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<h1 id="blochs-theorem">Bloch’s theorem</h1>
<p>In quantum mechanics, <em>Bloch’s theorem</em> states that, given a potential <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">V(\vec{r})</annotation></semantics></math> which is periodic on a lattice, i.e. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">V(\vec{r}) = V(\vec{r} + \vec{a})</annotation></semantics></math> for a primitive lattice vector <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>a</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math>, then it follows that the solutions <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\psi(\vec{r})</annotation></semantics></math> to the time-independent Schrödinger equation take the following form, where the function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">u(\vec{r})</annotation></semantics></math> is periodic on the same lattice, i.e. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">u(\vec{r}) = u(\vec{r} + \vec{a})</annotation></semantics></math>:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><menclose notation="box"><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><msup><mi>e</mi><mrow><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup></mrow></menclose></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{aligned}
\boxed{
\psi(\vec{r}) = u(\vec{r}) e^{i \vec{k} \cdot \vec{r}}
}
\end{aligned}
</annotation></semantics></math></p>
<p>In other words, in a periodic potential, the solutions are simply plane waves with a periodic modulation, known as <em>Bloch functions</em> or <em>Bloch states</em>.</p>
<p>This is suprisingly easy to prove: if the Hamiltonian <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>H</mi><mo accent="true">̂</mo></mover><annotation encoding="application/x-tex">\hat{H}</annotation></semantics></math> is lattice-periodic, then it will commute with the unitary translation operator <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\hat{T}(\vec{a})</annotation></semantics></math>, i.e. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">[</mo><mover><mi>H</mi><mo accent="true">̂</mo></mover><mo>,</mo><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo stretchy="false" form="postfix">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">[\hat{H}, \hat{T}(\vec{a})] = 0</annotation></semantics></math>. Therefore <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>H</mi><mo accent="true">̂</mo></mover><annotation encoding="application/x-tex">\hat{H}</annotation></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\hat{T}(\vec{a})</annotation></semantics></math> must share eigenstates <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\psi(\vec{r})</annotation></semantics></math>:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mover><mi>H</mi><mo accent="true">̂</mo></mover><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>E</mi><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mspace width="2.0em"></mspace><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>τ</mi><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{aligned}
\hat{H} \:\psi(\vec{r}) = E \:\psi(\vec{r})
\qquad
\hat{T}(\vec{a}) \:\psi(\vec{r}) = \tau \:\psi(\vec{r})
\end{aligned}
</annotation></semantics></math></p>
<p>Since <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>T</mi><mo accent="true">̂</mo></mover><annotation encoding="application/x-tex">\hat{T}</annotation></semantics></math> is unitary, its eigenvalues <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">\tau</annotation></semantics></math> must have the form <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup><annotation encoding="application/x-tex">e^{i \theta}</annotation></semantics></math>, with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>θ</mi><annotation encoding="application/x-tex">\theta</annotation></semantics></math> real. Therefore a translation by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>a</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> causes a phase shift, for some vector <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>k</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{k}</annotation></semantics></math>:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{aligned}
\psi(\vec{r} + \vec{a})
= \hat{T}(\vec{a}) \:\psi(\vec{r})
= e^{i \theta} \:\psi(\vec{r})
= e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r})
\end{aligned}
</annotation></semantics></math></p>
<p>Let us now define the following function, keeping our arbitrary choice of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>k</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{k}</annotation></semantics></math>:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{aligned}
u(\vec{r})
= e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r})
\end{aligned}
</annotation></semantics></math></p>
<p>As it turns out, this function is guaranteed to be lattice-periodic for any <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>k</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{k}</annotation></semantics></math>:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd><mtd columnalign="left"><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover></mrow></msup><msup><mi>e</mi><mrow><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex">
\begin{aligned}
u(\vec{r} + \vec{a})
&= e^{- i \vec{k} \cdot (\vec{r} + \vec{a})} \:\psi(\vec{r} + \vec{a})
\\
&= e^{- i \vec{k} \cdot \vec{r}} e^{- i \vec{k} \cdot \vec{a}} e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r})
\\
&= e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r})
\\
&= u(\vec{r})
\end{aligned}
</annotation></semantics></math></p>
<p>Then Bloch’s theorem follows from isolating the definition of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">u(\vec{r})</annotation></semantics></math> for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\psi(\vec{r})</annotation></semantics></math>.</p>
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