Bloch's theorem

The most common example of Bloch's theorem is describing electrons in a crystal, especially in characterizing the crystal's electronic properties, such as electronic band structure. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric structure in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the various forms of the dynamical theory of diffraction.

File:BlochWaves1D.svg vector. In all plots, blue is real part and red is imaginary part.]]

Suppose an electron is in a Bloch state

$$\backslash psi\; (\; \backslash mathbf\{r\}\; )\; =\; e^\{\; i\; \backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\; \}\; u\; (\; \backslash mathbf\{r\}\; )\; ,$$

where {{math|u}} is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by $\backslash psi$, not {{math|k}} or {{math|u}} directly. This is important because {{math|k}} and {{math|u}} are not unique. Specifically, if $\backslash psi$ can be written as above using {{math|k}}, it can also be written using {{math|(k + K)}}, where {{math|K}} is any reciprocal lattice vector (see figure at right). Therefore, wave vectors that differ by a reciprocal lattice vector are equivalent, in the sense that they characterize the same set of Bloch states.

The first Brillouin zone is a restricted set of values of {{math|k}} with the property that no two of them are equivalent, yet every possible {{math|k}} is equivalent to one (and only one) vector in the first Brillouin zone. Therefore, if we restrict {{math|k}} to the first Brillouin zone, then every Bloch state has a unique {{math|k}}. Therefore, the first Brillouin zone is often used to depict all of the Bloch states without redundancy, for example in a band structure, and it is used for the same reason in many calculations.

When {{math|k}} is multiplied by the reduced Planck constant, it equals the electron's crystal momentum. Related to this, the group velocity of an electron can be calculated based on how the energy of a Bloch state varies with {{math|k}}; for more details see crystal momentum.

For a detailed example in which the consequences of Bloch's theorem are worked out in a specific situation, see the article Particle in a one-dimensional lattice (periodic potential).

{{math theorem | name = Bloch's theorem | math_statement =

For electrons in a perfect crystal, there is a basis of wave functions with the following two properties:

• each of these wave functions is an energy eigenstate,
• each of these wave functions is a Bloch state, meaning that this wave function $\backslash psi$ can be written in the form $\backslash ;\backslash psi(\backslash mathbf\{r\})\; =\; e^\{i\backslash mathbf\{k\}\backslash cdot\backslash mathbf\{r\}\}\; u(\backslash mathbf\{r\}),$ where $u(\backslash mathbf\{r\})$ has the same periodicity as the atomic structure of the crystal, such that $$u\_\{\backslash mathbf\{k\}\}(\backslash mathbf\{x\})\; =\; u\_\{\backslash mathbf\{k\}\}(\backslash mathbf\{x\}\; +\; \backslash mathbf\{n\}\; \backslash cdot\; \backslash mathbf\{a\}).$$

}}

A second and equivalent way to state the theorem is the following{{cite book |last=Ziman |first=J. M. |date=1972 |edition=2nd |title=Principles of the theory of solids |publisher=Cambridge University Press |isbn=0521297338 |pages=17–20}}

{{math theorem | name = Bloch's theorem | math_statement =

For any wave function that satisfies the Schrödinger equation and for a translation of a lattice vector $\backslash mathbf\{a\}$, there exists at least one vector $\backslash mathbf\{k\}$ such that:

$$\backslash psi\_\{\backslash mathbf\{k\}\}(\backslash mathbf\{x\}+\backslash mathbf\{a\})\; =\; e^\{i\backslash mathbf\{k\}\backslash cdot\backslash mathbf\{a\}\}\backslash psi\_\{\backslash mathbf\{k\}\}(\backslash mathbf\{x\}).$$

}}

Bloch's theorem, being a statement about lattice periodicity, all the symmetries in this proof are encoded as translation symmetries of the wave function itself.

{{math proof | title = Proof Using lattice periodicity | proof =

Source:{{Harvnb|Ashcroft|Mermin|1976|p=134}}

The defining property of a crystal is translational symmetry, which means that if the crystal is shifted an appropriate amount, it winds up with all its atoms in the same places. (A finite-size crystal cannot have perfect translational symmetry, but it is a useful approximation.)

A three-dimensional crystal has three primitive lattice vectors {{math|a₁, a₂, a₃}}. If the crystal is shifted by any of these three vectors, or a combination of them of the form

$$n\_1\; \backslash mathbf\{a\}\_1\; +\; n\_2\; \backslash mathbf\{a\}\_2\; +\; n\_3\; \backslash mathbf\{a\}\_3,$$

where {{mvar|n_i}} are three integers, then the atoms end up in the same set of locations as they started.

Another helpful ingredient in the proof is the reciprocal lattice vectors. These are three vectors {{math|b₁, b₂, b₃}} (with units of inverse length), with the property that {{math|1=a_i · b_i = 2π}}, but {{math|1=a_i · b_j = 0}} when {{math|i ≠ j}}. (For the formula for {{math|b_i}}, see reciprocal lattice vector.)

