Bloch's Theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 30/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Bloch_wave


00:02:15 1 Applications and consequences
00:02:29 1.1 Applicability
00:02:41 1.2 Meaning and non-uniqueness of the parameter k
00:02:55 1.3 Detailed example
00:03:34 2 Proof of Bloch's theorem
00:06:03 2.1 Preliminaries: Crystal symmetries, lattice, and reciprocal lattice
00:06:27 2.2 Lemma about translation operators
00:07:44 2.3 Proof
00:09:33 3 History and related equations
00:09:53 4 See also
00:11:40 5 References
00:17:02 6 Further reading
00:17:10 Proof
00:19:08 History and related equations



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SUMMARY
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A Bloch wave (also called Bloch state or Bloch function or Bloch wavefunction), named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. A wavefunction ψ is a Bloch wave if it has the form:

  
    
      
        ψ
        (
        
          r
        
        )
        =
        
          
            e
          
          
            
              i
            
            
              k
            
            ⋅
            
              r
            
          
        
        u
        (
        
          r
        
        )
      
    
    {\displaystyle \psi (\mathbf {r} )=\mathrm {e} ^{\mathrm {i} \mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )}
  where r is position, ψ is the Bloch wave, u is a periodic function with the same periodicity as the crystal, k is a vector of real numbers called the crystal wave vector, e is Euler's number, and i is the imaginary unit. In other words, if we multiply a plane wave by a periodic function, we get a Bloch wave.
Bloch waves are important because of Bloch's theorem, which states that the energy eigenstates for an electron in a crystal can be written as Bloch waves. (More precisely, it states that the electron wave functions in a crystal have a basis consisting entirely of Bloch wave energy eigenstates.) This fact underlies the concept of electronic band structures.
These Bloch wave energy eigenstates are written with subscripts as ψn k, where n is a discrete index, called the band index, which is present because there are many different Bloch waves with the same k (each has a different periodic component u). Within a band (i.e., for fixed n), ψn k varies continuously with k, as does its energy. Also, for any reciprocal lattice vector K, ψn k = ψn,(k+K). Therefore, all distinct Bloch waves occur for k-values within the first Brillouin zone of the reciprocal lattice.