Editing Bloch's theorem (section)

=== 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:<ref name=":3">{{Harvnb|Ashcroft|Mermin|1976|p=134}}</ref>

==== 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'' {{math|'''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub>}}. If the crystal is shifted by any of these three vectors, or a combination of them of the form
<math display="block">n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,</math>
where {{mvar|n<sub>i</sub>}} 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 vector]]s''. These are three vectors {{math|'''b'''<sub>1</sub>, '''b'''<sub>2</sub>, '''b'''<sub>3</sub>}} (with units of inverse length), with the property that {{math|1='''a'''<sub>''i''</sub> · '''b'''<sub>''i''</sub> = 2''π''}}, but {{math|1='''a'''<sub>''i''</sub> · '''b'''<sub>''j''</sub> = 0}} when {{math|''i'' ≠ ''j''}}. (For the formula for {{math|'''b'''<sub>''i''</sub>}}, see [[reciprocal lattice vector]].)

==== Lemma about translation operators ====
Let <math> \hat{T}_{n_1,n_2,n_3} </math> denote a [[Translation operator (quantum mechanics)|translation operator]] that shifts every wave function by the amount {{math|''n''<sub>1</sub>'''a'''<sub>1</sub> + ''n''<sub>2</sub>'''a'''<sub>2</sub> + ''n''<sub>3</sub>'''a'''<sub>3</sub>}} (as above, {{mvar|n<sub>j</sub>}} 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|&psi;}} is an [[eigenfunction|eigenstate]] of all of the translation operators (simultaneously), then {{mvar|&psi;}} is a Bloch state.}}
{{math proof | title = Proof of Lemma | proof = Assume that we have a wave function {{mvar|&psi;}} which is an eigenstate of all the translation operators. As a special case of this, 
<math display="block">\psi(\mathbf{r}+\mathbf{a}_j) = C_j \psi(\mathbf{r})</math> 
for {{math|1=''j'' = 1, 2, 3}}, where {{mvar|C<sub>j</sub>}} are three numbers (the [[eigenvalue]]s) which do not depend on {{math|'''r'''}}. It is helpful to write the numbers {{mvar|C<sub>j</sub>}} in a different form, by choosing three numbers {{math|''θ''<sub>1</sub>, ''θ''<sub>2</sub>, ''θ''<sub>3</sub>}} with {{math|1=''e''<sup>2''πiθ''<sub>''j''</sub></sup> = ''C''<sub>''j''</sub>}}:
<math display="block">\psi(\mathbf{r}+\mathbf{a}_j) = e^{2 \pi i \theta_j} \psi(\mathbf{r})</math> 
Again, the {{mvar|θ<sub>j</sub>}} are three numbers which do not depend on {{math|'''r'''}}. Define {{math|1='''k''' = ''θ''<sub>1</sub>'''b'''<sub>1</sub> + ''θ''<sub>2</sub>'''b'''<sub>2</sub> + ''θ''<sub>3</sub>'''b'''<sub>3</sub>}}, where {{math|'''b'''<sub>''j''</sub>}} are the reciprocal lattice vectors (see above). Finally, define 
<math display="block">u(\mathbf{r}) = e^{-i \mathbf{k}\cdot\mathbf{r}} \psi(\mathbf{r})\,.</math>
Then
<math display="block">\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}</math>
This proves that {{mvar|u}} has the periodicity of the lattice. Since <math>\psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r}),</math> 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 <math> \hat{T}_{n_1,n_2,n_3} </math> denote a ''translation operator'' that shifts every wave function by the amount {{math|''n''<sub>1</sub>'''a'''<sub>1</sub> + ''n''<sub>2</sub>'''a'''<sub>2</sub> + ''n''<sub>3</sub>'''a'''<sub>3</sub>}}, where {{mvar|n<sub>i</sub>}} 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 [[Commuting matrices|simultaneous eigenbasis]] of the Hamiltonian operator and every possible <math> \hat{T}_{n_1,n_2,n_3} \!</math> 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).
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