Let $\backslash hat\{T\}\_\{n\_1,n\_2,n\_3\}$ denote a translation operator that shifts every wave function by the amount {{math|n₁a₁ + n₂a₂ + n₃a₃}} (as above, {{mvar|n_j}} are integers). The following fact is helpful for the proof of Bloch's theorem:

{{math theorem | name = Lemma | math_statement = If a wave function {{mvar|ψ}} is an eigenstate of all of the translation operators (simultaneously), then {{mvar|ψ}} is a Bloch state.}}

{{math proof | title = Proof of Lemma | proof = Assume that we have a wave function {{mvar|ψ}} which is an eigenstate of all the translation operators. As a special case of this,

$$\backslash psi(\backslash mathbf\{r\}+\backslash mathbf\{a\}\_j)\; =\; C\_j\; \backslash psi(\backslash mathbf\{r\})$$

for {{math|1=j = 1, 2, 3}}, where {{mvar|C_j}} are three numbers (the eigenvalues) which do not depend on {{math|r}}. It is helpful to write the numbers {{mvar|C_j}} in a different form, by choosing three numbers {{math|θ₁, θ₂, θ₃}} with {{math|1=e^2πiθ_j = C_j}}:

$$\backslash psi(\backslash mathbf\{r\}+\backslash mathbf\{a\}\_j)\; =\; e^\{2\; \backslash pi\; i\; \backslash theta\_j\}\; \backslash psi(\backslash mathbf\{r\})$$

Again, the {{mvar|θ_j}} are three numbers which do not depend on {{math|r}}. Define {{math|1=k = θ₁b₁ + θ₂b₂ + θ₃b₃}}, where {{math|b_j}} are the reciprocal lattice vectors (see above). Finally, define

$$u(\backslash mathbf\{r\})\; =\; e^\{-i\; \backslash mathbf\{k\}\backslash cdot\backslash mathbf\{r\}\}\; \backslash psi(\backslash mathbf\{r\})\backslash ,.$$

Then

$$\backslash begin\{align\}$$

u(\mathbf{r} + \mathbf{a}_j) &= e^{-i\mathbf{k} \cdot (\mathbf{r} + \mathbf{a}_j)} \psi(\mathbf{r}+\mathbf{a}_j) \\

&= \big( e^{-i\mathbf{k} \cdot \mathbf{r}} e^{-i\mathbf{k}\cdot \mathbf{a}_j} \big) \big( e^{2\pi i \theta_j} \psi(\mathbf{r}) \big) \\

&= e^{-i\mathbf{k} \cdot \mathbf{r}} e^{-2\pi i \theta_j} e^{2\pi i \theta_j} \psi(\mathbf{r}) \\

&= u(\mathbf{r}).

\end{align}

This proves that {{mvar|u}} has the periodicity of the lattice. Since $\backslash psi(\backslash mathbf\{r\})\; =\; e^\{i\; \backslash mathbf\{k\}\backslash cdot\backslash mathbf\{r\}\}\; u(\backslash mathbf\{r\}),$ that proves that the state is a Bloch state.}}

Finally, we are ready for the main proof of Bloch's theorem which is as follows.

As above, let $\backslash hat\{T\}\_\{n\_1,n\_2,n\_3\}$ denote a translation operator that shifts every wave function by the amount {{math|n₁a₁ + n₂a₂ + n₃a₃}}, where {{mvar|n_i}} are integers. Because the crystal has translational symmetry, this operator commutes with the Hamiltonian operator. Moreover, every such translation operator commutes with every other. Therefore, there is a simultaneous eigenbasis of the Hamiltonian operator and every possible $\backslash hat\{T\}\_\{n\_1,n\_2,n\_3\}\; \backslash !$ operator. This basis is what we are looking for. The wave functions in this basis are energy eigenstates (because they are eigenstates of the Hamiltonian), and they are also Bloch states (because they are eigenstates of the translation operators; see Lemma above).

}}

In this proof all the symmetries are encoded as commutation properties of the translation operators

{{math proof | title = Proof using operators | proof =

Source:{{Harvnb|Ashcroft|Mermin|1976|p=137}}

We define the translation operator

$$\backslash begin\{align\}$$

\hat{\mathbf{T}}_{\mathbf{n}}\psi(\mathbf{r})&= \psi(\mathbf{r}+\mathbf{T}_{\mathbf{n}}) \\

&= \psi(\mathbf{r}+n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_3) \\

&= \psi(\mathbf{r}+\mathbf{A}\mathbf{n})

\end{align}

with

\mathbf{A} = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \end{bmatrix}, \quad \mathbf{n} = \begin{pmatrix} n_1 \\ n_2 \\ n_3 \end{pmatrix}

