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Diffstat (limited to 'static/know/concept/blochs-theorem')
| -rw-r--r-- | static/know/concept/blochs-theorem/index.html | 2 |
1 files changed, 1 insertions, 1 deletions
diff --git a/static/know/concept/blochs-theorem/index.html b/static/know/concept/blochs-theorem/index.html index 6e5767c..f977739 100644 --- a/static/know/concept/blochs-theorem/index.html +++ b/static/know/concept/blochs-theorem/index.html @@ -59,7 +59,7 @@ \end{aligned} \]</span></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 <span class="math inline">\(\hat{H}\)</span> is lattice-periodic, then it will commute with the unitary translation operator <span class="math inline">\(\hat{T}(\vec{a})\)</span>, i.e. <span class="math inline">\(\comm{\hat{H}}{\hat{T}(\vec{a})} = 0\)</span>. Therefore <span class="math inline">\(\hat{H}\)</span> and <span class="math inline">\(\hat{T}(\vec{a})\)</span> must share eigenstates <span class="math inline">\(\psi(\vec{r})\)</span>:</p> +<p>This is suprisingly easy to prove: if the Hamiltonian <span class="math inline">\(\hat{H}\)</span> is lattice-periodic, then it will commute with the unitary translation operator <span class="math inline">\(\hat{T}(\vec{a})\)</span>, i.e. <span class="math inline">\([\hat{H}, \hat{T}(\vec{a})] = 0\)</span>. Therefore <span class="math inline">\(\hat{H}\)</span> and <span class="math inline">\(\hat{T}(\vec{a})\)</span> must share eigenstates <span class="math inline">\(\psi(\vec{r})\)</span>:</p> <p><span class="math display">\[ \begin{aligned} \hat{H} \:\psi(\vec{r}) = E \:\psi(\vec{r}) |