Using lattice periodicity 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:
Preliminaries: Crystal symmetries, lattice, and reciprocal lattice 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 . If the crystal is shifted by any of these three vectors, or a combination of them of the form n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3, where 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 (with units of inverse length), with the property that , but when . (For the formula for , see
reciprocal lattice vector.)
Lemma about translation operators Let \hat{T}_{n_1,n_2,n_3} denote a
translation operator that shifts every wave function by the amount (as above, are integers). The following fact is helpful for the proof of Bloch's theorem: {{math proof | title = Proof of Lemma | proof = Assume that we have a wave function which is an eigenstate of all the translation operators. As a special case of this, \psi(\mathbf{r}+\mathbf{a}_j) = C_j \psi(\mathbf{r}) for , where are three numbers (the
eigenvalues) which do not depend on . It is helpful to write the numbers in a different form, by choosing three numbers with : \psi(\mathbf{r}+\mathbf{a}_j) = e^{2 \pi i \theta_j} \psi(\mathbf{r}) Again, the are three numbers which do not depend on . Define , where are the reciprocal lattice vectors (see above). Finally, define u(\mathbf{r}) = e^{-i \mathbf{k}\cdot\mathbf{r}} \psi(\mathbf{r})\,. Then \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 has the periodicity of the lattice. Since \psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\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 \hat{T}_{n_1,n_2,n_3} denote a
translation operator that shifts every wave function by the amount , where 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 \hat{T}_{n_1,n_2,n_3} \! 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). }}
Using operators In this proof all the symmetries are encoded as commutation properties of the translation operators {{math proof | title = Proof using operators | proof = Source: We define the translation operator \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(\mathbf{x}+\mathbf{T}_{\mathbf{n}})= U(\mathbf{x}) and the
independent electron approximation with a Hamiltonian \hat{H}=\frac{\hat{\mathbf{p}}^2}{2m}+U(\mathbf{x}) Given the Hamiltonian is invariant for translations it shall commute with the translation operator [\hat{H},\hat{\mathbf{T}}_{\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: \hat{\mathbf{T}}_{\mathbf{n}}\psi(\mathbf{x})=\lambda_{\mathbf{n}}\psi(\mathbf{x}) Given \hat{\mathbf{T}}_{\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 \psi(\mathbf{x}) we have \lambda_{\mathbf{n}_1} \lambda_{\mathbf{n}_2} = \lambda_{\mathbf{n}_1 + \mathbf{n}_2} This is true for \lambda_{\mathbf{n}} = e^{s \mathbf{n} \cdot \mathbf{a} } where s \in \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} = and therefore 1 = |\lambda_{\mathbf{n}}|^2 and s = i k where k \in \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 \psi_{\mathbf{k}}(\mathbf{x}) = e^{i \mathbf{k} \cdot \mathbf{x} } u_{\mathbf{k}}(\mathbf{x}) with u_{\mathbf{k}}(\mathbf{x}) = u_{\mathbf{k}}(\mathbf{x} + \mathbf{A}\mathbf{n}) }}
Using group theory 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. 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 All
translations are
unitary and
abelian. Translations can be written in terms of unit vectors \boldsymbol{\tau} = \sum_{i=1}^3 n_i \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 \hat{\boldsymbol{\tau}}_i = n_i \hat{\mathbf{a}}_i The commutativity of the \hat{\boldsymbol{\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. 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 \gamma which shall obey to \gamma^n = 1, and therefore the character \chi(\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 \to \infty where the character remains finite. Given the character is a root of unity, for each subgroup the character can be then written as \chi_{k_1}(\hat{\boldsymbol{\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 \left(\mathbf {r} +\sum_i N_{i} \mathbf {a}_{i}\right) = V (\mathbf {r} +\mathbf{L}) = V (\mathbf {r} ) where
L is a macroscopic periodicity in the direction \mathbf{a} that can also be seen as a multiple of a_i where \mathbf{L} = \sum_i N_{i}\mathbf {a}_{i} This substituting in the time independent
Schrödinger equation with a simple effective Hamiltonian \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) induces a periodicity with the wave function: \psi \left(\mathbf {r} + \sum_i N_{i}\mathbf {a}_{i}\right) = \psi (\mathbf {r} ) And for each dimension a translation operator with a period
L \hat{P}_{\varepsilon|\tau_i + L_i} = \hat{P}_{\varepsilon|\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 \pi m_1 where m_1 \in \mathbb{Z} is an integer and k_1=\frac {2 \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 \chi_{k_1}(n_1,a_1)\chi_{k_2}(n_2,a_2)\chi_{k_3}(n_3,a_3) = e^{i\mathbf{k} \cdot \boldsymbol{\tau}} and the generic formula for the wave function becomes: \hat{P}_R\psi_j = \sum_{\alpha} \psi_{\alpha} \chi_{\alpha j}(R) i.e. specializing it for a translation \hat{P}_{\varepsilon|\boldsymbol{\tau}} \psi(\mathbf{r}) =\psi(\mathbf{r}) e^{i \mathbf{k} \cdot \boldsymbol{\tau}} = \psi(\mathbf{r} + \boldsymbol{\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. == Velocity and effective mass ==