation} \gamma_\nu = \eta_{\nu \mu} \gamma^\mu % \thinspace . \end{equation}\] Using the Dirac matrices \(\gamma^\mu\), the Dirac equation can be written in covariant form: \[\begin{equation} \label{eq:Dirac_equation_gamma_matrices} \qty( % -i \hbar \gamma^\mu \partial_\mu % + m_e c % ) \Psi(x) = 0 % \thinspace . \end{equation}\] It can be shown that this equation is Lorentz covariant (i.e. it has the same form in two interial frames related through Lorentz transformations), which means that even though \(\gamma^\mu\) are not the components of a contravariant vector, the combination \(\gamma^\mu \partial_\mu \Psi\) is Lorentz covariant, so we can treat \(\gamma^\mu \partial_\mu\) as a Lorentz scalar.
If we take the Hermitian adjoint of this equation, and subsequently multiply by \(\gamma^0\) on the right, we obtain the Dirac adjoint equation: \[\begin{equation} i \partial_{\mu} \bar{\Psi}(x) \gamma^{\mu} % + \frac{m_e c}{\hbar} \bar{\Psi}(x) = 0 % \thinspace , \end{equation}\] in which we have defined the Dirac adjoint (Bransden and Joachain 2000) as \[\begin{equation} \bar{\Psi}(x) = \Psi^\dagger(x) \gamma^0 % \thinspace . \end{equation}\] By multiplying the Dirac equation on the left with \(\bar{\Psi}\) and the Dirac adjoint equation on the right with \(\Psi\), we find a continuity equation: \[\begin{equation} \partial_\mu j^\mu(x) = 0 % \thinspace , \end{equation}\] if we define the \(4\)-current as: \[\begin{equation} j^\mu(x) = \bar{\Psi}(x) \gamma^\mu \Psi(x) % \thinspace . \end{equation}\] In terms of the Dirac \(\alpha\) and \(\beta\)-matrices, the time component gives the Dirac distribution: \[\begin{equation} \frac{j^0(x)}{c} = \rho(\vb{r}, t) % = \Psi^\dagger(\vb{r}, t) \Psi(\vb{r}, t) \end{equation}\] and the spatial components give the Dirac 3-current density: \[\begin{equation} \vb{j}(\vb{r}, t) = % c \Psi^\dagger(\vb{r}, t) % \boldsymbol{\alpha} % \Psi(\vb{r}, t) % \thinspace , \end{equation}\] which are related through the continuity equation: \[\begin{equation} \pdv{ % \rho(\vb{r}, t) % }{t} % + \boldsymbol{\nabla} \vdot \vb{j}(\vb{r}, t) = 0 % \thinspace . \end{equation}\]
The Dirac distribution and current density can be written in spinor (two-component) and bispinor (four-component) form as follows (G. Dyall and Faegri 2007): \[\begin{align} \rho(\vb{r}, t) &= \Psi^\dagger(\vb{r}, t) \Psi(\vb{r}, t) % \label{eq:Dirac_distribution} \\ &= \Psi^{\text{L} \dagger}(\vb{r}, t) \Psi^{\text{L}}(\vb{r}, t) % + \Psi^{\text{S} \dagger}(\vb{r}, t) \Psi^{\text{S}}(\vb{r}, t) \\ &= \sum_u^4 \Psi^{(u) \dagger}(\vb{r}, t) \Psi^{(u)}(\vb{r}, t) \end{align}\] and \[\begin{align} \vb{j}(\vb{r}, t) % &= c \Psi^\dagger(\vb{r}, t) % \boldsymbol{\alpha} % \Psi(\vb{r}, t) \\ &= c % \qty( % \Psi^{\text{L} \dagger}(\vb{r}, t) % \boldsymbol{\sigma} % \Psi^{\text{S}}(\vb{r}, t) % + \Psi^{\text{S} \dagger}(\vb{r}, t) % \boldsymbol{\sigma} % \Psi^{\text{L}}(\vb{r}, t) % ) \\ &= c \begin{pmatrix} \Psi^{(1)*} \Psi^{(4)} % + \Psi^{(2)*} \Psi^{(3)} % + \Psi^{(3)*} \Psi^{(2)} % + \Psi^{(4)*} \Psi^{(1)} \\ % -i \Psi^{(1)*} \Psi^{(4)} % + i \Psi^{(2)*} \Psi^{(3)} % - i \Psi^{(3)*} \Psi^{(2)} % + i \Psi^{(4)*} \Psi^{(1)} \\ % \Psi^{(1)*} \Psi^{(3)} % - \Psi^{(2)*} \Psi^{(4)} % + \Psi^{(3)*} \Psi^{(1)} % - \Psi^{(4)*} \Psi^{(2)} \end{pmatrix} \thinspace , \end{align}\] in which we have not written the explicit time- and position-dependence of the components of the bispinor in the last equation.
