From 53bfc618a0ce4d0211d6ed2eb5e045bdb089adc1 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 20 Feb 2021 10:14:30 +0100 Subject: Restructure knowledge base + consistency for blog --- content/blog/_index.md | 4 +- content/know/_index.md | 26 +++++- content/know/category/_index.md | 13 +++ content/know/category/quantum-mechanics/_index.md | 10 +++ content/know/concept/_index.md | 10 +++ static/know/blochs-theorem/index.html | 103 ---------------------- static/know/concept/blochs-theorem/index.html | 103 ++++++++++++++++++++++ 7 files changed, 163 insertions(+), 106 deletions(-) create mode 100644 content/know/category/_index.md create mode 100644 content/know/category/quantum-mechanics/_index.md create mode 100644 content/know/concept/_index.md delete mode 100644 static/know/blochs-theorem/index.html create mode 100644 static/know/concept/blochs-theorem/index.html diff --git a/content/blog/_index.md b/content/blog/_index.md index ca75248..a021305 100644 --- a/content/blog/_index.md +++ b/content/blog/_index.md @@ -2,7 +2,9 @@ title = "Blog" +++ -Random ramblings. +## Blog + +Just a bunch of random ramblings. 2020: * [Setting up an email server in 2020 with OpenSMTPD and Dovecot](/blog/2020/email-server/) diff --git a/content/know/_index.md b/content/know/_index.md index ff51273..6941f07 100644 --- a/content/know/_index.md +++ b/content/know/_index.md @@ -2,5 +2,27 @@ title = "Knowledge base" +++ -Work in progress... -* [Bloch's theorem](/know/blochs-theorem/) +## Knowledge base + +Welcome to my knowledge base! + +Over the years, I've learned a lot about physics, mathematics, and computing. +These are vast, difficult subjects, and I've spent many hours of my life +trying to make sense of obscure, poorly explained concepts. +To help me remember what I learn, I write notes in LaTeX. + +This knowledge base is based on my notes, and +is freely available to anyone who needs it. +I hope it helps you in your studies or work. +Currently there's isn't much here yet, +but I have over 200 pages of LaTeX waiting to be converted. + +Keep in mind that I'm only human, +so there are probably mistakes in my work. +I take no responsibility for any injuries incurred as a consequence. +If you're working on something important, +you should check things yourself! + +The contents of this knowledge base are found here: +* [List of concepts](/know/concept/) +* [List of categories](/know/category/) diff --git a/content/know/category/_index.md b/content/know/category/_index.md new file mode 100644 index 0000000..b8f0456 --- /dev/null +++ b/content/know/category/_index.md @@ -0,0 +1,13 @@ ++++ +title = "List of categories" ++++ + +# + +Alphabetical list of categories in this knowledge base. + +Note that these categories might overlap, +i.e. share certain concepts. + +## Q +* [Quantum mechanics](/know/category/quantum-mechanics/) diff --git a/content/know/category/quantum-mechanics/_index.md b/content/know/category/quantum-mechanics/_index.md new file mode 100644 index 0000000..cf1d7ba --- /dev/null +++ b/content/know/category/quantum-mechanics/_index.md @@ -0,0 +1,10 @@ ++++ +title = "Category: quantum mechanics" ++++ + +# + +Alphabetical list of concepts in this category. + +## B +* [Bloch's theorem](/know/concept/blochs-theorem/) diff --git a/content/know/concept/_index.md b/content/know/concept/_index.md new file mode 100644 index 0000000..5db6133 --- /dev/null +++ b/content/know/concept/_index.md @@ -0,0 +1,10 @@ ++++ +title = "List of concepts" ++++ + +# + +Alphabetical list of concepts in this knowledge base. + +## B +* [Bloch's theorem](/know/concept/blochs-theorem/) diff --git a/static/know/blochs-theorem/index.html b/static/know/blochs-theorem/index.html deleted file mode 100644 index 26e3480..0000000 --- a/static/know/blochs-theorem/index.html +++ /dev/null @@ -1,103 +0,0 @@ - - - - - - - Prefetch | Bloch’s theorem - - - -
-
PREFETCH
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- -

Bloch’s theorem

-

In quantum mechanics, Bloch’s theorem states that, given a potential V(r⃗)V(\vec{r}) which is periodic on a lattice, i.e. V(r⃗)=V(r⃗+a⃗)V(\vec{r}) = V(\vec{r} + \vec{a}) for a primitive lattice vector a⃗\vec{a}, then it follows that the solutions ψ(r⃗)\psi(\vec{r}) to the time-independent Schrödinger equation take the following form, where the function u(r⃗)u(\vec{r}) is periodic on the same lattice, i.e. u(r⃗)=u(r⃗+a⃗)u(\vec{r}) = u(\vec{r} + \vec{a}):

-

ψ(r⃗)=u(r⃗)eik⃗⋅r⃗ -\begin{aligned} - \boxed{ - \psi(\vec{r}) = u(\vec{r}) e^{i \vec{k} \cdot \vec{r}} - } -\end{aligned} -

-

In other words, in a periodic potential, the solutions are simply plane waves with a periodic modulation, known as Bloch functions or Bloch states.