We use the hypothesis of a mean periodic potential

$$U(\backslash mathbf\{x\}+\backslash mathbf\{T\}\_\{\backslash mathbf\{n\}\})=\; U(\backslash mathbf\{x\})$$

and the independent electron approximation with an Hamiltonian

$$\backslash hat\{H\}=\backslash frac\{\backslash hat\{\backslash mathbf\{p\}\}^2\}\{2m\}+U(\backslash mathbf\{x\})$$

Given the Hamiltonian is invariant for translations it shall commute with the translation operator

$$[\backslash hat\{H\},\backslash hat\{\backslash mathbf\{T\}\}\_\{\backslash mathbf\{n\}\}]\; =\; 0$$

and the two operators shall have a common set of eigenfunctions.

Therefore, we start to look at the eigen-functions of the translation operator:

$$\backslash hat\{\backslash mathbf\{T\}\}\_\{\backslash mathbf\{n\}\}\backslash psi(\backslash mathbf\{x\})=\backslash lambda\_\{\backslash mathbf\{n\}\}\backslash psi(\backslash mathbf\{x\})$$

Given $\backslash hat\{\backslash mathbf\{T\}\}\_\{\backslash mathbf\{n\}\}$ is an additive operator

\hat{\mathbf{T}}_{\mathbf{n}_1} \hat{\mathbf{T}}_{\mathbf{n}_2}\psi(\mathbf{x}) =

\psi(\mathbf{x} + \mathbf{A} \mathbf{n}_1 + \mathbf{A} \mathbf{n}_2) = \hat{\mathbf{T}}_{\mathbf{n}_1 + \mathbf{n}_2} \psi(\mathbf{x})

If we substitute here the eigenvalue equation and dividing both sides for $\backslash psi(\backslash mathbf\{x\})$ we have

\lambda_{\mathbf{n}_1} \lambda_{\mathbf{n}_2} =

\lambda_{\mathbf{n}_1 + \mathbf{n}_2}

This is true for

$$\backslash lambda\_\{\backslash mathbf\{n\}\}\; =\; e^\{s\; \backslash mathbf\{n\}\; \backslash cdot\; \backslash mathbf\{a\}\; \}$$

where $s\; \backslash in\; \backslash Complex$ if we use the normalization condition over a single primitive cell of volume V

1 = \int_V |\psi(\mathbf{x})|^2 d \mathbf{x} =

\int_V \left|\hat\mathbf{T}_\mathbf{n} \psi(\mathbf{x})\right|^2 d \mathbf{x} =

|\lambda_{\mathbf{n}}|^2 \int_V |\psi(\mathbf{x})|^2 d \mathbf{x}

and therefore

$$1\; =\; |\backslash lambda\_\{\backslash mathbf\{n\}\}|^2$$ and $$s\; =\; i\; k$$ where $k\; \backslash in\; \backslash mathbb\{R\}$. Finally,

\mathbf{\hat{T}_n}\psi(\mathbf{x})=

\psi(\mathbf{x} + \mathbf{n} \cdot \mathbf{a} ) =

e^{i k \mathbf{n} \cdot \mathbf{a} }\psi(\mathbf{x})

,

which is true for a Bloch wave i.e. for $\backslash psi\_\{\backslash mathbf\{k\}\}(\backslash mathbf\{x\})\; =\; e^\{i\; \backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\}\; \}\; u\_\{\backslash mathbf\{k\}\}(\backslash mathbf\{x\})$ with $u\_\{\backslash mathbf\{k\}\}(\backslash mathbf\{x\})\; =\; u\_\{\backslash mathbf\{k\}\}(\backslash mathbf\{x\}\; +\; \backslash mathbf\{A\}\backslash mathbf\{n\})$

}}

Apart from the group theory technicalities this proof is interesting because it becomes clear how to generalize the Bloch theorem for groups that are not only translations.

This is typically done for space groups which are a combination of a translation and a point group and it is used for computing the band structure, spectrum and specific heats of crystals given a specific crystal group symmetry like FCC or BCC and eventually an extra basis.{{rp|pp=365–367}}The vibrational spectrum and specific heat of a face centered cubic crystal, Robert B. Leighton [https://authors.library.caltech.edu/47755/1/LEIrmp48.pdf]

In this proof it is also possible to notice how it is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian.

{{math proof

| title = Proof with character theory{{Cite web|last=Dresselhaus|first=M. S. | author-link=Mildred Dresselhaus |date=2002|title=Applications of Group Theory to the Physics of Solids|url=http://web.mit.edu/course/6/6.734j/www/group-full02.pdf | url-status=live | archive-url=https://web.archive.org/web/20191101074639/http://web.mit.edu/course/6/6.734j/www/group-full02.pdf | archive-date=1 November 2019|access-date=12 September 2020 | website=MIT}}{{rp|pp=345–348}}

| proof =

All translations are unitary and abelian.

