Breit equation · Lablypedia

Related on Lably

Through Spin (physics), Electron and Electric field

Lab · Class 10–12Electric Motor Lab · Class 6–12Atomic Spectra & Starlight Lab · Class 11Respiration in Plants — Animated Chapter Film Lab · Class 9Structure of the Atom

Introduction

The Breit equation is not only an approximation in terms of quantum mechanics, but also in terms of relativity theory as it is not completely invariant with respect to the Lorentz transformation. Just as does the Dirac equation, it treats nuclei as point sources of an external field for the particles it describes. For N particles, the Breit equation has the form (r_ij is the distance between particle i and j):

$i ℏ \frac{\partial Ψ}{\partial t} = (\sum_{i} {\hat{H}}_{D} (i) + \sum_{i > j} \frac{1}{r_{i j}} + \sum_{i > j} {\hat{B}}_{i j}) Ψ$

where ${\hat{H}}_{D} (i) = [q_{i} ϕ (r_{i}) + c \sum_{s = x, y, z} α_{s} (i) π_{s} (I) + α_{0} (I) m_{0} c^{2}]$ is the Dirac Hamiltonian (see Dirac equation) for particle i at position $r_{i}$ and $ϕ (r_{i})$ is the scalar potential at that position; q_i is the charge of the particle, thus for electrons q_i = −e. The one-electron Dirac Hamiltonians of the particles, along with their instantaneous Coulomb interactions 1/r_ij, form the Dirac–Coulomb operator. To this, Breit added the operator (now known as the (frequency-independent) Breit operator): ${\hat{B}}_{i j} = - \frac{1}{2 r_{i j}} [\vec{α} (i) \cdot \vec{α} (j) + \frac{(\vec{α} (i) \cdot r_{i j}) (\vec{α} (j) \cdot r_{i j})}{r_{i j}^{2}}],$ where the Dirac matrices for electron i: α(i) = [α_x(i), α_y(i), α_z(i)]. The two terms in the Breit operator account for retardation effects to the first order. The wave function Ψ in the Breit equation is a spinor with 4^N elements, since each electron is described by a Dirac spinor with 4 elements as in the Dirac equation, and the total wave function is the tensor product of these.

One more chapter is free with a Lably account.

Breit Hamiltonians

The total Hamiltonian of the Breit equation, sometimes called the Dirac–Coulomb–Breit Hamiltonian (H_DCB) can be decomposed into the following practical energy operators for electrons in electric and magnetic fields (also called the Breit–Pauli Hamiltonian), which have well-defined meanings in the interaction of molecules with magnetic fields (for instance for nuclear magnetic resonance): ${\hat{B}}_{i j} = {\hat{H}}_{0} + {\hat{H}}_{1} + \dots + {\hat{H}}_{6},$ in which the consecutive partial operators are:

${\hat{H}}_{0} = \sum_{i} \frac{{\hat{p}}_{i}^{2}}{2 m_{i}} + V$ is the nonrelativistic Hamiltonian ( $m_{i}$ is the stationary mass of particle i).
${\hat{H}}_{1} = - \frac{1}{8 c^{2}} \sum_{i} \frac{{\hat{p}}_{i}^{4}}{m_{i}^{3}}$ is connected to the dependence of mass on velocity: $E_{k i n}^{2} - {(m_{0} c^{2})}^{2} = m^{2} v^{2} c^{2}$ .
${\hat{H}}_{2} = - \sum_{i > j} \frac{q_{i} q_{j}}{2 r_{i j} m_{i} m_{j} c^{2}} [{\hat{p}}_{i} \cdot {\hat{p}}_{j} + \frac{(r_{i j} \cdot {\hat{p}}_{i}) (r_{i j} \cdot {\hat{p}}_{j})}{r_{i j}^{2}}]$ is a correction that partly accounts for retardation and can be described as the interaction between the magnetic dipole moments of the particles, which arise from the orbital motion of charges (also called orbit–orbit interaction).
${\hat{H}}_{3} = \frac{μ_{B}}{c} \sum_{i} \frac{1}{m_{i}} s_{i} \cdot [F (r_{i}) \times {\hat{p}}_{i} + \sum_{j > i} \frac{2 q_{i}}{r_{i j}^{3}} r_{i j} \times {\hat{p}}_{j}]$ is the classical interaction between the orbital magnetic moments (from the orbital motion of charge) and spin magnetic moments (also called spin–orbit interaction). The first term describes the interaction of a particle's spin with its own orbital moment (F(r_i) is the electric field at the particle's position), and the second term between two different particles.
${\hat{H}}_{4} = \frac{i h}{8 π c^{2}} \sum_{i} \frac{q_{i}}{m_{i}^{2}} {\hat{p}}_{i} \cdot F (r_{i})$ is a nonclassical term characteristic for Dirac theory, sometimes called the Darwin term.
${\hat{H}}_{5} = 4 μ_{B}^{2} \sum_{i > j} {- \frac{8 π}{3} (s_{i} \cdot s_{j}) δ (r_{i j}) + \frac{1}{r_{i j}^{3}} [s_{i} \cdot s_{j} - \frac{3 (s_{i} \cdot r_{i j}) (s_{j} \cdot r_{i j})}{r_{i j}^{2}}]}$ is the magnetic moment spin-spin interaction. The first term is called the contact interaction, because it is nonzero only when the particles are at the same position; the second term is the interaction of the classical dipole-dipole type.
${\hat{H}}_{6} = 2 μ_{B} \sum_{i} [H (r_{i}) \cdot s_{i} + \frac{q_{i}}{m_{i} c} A (r_{i}) \cdot {\hat{p}}_{i}]$ is the interaction between spin and orbital magnetic moments with an external magnetic field H.

where: $V = \sum_{i > j} \frac{q_{i} q_{j}}{r_{i j}}$ and $μ_{B} = \frac{e ℏ}{2 m c}$ is the Bohr magneton.

More in Physics

PhysicsScientific field of study UltravioletForm of electromagnetic radiation IonParticle, atom or molecule with a net electrical charge ElectronElementary particle with negative charge Quantum mechanicsDescription of physical properties at the atomic and subatomic scale Digital object identifierISO standard unique string identifier for a digital object