Abstract
We have already seen in Chap. 10 that basic properties of electron states in materials are determined by quantum effects. This impacts all properties of materials, including their mechanical properties, electrical and thermal conductivities, and optical properties. Examples of the inherently quantum mechanical nature of electromagnetic properties of materials are provided by the role of virtual intermediate states in the polarizability tensor in Sect. 15.3, and the importance of exchange interactions for magnetism in materials, as discussed in Sect. 17.7.
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Notes
- 1.
This assumes that we use Coulomb gauge for the electromagnetic fields, see Chap. 18. Otherwise, we would have to use exchange of virtual longitudinal photons for the description of the dominant electromagnetic interaction between non-relativistic charged particles, which would be clumsy. Furthermore, strictly speaking, the assertion that the dominant interaction between non-relativistic nuclei is due to their Coulomb repulsion assumes that the nuclei are at least 1 fm apart. Otherwise they would be dominated by the nuclear force. Sections 23.2 and 23.4 contain explicit demonstrations that electromagnetic interactions of non-relativistic particles in Coulomb gauge are dominated by the Coulomb potential.
- 2.
We would have to be more careful if we would also discuss expectation values, because exchange integrals appear in the expectation values of potential terms, see Sect. 17.7.
- 3.
We have seen the corresponding one-dimensional equations in (10.1)–(10.5). However, when comparing Eqs. (20.68) and (20.69) with (10.1)–(10.5) please keep in mind that the continuous variables κi play the role of x there, while the discrete lattice sites ℓ = niai compare to the discrete momenta 2πn∕a in Eqs. (10.1)–(10.5), see also (10.19).
- 4.
- 5.
Note that we use the symbol \(\tilde {\boldsymbol {Q}}^+\) both for the hermitian adjoint row vector and for the complex conjugate column vector, since the position in a scalar product or tensor product makes it clear which of the two versions is meant. This is in agreement with the common convention to not explicitly mark the transposed row vectors as \(\tilde {\boldsymbol {Q}}^T\), see e.g. Eq. (20.185).
- 6.
You also have to use that the matrix \( \underline {\tilde {\Omega }^2}(\boldsymbol {k})\) has a positive semi-definite square root \( \underline {\tilde {\Omega }}(\boldsymbol {k})\), see Problem 20.3. Therefore we also have e.g.
$$\displaystyle \begin{aligned} \sum_{A,B}\hat{\boldsymbol{Q}}_{I,A}(\boldsymbol{k})\cdot \underline{\tilde{\Omega}^2}_{A,B}(-\,\boldsymbol{k})\cdot \hat{\boldsymbol{Q}}_{J,B}(-\,\boldsymbol{k}) =\omega_I(\boldsymbol{k})\omega_J(-\,\boldsymbol{k})\sum_{A} \hat{\boldsymbol{Q}}_{I,A}(\boldsymbol{k})\hat{\boldsymbol{Q}}_{J,A}(-\,\boldsymbol{k}). \end{aligned} $$(20.200) - 7.
The Hamiltonian H0 + He−q is often addressed as the second quantized Hamiltonian of the electron-phonon system, but it is actually a mixed (“1.5th”) level of quantization, because the electrons are treated at second quantized level, but the phonons correspond to a first quantized description of the vibrations of the ion cores.
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Dick, R. (2020). Quantum Aspects of Materials II. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-57870-1_20
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