Translations can be written in terms of unit vectors

$$\backslash boldsymbol\{\backslash tau\}\; =\; \backslash sum\_\{i=1\}^3\; n\_i\; \backslash mathbf\{a\}\_i$$

We can think of these as commuting operators

\hat{\boldsymbol{\tau}} =

\hat{\boldsymbol{\tau}}_1 \hat{\boldsymbol{\tau}}_2 \hat{\boldsymbol{\tau}}_3

where $$\backslash hat\{\backslash boldsymbol\{\backslash tau\}\}\_i\; =\; n\_i\; \backslash hat\{\backslash mathbf\{a\}\}\_i$$

The commutativity of the $\backslash hat\{\backslash boldsymbol\{\backslash tau\}\}\_i$ operators gives three commuting cyclic subgroups (given they can be generated by only one element) which are infinite, 1-dimensional and abelian. All irreducible representations of abelian groups are one dimensional.{{cite web |last=Roy |first=Ricky |title=Representation Theory |date=May 2, 2010 |url=http://buzzard.pugetsound.edu/courses/2010spring/projects/roy-representation-theory-ups-434-2010.pdf |publisher=University of Puget Sound}}

Given they are one dimensional the matrix representation and the character are the same. The character is the representation over the complex numbers of the group or also the trace of the representation which in this case is a one dimensional matrix.

All these subgroups, given they are cyclic, they have characters which are appropriate roots of unity. In fact they have one generator $\backslash gamma$ which shall obey to $\backslash gamma^n\; =\; 1$, and therefore the character $\backslash chi(\backslash gamma)^n\; =\; 1$. Note that this is straightforward in the finite cyclic group case but in the countable infinite case of the infinite cyclic group (i.e. the translation group here) there is a limit for $n\; \backslash to\; \backslash infty$ where the character remains finite.

Given the character is a root of unity, for each subgroup the character can be then written as

$$\backslash chi\_\{k\_1\}(\backslash hat\{\backslash boldsymbol\{\backslash tau\}\}\_1\; (n\_1,a\_1))\; =\; e^\{i\; k\_1\; n\_1\; a\_1\}$$

If we introduce the Born–von Karman boundary condition on the potential:

$$V\; \backslash left(\backslash mathbf\; \{r\}\; +\backslash sum\_i\; N\_\{i\}\; \backslash mathbf\; \{a\}\_\{i\}\backslash right)\; =\; V\; (\backslash mathbf\; \{r\}\; +\backslash mathbf\{L\})\; =\; V\; (\backslash mathbf\; \{r\}\; )$$

where L is a macroscopic periodicity in the direction $\backslash mathbf\{a\}$ that can also be seen as a multiple of $a\_i$ where $\backslash mathbf\{L\}\; =\; \backslash sum\_i\; N\_\{i\}\backslash mathbf\; \{a\}\_\{i\}$

This substituting in the time independent Schrödinger equation with a simple effective Hamiltonian

$$\backslash hat\{H\}\; =\; -\backslash frac\{\backslash hbar^2\}\{2m\}\backslash nabla^2\; +\; V(\backslash mathbf\{r\})$$

induces a periodicity with the wave function:

$$\backslash psi\; \backslash left(\backslash mathbf\; \{r\}\; +\; \backslash sum\_i\; N\_\{i\}\backslash mathbf\; \{a\}\_\{i\}\backslash right)\; =\; \backslash psi\; (\backslash mathbf\; \{r\}\; )$$

And for each dimension a translation operator with a period L

$$\backslash hat\{P\}\_\{\backslash varepsilon|\backslash tau\_i\; +\; L\_i\}\; =\; \backslash hat\{P\}\_\{\backslash varepsilon|\backslash tau\_i\}$$

From here we can see that also the character shall be invariant by a translation of $L\_i$:

$$e^\{i\; k\_1\; n\_1\; a\_1\}\; =\; e^\{i\; k\_1\; (\; n\_1\; a\_1\; +\; L\_1)\}$$

and from the last equation we get for each dimension a periodic condition:

$$k\_1\; n\_1\; a\_1\; =\; k\_1\; (\; n\_1\; a\_1\; +\; L\_1)\; -\; 2\; \backslash pi\; m\_1$$

where $m\_1\; \backslash in\; \backslash mathbb\{Z\}$ is an integer and $k\_1=\backslash frac\; \{2\; \backslash pi\; m\_1\}\{L\_1\}$

The wave vector $k\_1$ identify the irreducible representation in the same manner as $m\_1$, and $L\_1$ is a macroscopic periodic length of the crystal in direction $a\_1$. In this context, the wave vector serves as a quantum number for the translation operator.