Negative-energy states
Let us now turn to the solution of the Dirac equation. For a particle at rest, the kinetic contributions in the Dirac equation vanish, so we are left with \[\begin{equation} i \hbar \pdv{t} \Psi(\vb{r}, t) % = \beta m_e c^2 \Psi(\vb{r}, t) % \thinspace , \end{equation}\] which as four solutions: \[\begin{align} & \Psi_1^{(+)} = \exp(-i \frac{m_e c^2}{\hbar} t) % \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \\ % & \Psi_2^{(+)} = \exp(-i \frac{m_e c^2}{\hbar} t) % \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} \\ % & \Psi_1^{(-)} = \exp(i \frac{m_e c^2}{\hbar} t) % \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \\ % & \Psi_2^{(-)} = \exp(i \frac{m_e c^2}{\hbar} t) % \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \thinspace . \end{align}\] The form of these solutions is familiar, because they are just plane waves, which is what we would expect because the Dirac equation implies the Klein-Gordon equation. The solutions of the Dirac equation also belong to two energies: \[\begin{align} & E^{(+)} % = \ev*{ % \beta m_e c^2 % }{\Psi_{1,2}^{(+)}} % = m_e c^2 % \label{eq:E+_Dirac} \\ % & E^{(-)} % = \ev*{ % \beta m_e c^2 % }{\Psi_{1,2}^{(-)}} % = - m_e c^2 % \thinspace . \end{align}\] We can thus see that the \(\beta\)-matrix divides the four solutions into two classes of solutions. The first class is spanned by \(\set{\Psi_{1}^{(+)}, \Psi_{2}^{(+)}}\) and has a positive energy \(E^{(+)} = m_e c^2\), while the second class of solutions is spanned by \(\set{\Psi_{1}^{(-)}, \Psi_{2}^{(-)}}\) and has a negative energy \(E^{(-)} = -m_e c^2\). Is is these negative-energy solutions that play the most important role when switching from a non-relativistic quantum chemical theory to a relatistivic one.
The Dirac electron in an external electromagnetic field
If we want to describe a relativistic electron in an external electromagnetic field, in order for the Dirac equation to remain Lorentz covariant, the electromagnetic interaction should be represented by a Lorentz scalar. The simplest choice is called minimal coupling: \[\begin{equation} - i \hbar \partial_\mu % \rightarrow % -i \hbar \partial_\mu + q_e A_\mu(x) % \thinspace , \end{equation}\] and we are assured that the electromagnetic interaction term \[\begin{equation*} q_e A_\mu(x) \end{equation*}\] becomes a Lorentz scalar when combined with the \(\gamma^\mu\) that is in front of it. By plugging in this minimal coupling, we obtain the Dirac equation for an electron in an electromagnetic field characterized by the four-potential \(A_\mu(x)\): \[\begin{equation} \label{eq:Dirac_electron_external_field_gamma_matrices} \qty[ % \gamma^\mu \qty( % -i \hbar \partial_\mu + q_e A_\mu(x) % ) + m_e c % ] \Psi(x) = 0 \thinspace , \end{equation}\] which is by Lorentz covariant, and we can show that it is also gauge invariant under the the gauge transformation \[\begin{align} & A'_{\mu}(x) = A_\mu(x) - \partial_\mu \chi(x) \\ & \Psi'(x) = \exp( \frac{-iq_e}{\hbar} \chi(x) ) \Psi(x) % \thinspace . \end{align}\] Written in terms of the Dirac \(\alpha\)- and \(\beta\)-matrices, we find: \[\begin{equation} \label{eq:Dirac_equation_external_field_after_minimal_coupling} i \hbar \pdv{t} \Psi(\vb{r}, t) = % \qty[ % c \boldsymbol{\alpha} \vdot \qty( % \vb{p}^c - q_e \vb{A}^c_\text{ext}(\vb{r}, t) % ) % + \beta m_e c^2 % + q_e \phi^c_\text{ext}(\vb{r}, t) % ] \Psi(\vb{r}, t) % \thinspace . \end{equation}\] We will now