Abstract
We have already seen in Chapter ?? 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. An example of the inherently quantum mechanical nature of electrical properties is provided by the role of virtual intermediate states in the polarizability tensor in Section ??.
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Notes
- 1.
Ø. Burrau, Naturwissenschaften 15, 16 (1927); K. Danske Vidensk. Selsk., Mat.-fys. Medd. 7(14) (1927).
- 2.
W. Heitler & F. London, Z. Phys. 44, 455 (1927).
- 3.
A.H. Wilson, Proc. Roy. Soc. London A 118, 617, 635 (1928); E. Teller, Z. Phys. 61, 458 (1930); E.A. Hylleraas, Z. Phys. 71, 739 (1931); G. Jaffé, Z. Phys. 87, 535 (1934).
- 4.
See e.g. G. Hunter & H.O. Pritchard, J. Chem. Phys. 46, 2146 (1967); M. Aubert, N. Bessis & G. Bessis, Phys. Rev. A 10, 51 (1974); T.C. Scott, M. Aubert-Frécon & J. Grotendorf, Chem. Phys. 324, 323 (2006).
- 5.
B. Grémaud, D. Delande & N. Billy, J. Phys. B 31, 383 (1998); M.M. Cassar & G.W.F. Drake, J. Phys. B 37, 2485 (2004); H. Li, J. Wu, B.-L. Zhou, J.-M. Zhu & Z.-C. Yan, Phys. Rev. A 75, 012504 (2007).
- 6.
We would have to be more careful if we would discuss expectation values, because exchange integrals appear in the expectation values of potential terms, see Section 17.7.
- 7.
M. Born & J.R. Oppenheimer, Annalen Phys. 84, 457 (1927).
- 8.
M. Aubert, N. Bessis & G. Bessis, Phys. Rev. A 10, 51 (1974).
- 9.
T. Kato, Commun. Pure Appl. Math. 10, 151 (1957). See also R.T. Pack & W.B. Brown, J. Chem. Phys. 45, 556 (1966) and Á. Nagy & C. Amovilli, Phys. Rev. A 82, 042510 (2010).
- 10.
We have seen the corresponding one-dimensional equations in (10.1-10.4). However, when comparing equations (19.36) and (19.37) with (10.1-10.4) please keep in mind that the continuous variables κ i play the role of xthere, while the discrete lattice sites ℓ = n i a i compare to the discrete momenta 2πn ∕ ain equations (10.1-10.4), see also (10.8).
- 11.
J. Hubbard, Proc. Roy. Soc. London A 276, 238 (1963), see also M.C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).
- 12.
See e.g. J.E. Hirsch, Phys. Rev. B 31, 4403 (1985); I. Affleck & J.B. Marston, Phys. Rev. B 37, 3774 (1988); Y.M. Vilk & A.-M.S. Tremblay, J. Physique I 7, 1309 (1997). More comprehensive textbook discussions can be found in references [4, 10].
- 13.
You also have to use that the matrix \(\underline{\tilde{{\Omega }}^{2}}(k)\)has a positive semi-definite square root \(\underline{\tilde{\Omega }}(k)\), see Problem 2. Therefore we also have e.g.
$${\sum \nolimits }_{A,B}\hat{{Q}}_{I,A}(k) \cdot {\underline{\tilde{{\Omega }}^{2}}}_{ A,B}(-k) \cdot \hat{ {Q}}_{J,B}(-k) = {\omega }_{I}(k){\omega }_{J}(-k){ \sum \nolimits }_{A}\hat{{Q}}_{I,A}(k)\hat{{Q}}_{J,A}(-k).$$ - 14.
J. Bardeen, L.N. Cooper & J.R. Schrieffer, Phys. Rev. 108, 1175 (1957); see also H. Fröhlich, Phys. Rev. 79, 845 (1950) and J. Bardeen & D. Pines, Phys. Rev. 99, 1140 (1955).
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Dick, R. (2012). Quantum Aspects of Materials II. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8077-9_19
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