We can generalize this for 3 dimensions

$\backslash chi\_\{k\_1\}(n\_1,a\_1)\backslash chi\_\{k\_2\}(n\_2,a\_2)\backslash chi\_\{k\_3\}(n\_3,a\_3)\; =\; e^\{i\backslash mathbf\{k\}\; \backslash cdot\; \backslash boldsymbol\{\backslash tau\}\}$

and the generic formula for the wave function becomes:

$$\backslash hat\{P\}\_R\backslash psi\_j\; =\; \backslash sum\_\{\backslash alpha\}\; \backslash psi\_\{\backslash alpha\}\; \backslash chi\_\{\backslash alpha\; j\}(R)$$

i.e. specializing it for a translation

$$\backslash hat\{P\}\_\{\backslash varepsilon|\backslash boldsymbol\{\backslash tau\}\}\; \backslash psi(\backslash mathbf\{r\})\; =\backslash psi(\backslash mathbf\{r\})\; e^\{i\; \backslash mathbf\{k\}\; \backslash cdot\; \backslash boldsymbol\{\backslash tau\}\}\; =\; \backslash psi(\backslash mathbf\{r\}\; +\; \backslash boldsymbol\{\backslash tau\})$$

and we have proven Bloch’s theorem.

}}

In the generalized version of the Bloch theorem, the Fourier transform, i.e. the wave function expansion, gets generalized from a discrete Fourier transform which is applicable only for cyclic groups, and therefore translations, into a character expansion of the wave function where the characters are given from the specific finite point group.

Also here is possible to see how the characters (as the invariants of the irreducible representations) can be treated as the fundamental building blocks instead of the irreducible representations themselves.Group Representations

and Harmonic Analysis from Euler to Langlands, Part II [https://web.archive.org/web/20190305032503/http://pdfs.semanticscholar.org/ce73/4a226c19a412148dadbc2094fb75a7a609a4.pdf]

If we apply the time-independent Schrödinger equation to the Bloch wave function we obtain

$$\backslash hat\{H\}\_\backslash mathbf\{k\}\; u\_\backslash mathbf\{k\}(\backslash mathbf\{r\})\; =$$

\left[ \frac{\hbar^2}{2m} \left( -i \nabla + \mathbf{k} \right)^2 + U(\mathbf{r}) \right] u_\mathbf{k}(\mathbf{r}) =

\varepsilon_\mathbf{k} u_\mathbf{k}(\mathbf{r})

with boundary conditions

$$u\_\backslash mathbf\{k\}(\backslash mathbf\{r\})\; =\; u\_\backslash mathbf\{k\}(\backslash mathbf\{r\}\; +\; \backslash mathbf\{R\})$$

Given this is defined in a finite volume we expect an infinite family of eigenvalues; here $\{\backslash mathbf\{k\}\}$ is a parameter of the Hamiltonian and therefore we arrive at a "continuous family" of eigenvalues $\backslash varepsilon\_n(\backslash mathbf\{k\})$ dependent on the continuous parameter $\{\backslash mathbf\{k\}\}$ and thus at the basic concept of an electronic band structure.

{{math proof

| title = Proof{{Harvnb|Ashcroft|Mermin|1976|p=140}}

| proof =

E_\mathbf{k} \left(e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})\right)

=

\left[\frac{- \hbar^2}{2m} \nabla^2 + U(\mathbf{x} ) \right]

\left(e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})\right)

We remain with

$$\backslash begin\{align\}$$

E_\mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})

&=

\frac{- \hbar^2}{2m} \nabla \cdot

\left( i \mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})

+ e^{i \mathbf{k} \cdot \mathbf{x} } \nabla u_\mathbf{k}(\mathbf{x}) \right)

+ U(\mathbf{x}) e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) \\[1.2ex]

E_\mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})

&=

\frac{- \hbar^2}{2m}

\left( i \mathbf{k} \cdot

\left( i \mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})

+ e^{i \mathbf{k} \cdot \mathbf{x} } \nabla u_\mathbf{k}(\mathbf{x}) \right)

+ i \mathbf{k} \cdot e^{i \mathbf{k} \cdot \mathbf{x} } \nabla u_\mathbf{k}(\mathbf{x})

+ e^{i \mathbf{k} \cdot \mathbf{x} } \nabla^2 u_\mathbf{k}(\mathbf{x}) \right)

+ U(\mathbf{x}) e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) \\[1.2ex]

E_\mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})

&=

\frac{ \hbar^2}{2m}

\left(\mathbf{k}^2 e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})

- 2i \mathbf{k} \cdot e^{i \mathbf{k} \cdot \mathbf{x} } \nabla u_\mathbf{k}(\mathbf{x})

- e^{i \mathbf{k} \cdot \mathbf{x} } \nabla^2 u_\mathbf{k}(\mathbf{x}) \right)

+ U(\mathbf{x}) e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) \\[1.2ex]