introduce the kinematic momentum operator \(\boldsymbol{\pi}^c(\vb{r}, t)\) as \[\begin{equation} \boldsymbol{\pi}^c(\vb{r}) % = \vb{p}^c - q_e \vb{A}^c_\text{ext}(\vb{r}) % \thinspace , \end{equation}\] which represents the real kinetic energy of the electron in the external electromagnetic field. \(\vb{p}^c\) is then just called the canonical momentum. We should already note that the components of the kinematic momentum operator do not commute: \[\begin{equation} \comm{ % \pi^c_i(\vb{r}, t) % }{ % \pi^c_j(\vb{r}) % } % = i \hbar q_e \epsilon_{ijk} B_k(\vb{r}, t) % \thinspace , \end{equation}\] We can then write the Dirac equation as: \[\begin{align} i \hbar \pdv{t} \Psi(\vb{r}, t) % &= \qty( % c \boldsymbol{\alpha} \vdot \boldsymbol{\pi}^c(\vb{r}, t) % + \beta m_e c^2 % + q_e \phi^c_\text{ext}(\vb{r}, t) % ) \Psi(\vb{r}, t) % \label{eq:Dirac_equation_emf_alpha_beta} \\ % i \hbar \pdv{t} % \begin{pmatrix} \Psi^\text{L}(\vb{r}, t) \\ \Psi^\text{S}(\vb{r}, t) \end{pmatrix} &= % \begin{pmatrix} m_e c^2 + q_e \phi^c_\text{ext}(\vb{r}, t) % & c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c(\vb{r}, t) \\ % c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c(\vb{r}, t) % & - m_e c^2 + q_e \phi^c_\text{ext}(\vb{r}, t) \end{pmatrix} \begin{pmatrix} \Psi^\text{L}(\vb{r}, t) \\ \Psi^\text{S}(\vb{r}, t) \end{pmatrix} \thinspace . \end{align}\] If we rewrite equation \(\eqref{eq:Dirac_equation_external_field_after_minimal_coupling}\) as \[\begin{equation} i \hbar \pdv{t} \Psi(\vb{r}, t) = \qty( % c \boldsymbol{\alpha} \vdot \vb{p}^c % + \beta m_e c^2 % + q_e \phi(\vb{r}, t) % - q_e c \boldsymbol{\alpha} \vdot \vb{A}(\vb{r}, t) % ) \Psi(\vb{r}, t) % \end{equation}\] and compare this equation with the interaction Lagrangian for two moving charged particles \(\eqref{eq:interaction_energy_moving_charge_external_field}\), we can seemingly set up a Dirac correspondence principle: \[\begin{equation} \label{eq:Dirac_correspondence_principle} \dot{\vb{r}} = c \boldsymbol{\alpha} % \thinspace . \end{equation}\]
After some manipulation we can find the corresponding Dirac adjoint equation in covariant form: \[\begin{equation} \qty( % i \hbar \partial_{\mu} % + q_e A_{\mu}(x) % ) \bar{\Psi} \gamma^\mu % + m_e c \bar{\Psi}(x) = 0 % \thinspace . \end{equation}\] By multiplying the Dirac equation on the left with \(\bar{\Psi}(x)\) and the Dirac adjoint equation on the right with \(\Psi(x)\), we find a continuity equation: \[\begin{equation} \partial_\mu j^\mu(x) = 0 % \thinspace , \end{equation}\] with the \(4\)-current given by \[\begin{equation} j^\mu(x) = \bar{\Psi}(x) \gamma^\mu \Psi(x) % \thinspace . \end{equation}\] We sometimes define the commutator of the Dirac gamma matrices as \[\begin{equation} \Sigma^{\mu \nu} = \frac{i}{2} \comm{\gamma^\mu}{\gamma^\nu} % \thinspace , \end{equation}\] which, together with the anticommutator between the gamma matrices \(\eqref{eq:anticommutator_gamma_matrices}\) gives: \[\begin{equation} \gamma^{\mu} \gamma^{\nu} = \eta^{\mu \nu} - i \Sigma^{\mu \nu} \end{equation}\] We can then write the \(4\)-current density as: \[\begin{equation} j^{\mu} % = \frac{-i \hbar}{2 m_e} \qty( % (\partial^\mu \bar{\Psi}) \Psi % - \bar{\Psi} (\partial^\mu \Psi) % ) % + \frac{\hbar}{2 m_e} \partial_\nu ( % \bar{\Psi} \Sigma^{\mu \nu} \Psi % ) % - \frac{q_e}{m_e} A^\mu \bar{\Psi} \Psi % \thinspace , \end{equation}\] which is the so-called Gordon decomposition of the Dirac 4-current.
