# What is Bloch theorem: Definition and 17 Discussions

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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1. ### A Bloch momentum-space wave functions

Hello, I wonder if it is possible to write Bloch wave functions in momentum space. To be more specific, it would calculate something like (using Sakurai's notation): $$\phi(\vec k) = \langle \vec k | \alpha \rangle$$ Moving forward in a few steps: Expanding:  \phi(\vec k) = \int d^3\vec r...
2. ### A potential well with 3-fold reflection symmetry

When we are talking about Bloch's theorem and also the tight-binding approximation, we can use them to help finding eigenstates of a system. However, I am so confused how to apply it in this case (below is my homework) and don't even know how to start it... All I understand about the Bloch's...
3. E

### B Don't understand proof of Bloch theorem

The potential inside the crystal is periodic ##U(\vec{r} + \vec{R}) = U(\vec{r})## for lattice vectors ##\vec{R} = n_i \vec{a}_i##, ##n_i \in \mathbb{Z}## (where the ##\vec{a}_i## are the crystal basis), and Hamiltonian for an electron in the crystal is ##\hat{H} = \left( -\frac{\hbar^2}{2m}...
4. ### Derivation of the Bloch theorem

hi guys our solid state professor gave us a series of power point slides that contains the derivation of Bloch theorem , but some points is not clear to me , and when i asked him his answer was also not clear : in the first part i understand the he represented both the potential energy and the...
5. ### Can These Equations Be Represented in Bloch Form?

Homework Statement Given: sin(Πx/a)e6Πix/Na and e2Πi/a(7/N+4)x can these equations be represented in Bloch form?[/B] Homework Equations Given that Bloch form can be represented as: Ψ(x) = u(x) eikx[/B] The Attempt at a Solution sin(Πx/a)eikx w/n = 3 and...
6. ### A Bloch theorem proof with V(x)=V(x+ma)

In Grosso's Solid State Physics, chapter 1, page 2, The author said that: Therefore, I plug (4) into (1), and I expect that I can get the following relationship, which proves that ##H\left|W_{k}(x)\right\rangle## belongs to the subspace ##\mathbf{S}_{k}## of plane waves of wavenumbers...
7. ### A How does parity transformation affect Bloch states?

Hello! I want to know how does a parity transformation affect Bloch states! I always knew that parity takes the position vector to minus itself (in odd number of dimensions), but I have read that it also takes the Bloch wave vector to minus itself but I have not found a satisfactory proof of...
8. ### MHB The Integers as an Ordered Integral Domain .... Bloch Theorem 1.4.6 ....

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...
9. ### MHB Properties of "less Than" & "Less Than or Equals" - Bloch Theorem 1.2.9 - Peter

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.2.9 (6) ... Theorem 1.2.9 reads as follows: In the above proof of (6) we read the...
10. ### Electrons in periodic potentials -- lesson

Hi, I need to teach a lesson on electrons in periodic potentials for Bachelor Physics students in just 20 minutes Any ideas on how to organize the lesson (pre-concepts they should know, relevant message and consequences) would be very much appreciated
11. ### Fourier transform of periodic potential in crystal lattice

Homework Statement Homework Equations I'm not sure. The Attempt at a Solution I started on (i) -- this is where I've gotten so far. I am asked to compute the Fourier transform of a periodic potential, ##V(x)=\beta \cos(\frac{2\pi x}{a})## such that...
12. ### I Bloch's theorem infinite system?

Hi, Does anyone know why k has to be real in an infinite system for bloch's theorem. I understand that the wavefunction becomes unphysical in an infinite system as it diverges. Why does that mean k has to be real? f(x)=u(x)exp(ikx)
13. ### How big should a defected supercell be? DFT

When studying defects using plane-wave basis density functional theory, it is necessary to ensure that the size of the supercell in which the defect is located is large enough to ensure that there is no interaction between the defect in question, and the periodically repeated defects that are a...
14. ### Silicon: Lattice spacing <111><100> Thermodynamics and Bloch theorem

Hi, I had a question that maybe someone might know, and that although I have been researching it I am not finding enough information on the web that would solve the issue. (it's the end of the month, too... and being broke and in a hurry is a problem too...) The project is aimed at...
15. ### Simultaneous diagonalisation of non-hermitian operator (Bloch theorem)

A valuable math result for quantum mechanics is that if two hermitian operators (physical observables) commute, then a simultaneous basis of eigenvectors exists. Nevertheless, there are cases in which two operators commute without being both hermitians -- a really common one is when one operator...
16. ### Uncovering the Mysteries of Bloch Theorem

Hi guys I have these question, please someone help me to answer? What is Bloch theorem? Why we use it? Explain? What is the consequence of this theorem?
17. ### Understanding the Bloch Theorem

This is not any homework problem but just something I don't understand. The Bloch theorem states that \psi(\textbf{r}+\textbf{R})=e^{i\textbf{k}\cdot \textbf{R}}\psi(\textbf{r}) Now the k is a vector in the reciprocal lattice (usually in the first Brillouin zone), which is defined as the set...