E_\mathbf{k} u_\mathbf{k}(\mathbf{x})

&=

\frac{ \hbar^2}{2m}

\left(-i \nabla + \mathbf{k}\right)^2 u_\mathbf{k}(\mathbf{x})

+ U(\mathbf{x}) u_\mathbf{k}(\mathbf{x})

\end{align}

}}

This shows how the effective momentum can be seen as composed of two parts,

$$\backslash hat\{\backslash mathbf\{p\}\}\_\backslash text\{eff\}\; =\; -i\; \backslash hbar\; \backslash nabla\; +\; \backslash hbar\; \backslash mathbf\{k\}\; ,$$

a standard momentum $-i\; \backslash hbar\; \backslash nabla$ and a crystal momentum $\backslash hbar\; \backslash mathbf\{k\}$. More precisely the crystal momentum is not a momentum but it stands for the momentum in the same way as the electromagnetic momentum in the minimal coupling, and as part of a canonical transformation of the momentum.

For the effective velocity we can derive

{{Equation box 1

|indent=:

|title=mean velocity of a Bloch electron

|equation=$\backslash frac\{\backslash partial\; \backslash varepsilon\_n\}\{\backslash partial\; \backslash mathbf\{k\}\}\; =\; \backslash frac\; \{\backslash hbar^2\}\{m\}\; \backslash int$

d\mathbf{r}\, \psi^{*}_{n\mathbf{k}} (-i \nabla)\psi_{n\mathbf{k}} = \frac {\hbar}{m}\langle\hat{\mathbf{p}}\rangle = \hbar \langle\hat{\mathbf{v}}\rangle

|cellpadding

|border

|border colour = rgb(80,200,120)

|background colour = rgb(80,200,120,10%)}}

{{math proof

| title = Proof{{Harvnb|Ashcroft|Mermin|1976|p=765 Appendix E}}

| proof =

We evaluate the derivatives $\backslash frac\{\backslash partial\; \backslash varepsilon\_n\}\{\backslash partial\; \backslash mathbf\{k\}\}$ and $\backslash frac\{\backslash partial^2\; \backslash varepsilon\_n(\backslash mathbf\{k\})\}\{\backslash partial\; k\_i\; \backslash partial\; k\_j\}$

given they are the coefficients of the following expansion in {{math|q}} where {{math|q}} is considered small with respect to {{math|k}}

\varepsilon_n(\mathbf{k} + \mathbf{q}) = \varepsilon_n(\mathbf{k}) +

\sum_i \frac{\partial \varepsilon_n}{\partial k_i} q_i +

\frac{1}{2} \sum_{ij} \frac{\partial^2 \varepsilon_n}{\partial k_i \partial k_j} q_i q_j +

O(q^3)

Given $\backslash varepsilon\_n(\backslash mathbf\{k\}+\backslash mathbf\{q\})$ are eigenvalues of $\backslash hat\{H\}\_\{\backslash mathbf\{k\}+\backslash mathbf\{q\}\}$

We can consider the following perturbation problem in q:

\hat{H}_{\mathbf{k}+\mathbf{q}} =

\hat{H}_\mathbf{k} +

\frac{\hbar^2}{m} \mathbf{q} \cdot ( -i\nabla + \mathbf{k} ) +

\frac{\hbar^2}{2m} q^2

Perturbation theory of the second order states that

E_n =E^0_n + \int d\mathbf{r}\, \psi^{*}_n \hat{V} \psi_n +

\sum_{n' \neq n} \frac{|\int d\mathbf{r} \,\psi^{*}_n \hat{V} \psi_n|^2}{E^0_n - E^0_{n'}} + ...

To compute to linear order in {{math|q}}

\sum_i \frac{\partial \varepsilon_n}{\partial k_i} q_i =

\sum_i \int d\mathbf{r}\, u_{n\mathbf{k}}^{*} \frac{\hbar^2}{m} ( -i\nabla + \mathbf{k} )_i q_i u_{n\mathbf{k}}

where the integrations are over a primitive cell or the entire crystal, given if the integral

$$\backslash int\; d\backslash mathbf\{r\}\backslash ,\; u\_\{n\backslash mathbf\{k\}\}^\{*\}\; u\_\{n\backslash mathbf\{k\}\}$$

is normalized across the cell or the crystal.

