Bloch's theorem

In <a href="/facts/Condensed_matter_physics/EBVI5nmL">condensed matter physics</a>, Bloch's theorem states that solutions to the <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a> in a periodic potential can be expressed as <a href="/facts/Plane_wave/0aDUbNhQ">plane waves</a> modulated by <a href="/facts/Periodic_function/Sm8lJQsF">periodic functions</a>. The theorem is named after the Swiss physicist <a href="/facts/Felix_Bloch/yrtqNe9a">Felix Bloch</a>, who discovered the theorem in 1929. Mathematically, they are written

Bloch function

 
 
 
 ψ
 (
 
 r
 
 )
 =
 
 e
 
 i
 
 k
 
 ⋅
 
 r
 
 
 
 u
 (
 
 r
 
 )
 
 
 {\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )}

where 
 
 
 
 
 r
 
 
 
 {\displaystyle \mathbf {r} }
 
 is position, 
 
 
 
 ψ
 
 
 {\displaystyle \psi }
 
 is the <a href="/facts/Wave_function/KgM5ykjY">wave function</a>, 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is a <a href="/facts/Periodic_function/Sm8lJQsF">periodic function</a> with the same periodicity as the crystal, the <a href="/facts/Wave_vector/Jk8wUEft">wave vector</a> 
 
 
 
 
 k
 
 
 
 {\displaystyle \mathbf {k} }
 
 is the <a href="/facts/Crystal_momentum/GzTGXyUj">crystal momentum vector</a>, 
 
 
 
 e
 
 
 {\displaystyle e}
 
 is <a href="/facts/E_(mathematical_constant)/coEtSBk8">Euler's number</a>, and 
 
 
 
 i
 
 
 {\displaystyle i}
 
 is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>.
Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable <a href="/facts/Basis_function/Bt7jqjY2">basis</a> for the <a href="/facts/Wave_functions/KgM5ykjY">wave functions</a> or <a href="/facts/Quantum_states/7jbxd7Dx">states</a> of electrons in <a href="/facts/Crystal/e2R2mk3E">crystalline solids</a>.
The description of electrons in terms of Bloch functions, termed Bloch electrons (or less often Bloch Waves), underlies the concept of <a href="/facts/Electronic_band_structure/W5o7oE2R">electronic band structures</a>.
These eigenstates are written with subscripts as 
 
 
 
 
 ψ
 
 n
 
 k
 
 
 
 
 
 {\displaystyle \psi _{n\mathbf {k} }}
 
, where 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is a discrete index, called the <a href="/facts/Energy_band/W5o7oE2R">band index</a>, which is present because there are many different wave functions with the same 
 
 
 
 
 k
 
 
 
 {\displaystyle \mathbf {k} }
 
 (each has a different periodic component 
 
 
 
 u
 
 
 {\displaystyle u}
 
). Within a band (i.e., for fixed 
 
 
 
 n
 
 
 {\displaystyle n}
 
), 
 
 
 
 
 ψ
 
 n
 
 k
 
 
 
 
 
 {\displaystyle \psi _{n\mathbf {k} }}
 
 varies continuously with 
 
 
 
 
 k
 
 
 
 {\displaystyle \mathbf {k} }
 
, as does its energy. Also, 
 
 
 
 
 ψ
 
 n
 
 k
 
 
 
 
 
 {\displaystyle \psi _{n\mathbf {k} }}
 
 is unique only up to a constant <a href="/facts/Reciprocal_lattice/jYCjcEpH">reciprocal lattice</a> vector 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbf {K} }
 
, or, 
 
 
 
 
 ψ
 
 n
 
 k
 
 
 
 =
 
 ψ
 
 n
 (
 
 k
 +
 K
 
 )
 
 
 
 
 {\displaystyle \psi _{n\mathbf {k} }=\psi _{n(\mathbf {k+K} )}}
 
. Therefore, the wave vector 
 
 
 
 
 k
 
 
 
 {\displaystyle \mathbf {k} }
 
 can be restricted to the first <a href="/facts/Brillouin_zone/V8unhAAl">Brillouin zone</a> of the reciprocal lattice <a href="/facts/Without_loss_of_generality/gVP4dmEe">without loss of generality</a>.

Bloch's theorem open-in-new

Bloch's theorem