Skip to main content
Physics LibreTexts

10.7: A continuum of quantum states - quantum numbers in a crystal

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    In a crystal quantum states can be indexed by the Bloch quantum number \(k\). In the LCAO approximation, there is a state for each possible atomic orbital at each value of \( {\bf k}\). As the number of electrons tends to infinity, the allowed \( {\bf k}\)’s form a continuum. The most important application of quantum mechanics in solid state physics is to understand the relationship between energy and momentum. A graph of energy vs momentum is called a band structure.

    States are occupied from the lowest energy upwards according to the exclusion principle. The set of momenta which correspond to the maximum allowed energy form a surface in the 3-d space - the so-called Fermi Surface.

    Shown is the valence “band structure” of dhcp potassium calculated using DFT and pseudopotentials: letters are crystallographic notation for values of \( {\bf k} (\Gamma = (0, 0, 0)\), others are on the edge of the Brilloiun zone). Note the free electron parabola around \(\Gamma\), as \(E = \hbar^2 k^2/2m\). This structure has four layers of atoms per unit cell, so on average there are two bands below the Fermi surface at each \(k\)-point (each is spin degenerate). There are lots of bands crossing the Fermi level, showing that electrons can move from one state to another without requiring energy: potassium is a metal. \(\Gamma\)-A is quite a short distance in k-space, corresponding to waves along the long direction in the unit cell: the band structure appears like a parabola “folded back” on itself.


    Figure \(\PageIndex{1}\): Band structure of potassium, energy scaled so that \(E_F=0\). x-axis labels denote a path through the 3d space of \( {\bf k}\)-vectors.

    This page titled 10.7: A continuum of quantum states - quantum numbers in a crystal is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

    • Was this article helpful?