We can simplify over {{math|q}} to obtain

\frac{\partial \varepsilon_n}{\partial \mathbf{k}} =

\frac{\hbar^2}{m} \int d\mathbf{r} \, u_{n\mathbf{k}}^{*}( -i\nabla + \mathbf{k} ) u_{n\mathbf{k}}

and we can reinsert the complete wave functions

\frac{\partial \varepsilon_n}{\partial \mathbf{k}} =

\frac{\hbar^2}{m} \int d\mathbf{r} \, \psi_{n\mathbf{k}}^{*}( -i\nabla) \psi_{n\mathbf{k}}

}}

For the effective mass

{{Equation box 1

|indent=:

|title=effective mass theorem

|equation=$$

\frac{\partial^2 \varepsilon_n(\mathbf{k})}{\partial k_i \partial k_j} =

\frac {\hbar^2}{m} \delta_{ij} +

\left(

\frac {\hbar^2}{m}

\right)^2

\sum_{n' \neq n}

\frac{

\langle n\mathbf{k} | -i \nabla_i | n'\mathbf{k} \rangle

\langle n'\mathbf{k} | -i \nabla_j | n\mathbf{k} \rangle

+

\langle n\mathbf{k} | -i \nabla_j | n'\mathbf{k} \rangle

\langle n'\mathbf{k} | -i \nabla_i | n\mathbf{k} \rangle

}{

\varepsilon_n(\mathbf{k}) - \varepsilon_{n'}(\mathbf{k})

}

|cellpadding

|border

|border colour = rgb(80,200,120)

|background colour = rgb(80,200,120,10%)}}

{{math proof

| title = Proof

| proof =

The second order term

\frac{1}{2} \sum_{ij} \frac{\partial^2 \varepsilon_n}{\partial k_i \partial k_j} q_i q_j =

\frac {\hbar^2}{2m} q^2 +

\sum_{n' \neq n}

\frac{|

\int d\mathbf{r} \,

u_{n\mathbf{k}}^{*}

\frac{\hbar^2}{m} \mathbf{q} \cdot (-i\nabla + \mathbf{k})

u_{n'\mathbf{k}}

|^2}

{\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}}}

Again with $\backslash psi\_\{n\backslash mathbf\{k\}\}\; =|\; n\backslash mathbf\{k\}\backslash rangle\; =\; e^\{i\backslash mathbf\{k\}\backslash mathbf\{x\}\}\; u\_\{n\backslash mathbf\{k\}\}$

\frac{1}{2} \sum_{ij} \frac{\partial^2 \varepsilon_n}{\partial k_i \partial k_j} q_i q_j =

\frac {\hbar^2}{2m} q^2 +

\sum_{n' \neq n}

\frac{|

\langle n\mathbf{k} |

\frac{\hbar^2}{m} \mathbf{q} \cdot (-i\nabla)

| n'\mathbf{k}\rangle

|^2}

{\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}}}

Eliminating $q\_i$ and $q\_j$ we have the theorem

\frac{\partial^2 \varepsilon_n(\mathbf{k})}{\partial k_i \partial k_j} =

\frac {\hbar^2}{m} \delta_{ij} +

\left(

\frac {\hbar^2}{m}

\right)^2

\sum_{n' \neq n}

\frac{

\langle n\mathbf{k} | -i \nabla_i | n'\mathbf{k} \rangle

\langle n'\mathbf{k} | -i \nabla_j | n\mathbf{k} \rangle

+

\langle n\mathbf{k} | -i \nabla_j | n'\mathbf{k} \rangle

\langle n'\mathbf{k} | -i \nabla_i | n\mathbf{k} \rangle

}{

\varepsilon_n(\mathbf{k}) - \varepsilon_{n'}(\mathbf{k})

}

}}

The quantity on the right multiplied by a factor$\backslash frac\{1\}\{\backslash hbar^2\}$ is called effective mass tensor $\backslash mathbf\{M\}(\backslash mathbf\{k\})${{Harvnb|Ashcroft|Mermin|1976|p=228}} and we can use it to write a semi-classical equation for a charge carrier in a band{{Harvnb|Ashcroft|Mermin|1976|p=229}}

{{Equation box 1

|indent=:

|title=Second order semi-classical equation of motion for a charge carrier in a band

|equation=$$

\mathbf{M}(\mathbf{k}) \mathbf{a} = \mp e \left(\mathbf {E} + \mathbf{v}(\mathbf{k}) \times \mathbf{B}\right)

|cellpadding

|border

|border colour = rgb(80,200,120)

|background colour = rgb(80,200,120,10%)}}

where $\backslash mathbf\{a\}$ is an acceleration. This equation is analogous to the de Broglie wave type of approximation{{Harvnb|Ashcroft|Mermin|1976|p=227}}

{{Equation box 1

|indent=:

|title=First order semi-classical equation of motion for electron in a band

|equation=$$

\hbar \dot{k} = - e \left(\mathbf {E} + \mathbf{v} \times \mathbf{B}\right)

|cellpadding

|border

|border colour = rgb(80,200,120)

|background colour = rgb(80,200,120,10%)}}

As an intuitive interpretation, both of the previous two equations resemble formally and are in a semi-classical analogy with Newton's second law for an electron in an external Lorentz force.

