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Quantum Mechanics

Quantum Matter Formulas

These formula cards collect compact expressions used in single-particle band theory, effective-mass approximations, density-of-states estimates, and topological band diagnostics. Use Quantum Matter Reference and Data when a card must be translated into a convention- and provenance-complete material-facing record; full physical models, material claims, and response derivations remain with their canonical quantum-matter and condensed-matter owners.

Cards

Section titled “Cards”
CardCore FormulaMain Assumption
Bloch Theoremψnk(r)=eik⋅runk(r)\psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r)Single-particle Hamiltonian with lattice-translation symmetry.
Density of StatesD(E)=∑αδ(E−Eα)D(E)=\sum_\alpha\delta(E-E_\alpha)State-counting convention and degeneracy factors are fixed.
Effective Mass(m−1)ij=ℏ−2∂i∂jEn(m^{-1})_{ij}=\hbar^{-2}\partial_i\partial_j E_nExpansion near a smooth band extremum or local wave-packet region.
Berry CurvatureΩij(n)=i Tr⁡(Pn[∂iPn,∂jPn])\Omega_{ij}^{(n)}=i\,\operatorname{Tr}(P_n[\partial_iP_n,\partial_jP_n])Isolated band or isolated band subspace.
Chern NumberCn=(2π)−1∫BZΩn d2kC_n=(2\pi)^{-1}\int_{\mathrm{BZ}}\Omega_n\,d^2kTwo-dimensional isolated band over an oriented Brillouin-zone torus.

Reading Rule

Section titled “Reading Rule”

Quantum-matter formulas hide assumptions about periodicity, normalization volume, Brillouin-zone convention, spin and valley degeneracy, band isolation, gauge choice, and orientation. State these before using a compact formula in a calculation or comparison.

References

Section titled “References”
  • N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
  • C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, 2004.
  • S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2013.
  • D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties”, Reviews of Modern Physics 82, 1959-2007, 2010.