The non-relativistic limit - the Pauli equation
Since the lowest possible energy of a nonrelativistic free particle is zero instead of \(+m_e c^2\) (cfr. equation \(\eqref{eq:E+_Dirac}\)), let us start by shifting the energy by \(-m_e c^2\), leading to \[\begin{equation} i \hbar \pdv{t} % \begin{pmatrix} \Psi^\text{L}(\vb{r}, t) \\ \Psi^\text{S}(\vb{r}, t) \end{pmatrix} = \begin{pmatrix} q_e \phi^c_\text{ext}(\vb{r}, t) % & c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c(\vb{r}, t) \\ % c \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c(\vb{r}, t) % & - 2m_e c^2 + q_e \phi^c_\text{ext}(\vb{r}, t) \end{pmatrix} \begin{pmatrix} \Psi^\text{L}(\vb{r}, t) \\ \Psi^\text{S}(\vb{r}, t) \end{pmatrix} \thinspace . \end{equation}\] For nonrelativistic energies, i.e. energies \[\begin{equation*} i \hbar \pdv{t} \rightarrow E \end{equation*}\] that are small compared to \(m_e c^2\), we find from the equation for the small component that: \[\begin{equation} \label{eq:magnetic_balance} \Psi^\text{S}(\vb{r}, t) % \approx \frac{ % \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c(\vb{r}, t) % }{2 m_e c} \Psi^\text{L}(\vb{r}, t) % \thinspace , \end{equation}\] which is called the kinetic (or magnetic balance condition, depending on if there is an external magnetic field) and now we can clearly reason the naming of the two components: the small component \(\Psi^\text{S}\) is smaller than the large component \(\Psi^\text{L}\) by a factor of \(1/c\).
Plugging in the magnetic balance equation and using Dirac’s relation to calculate \[\begin{equation} ( \boldsymbol{\sigma} \vdot \boldsymbol{\pi}^c(\vb{r}, t) )^2 % = \boldsymbol{\pi}^c(\vb{r}, t) % \vdot \boldsymbol{\pi}^c(\vb{r}, t) % - q_e \hbar \boldsymbol{\sigma} % \vdot \vb{B}^c(\vb{r}, t) % \thinspace , \end{equation}\] leads to the so-called Pauli equation for the large component: \[\begin{equation} i \hbar \pdv{t} \Psi^\text{L}(\vb{r}, t) % = \qty[ % \frac{ % \norm{\boldsymbol{\pi}^c(\vb{r}, t)}^2 % }{2 m_e} % - \frac{q_e \hbar}{2 m_e} % \boldsymbol{\sigma} \vdot \vb{B}^c(\vb{r}, t) % + q_e \phi^c(\vb{r}, t) % ] \Psi^\text{L}(\vb{r}, t) % \thinspace . \end{equation}\] Expanding the kinematic momentum operator in Coulomb gauge, introducing the angular momentum operator in coordinate representation as \[\begin{equation} \vb{l}^c(\vb{r}) = \vb{r}^c \cross \vb{p}^c % \thinspace , \end{equation}\] and introducing the spin operator \(\vb{s}\) as \[\begin{equation} \vb{s} = \frac{\hbar}{2} \boldsymbol{\sigma} \end{equation}\] eventually leads to the more familiar form of the Pauli equation: \[\begin{equation} i \hbar \pdv{t} \Psi^\text{L} % = \qty[ % \frac{ \norm{\vb{p}^c}^2 }{2 m_e} % + \frac{q_e^2}{2 m_e} \norm{ \vb{A}^c(\vb{r}) }^2 % - \frac{q_e}{2 m_e} % (\vb{l}^c(\vb{r}) + 2 \vb{s}) \vdot \vb{B}^c(\vb{r}) % + q_e \phi^c(\vb{r}) % ] \Psi^\text{L} \thinspace , \end{equation}\] in which the term quadratic in the vector potential is called the diamagnetic term, and the term linear (and imaginary) in the vector potential/magnetic field are called the paramagnetic terms.
This nonrelativistic approximation is justified for small nucleic charges (i.e. a small contribution \(q_e \phi^c_\text{ext}(\vb{r})\)). Furthermore, in the limit \(c \rightarrow + \infty\), the small component vanishes, such that all relativistic effects are ignored and the Schrödinger equation is recovered.