The concept of the Bloch state was developed by Felix Bloch in 1928{{cite journal|author=Felix Bloch|author-link=Felix Bloch|title=Über die Quantenmechanik der Elektronen in Kristallgittern|journal=Zeitschrift für Physik| volume=52 | issue=7–8| pages=555–600 |year=1928|doi=10.1007/BF01339455|bibcode = 1929ZPhy...52..555B |s2cid=120668259|language=de}} to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by George William Hill (1877),{{cite journal|doi=10.1007/BF02417081| author=George William Hill|author-link=George William Hill|title=On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon|journal=Acta Math.|volume=8|pages=1–36 |year=1886|url=https://zenodo.org/record/1691491|doi-access=free}} This work was initially published and distributed privately in 1877. Gaston Floquet (1883),{{cite journal|author=Gaston Floquet|author-link=Gaston Floquet | title=Sur les équations différentielles linéaires à coefficients périodiques|journal= Annales Scientifiques de l'École Normale Supérieure|volume=12|pages=47–88 |year=1883|doi=10.24033/asens.220|doi-access=free}} and Alexander Lyapunov (1892).{{cite book|author=Alexander Mihailovich Lyapunov|author-link=Aleksandr Lyapunov|title=The General Problem of the Stability of Motion|location=London|publisher= Taylor and Francis|year= 1992}} Translated by A. T. Fuller from Edouard Davaux's French translation (1907) of the original Russian dissertation (1892). As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory (or occasionally the Lyapunov–Floquet theorem). The general form of a one-dimensional periodic potential equation is Hill's equation:

{{cite book

|first1=W|last1= Magnus |author-link=Wilhelm Magnus|first2=S|last2= Winkler

|title=Hill's Equation

|year= 2004

|page=11

|publisher=Courier Dover

|isbn=0-486-49565-5

|url=https://books.google.com/books?id=ML5wm-T4RVQC&q=%22hill's+equation%22}}

$$\backslash frac\; \{d^2y\}\{dt^2\}+f(t)\; y=0,$$

where {{math|f(t)}} is a periodic potential. Specific periodic one-dimensional equations include the Kronig–Penney model and Mathieu's equation.

Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to spectral geometry.Kuchment, P.(1982), Floquet theory for partial differential equations, RUSS MATH SURV., 37, 1–60{{cite journal |author=Katsuda, A. |author2=Sunada, T |author2-link=Toshikazu Sunada |year=1987 |title=Homology and closed geodesics in a compact Riemann surface |journal=Amer. J. Math. |volume=110 |issue=1 |pages=145–156 |doi=10.2307/2374542| jstor=2374542 }}{{cite journal |author=Kotani M |author2=Sunada T. |year=2000 |title=Albanese maps and an off diagonal long time asymptotic for the heat kernel |journal=Comm. Math. Phys. |volume=209 |issue=3 |pages=633–670 |doi=10.1007/s002200050033 | bibcode = 2000CMaPh.209..633K |s2cid=121065949 }}

{{colbegin}}

• Bloch oscillations
• Bloch wave – MoM method
• Electronic band structure
• Nearly free electron model
• Periodic boundary conditions
• Symmetries in quantum mechanics
• Tight-binding model
• Wannier function

{{colend}}

{{reflist}}

{{Refbegin}}

• {{cite book|last1=Ashcroft|first1=Neil|author-link=Neil Ashcroft|last2=Mermin|first2=N. David|author-link2=N. David Mermin|title=Solid State Physics|publisher=Holt, Rinehart and Winston|location=New York|year=1976|isbn=978-0-03-083993-1|url=https://archive.org/details/solidstatephysic00ashc}}
• {{Cite book|last=Dresselhaus, M. S.|url=http://worldcat.org/oclc/692760083|title=Group theory: application to the physics of condensed matter|date=2010|publisher=Springer-Verlag|isbn=978-3-642-06945-1|oclc=692760083}}
• {{cite web|url=http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_1_4.html |title=Periodic Potentials and Bloch's Theorem – lectures in "Semiconductors I" |publisher=The University of Kiel|author=H. Föll}}
• {{cite book|author=M.S.P. Eastham|title=The Spectral Theory of Periodic Differential Equations|location=Edinburgh|publisher= Scottish Academic Press|series= Texts in Mathematics|year= 1973}}
• {{cite journal|doi=10.1016/j.wavemoti.2012.12.010|author=J. Gazalet|author2=S. Dupont|author3=J.C. Kastelik|author4=Q. Rolland|author5=B. Djafari-Rouhani|name-list-style=amp|title=A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains|journal=Wave Motion|volume=50|issue=3|pages=619–654 |year=2013|bibcode=2013WaMot..50..619G |url=https://hal.archives-ouvertes.fr/hal-00808153 }}

{{Refend}}

{{Condensed matter physics topics}}

{{Authority control}}

Category:Eponymous theorems of physics

Category:Theorems in quantum mechanics

Category:Condensed matter physics