Index
Chapter 7 · Item 7.2
Interactions between angular momenta
Hamiltonians that motivate the addition of angular momenta
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Guided reading

The section introduces interaction Hamiltonians before the algebra of addition. The purpose is physical: once a Hamiltonian contains more than one angular momentum, a basis built from only one momentum is not enough.

Isotropic and anisotropic exchange

The isotropic Heisenberg interaction is written as

\[\hat H=-J\vec{\hat S}_1\cdot\vec{\hat S}_2.\]

For \(J>0\), the spins are ordered parallel; for \(J<0\), they are ordered antiparallel. The parameter \(J\) regulates the intensity of the interaction.

The Ising Hamiltonian is the \(z\)-axis anisotropic case:

\[\hat H=-J\hat S_{1z}\hat S_{2z}.\]

The \(XY\) interaction keeps the \(x\) and \(y\) spin components:

\[\hat H=-J(\hat S_{1x}\hat S_{2x}+\hat S_{1y}\hat S_{2y}).\]
Antisymmetric and spin-orbit terms

The antisymmetric interaction is written as

\[\hat H=\vec d\cdot(\vec{\hat S}_1\times\vec{\hat S}_2).\]

The vector \(\vec d\) regulates the intensity of this interaction; this term is also known as the Dzialoshinsky-Moriya interaction.

The spin-orbit interaction couples the orbital angular momentum of an electron with its own spin:

\[\hat H=\zeta\vec{\hat S}\cdot\vec{\hat L}.\]
Local magnetocrystalline anisotropy

This Hamiltonian has anisotropic character and originates from the spin-orbit interaction. It describes the interaction of a spin with the crystal lattice:

\[\hat H=D\left(\hat S_z^2-\frac{1}{3}\hat S^2\right)+E(\hat S_x^2-\hat S_y^2).\]

The parameter \(D\) is axial and the parameter \(E\) is rhombic.

Zeeman interaction

The Zeeman case represents the interaction between spin and/or orbital angular momenta with a magnetic field:

\[\hat H=\mu_B(2\vec{\hat S}+\vec{\hat L})\cdot\vec B.\]

This term returns later when the chapter applies angular-momentum addition to magnetic moments and the Zeeman effect.

In real units, the Hamiltonians above have the prefactors described in Chapter 7. The displayed forms use dimensionless angular momenta, following the section.
Exercise-ready boundary

This page supports compact guided exercises on: the angular-momentum interactions that motivate the addition problem.

  • Use from this page: the definitions, physical setup, highlighted equations and quantum-number restrictions shown here.
  • Keep in Chapter 7: the complete demonstrations, long examples, historical notes and extended exercise solutions.
  • Boundary: do not introduce notation, Hamiltonians, coupling schemes or selection rules outside the sequence presented here.
Practice anchors

Use these anchors only within the material introduced on this page.

  • Focus: the angular-momentum interactions that motivate the addition problem.
  • Conceptual check: identify the basis or angular-momentum interaction before applying the equation.
  • Key equation 1:
    \[\hat H=-J\vec{\hat S}_1\cdot\vec{\hat S}_2\]
  • Key equation 2:
    \[\hat H_Z=\mu_B(2\vec{\hat S}+\vec{\hat L})\cdot\vec B\]
  • Typical task: apply one central equation and state what basis, quantum numbers or physical consequence it uses.
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Source note: Original auxiliary summary for this book-app, based on Chapter 7 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for complete demonstrations, examples and exercises.