Bloch theorem

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. Mathematically, they are written:

where




r



{\displaystyle \mathbf {r} }
is position,



ψ


{\displaystyle \psi }
is the wave function,



u


{\displaystyle u}
is a periodic function with the same periodicity as the crystal, the wave vector




k



{\displaystyle \mathbf {k} }
is the crystal momentum vector,




e



{\displaystyle \mathrm {e} }
is Euler's number, and




i



{\displaystyle \mathrm {i} }
is the imaginary unit.
Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids.
Named after Swiss physicist Felix Bloch, the description of electrons in terms of Bloch functions, termed Bloch electrons (or less often Bloch Waves), underlies the concept of electronic band structures.
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 band index, 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 reciprocal lattice 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 Brillouin zone of the reciprocal lattice without loss of generality.

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