-

This is suprisingly easy to prove: if the Hamiltonian Ĥ\hat{H} is lattice-periodic, then it will commute with the unitary translation operator T̂(a⃗)\hat{T}(\vec{a}), i.e. [Ĥ,T̂(a⃗)]=0[\hat{H}, \hat{T}(\vec{a})] = 0. Therefore Ĥ\hat{H} and T̂(a⃗)\hat{T}(\vec{a}) must share eigenstates ψ(r⃗)\psi(\vec{r}):

-

Ĥψ(r⃗)=Eψ(r⃗)T̂(a⃗)ψ(r⃗)=τψ(r⃗) -\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} -

-

Since T̂\hat{T} is unitary, its eigenvalues τ\tau must have the form eiθe^{i \theta}, with θ\theta real. Therefore a translation by a⃗\vec{a} causes a phase shift, for some vector k⃗\vec{k}:

-

ψ(r⃗+a⃗)=T̂(a⃗)ψ(r⃗)=eiθψ(r⃗)=eik⃗⋅a⃗ψ(r⃗) -\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} -

-

Let us now define the following function, keeping our arbitrary choice of k⃗\vec{k}:

-

u(r⃗)=e−ik⃗⋅r⃗ψ(r⃗) -\begin{aligned} - u(\vec{r}) - = e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) -\end{aligned} -

-

As it turns out, this function is guaranteed to be lattice-periodic for any k⃗\vec{k}:

-

u(r⃗+a⃗)=e−ik⃗⋅(r⃗+a⃗)ψ(r⃗+a⃗)=e−ik⃗⋅r⃗e−ik⃗⋅a⃗eik⃗⋅a⃗ψ(r⃗)=e−ik⃗⋅r⃗ψ(r⃗)=u(r⃗) -\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} -

-

Then Bloch’s theorem follows from isolating the definition of u(r⃗)u(\vec{r}) for ψ(r⃗)\psi(\vec{r}).

- -
-© "Prefetch". Licensed under CC BY-SA 4.0. - - diff --git a/static/know/concept/blochs-theorem/index.html b/static/know/concept/blochs-theorem/index.html new file mode 100644 index 0000000..26e3480 --- /dev/null +++ b/static/know/concept/blochs-theorem/index.html @@ -0,0 +1,103 @@ + + + + + + + Prefetch | Bloch’s theorem + + + +
+
PREFETCH
+
+blog  +code  +know +
+
+
+ +

Bloch’s theorem

+

In quantum mechanics, Bloch’s theorem states that, given a potential V(r⃗)V(\vec{r}) which is periodic on a lattice, i.e. V(r⃗)=V(r⃗+a⃗)V(\vec{r}) = V(\vec{r} + \vec{a}) for a primitive lattice vector a⃗\vec{a}, then it follows that the solutions ψ(r⃗)\psi(\vec{r}) to the time-independent Schrödinger equation take the following form, where the function u(r⃗)u(\vec{r}) is periodic on the same lattice, i.e. u(r⃗)=u(r⃗+a⃗)u(\vec{r}) = u(\vec{r} + \vec{a}):

+

ψ(r⃗)=u(r⃗)eik⃗⋅r⃗ +\begin{aligned} + \boxed{ + \psi(\vec{r}) = u(\vec{r}) e^{i \vec{k} \cdot \vec{r}} + } +\end{aligned} +

+

In other words, in a periodic potential, the solutions are simply plane waves with a periodic modulation, known as Bloch functions or Bloch states.

+

This is suprisingly easy to prove: if the Hamiltonian Ĥ\hat{H} is lattice-periodic, then it will commute with the unitary translation operator T̂(a⃗)\hat{T}(\vec{a}), i.e. [Ĥ,T̂(a⃗)]=0[\hat{H}, \hat{T}(\vec{a})] = 0. Therefore Ĥ\hat{H} and T̂(a⃗)\hat{T}(\vec{a}) must share eigenstates ψ(r⃗)\psi(\vec{r}):

+

Ĥψ(r⃗)=Eψ(r⃗)T̂(a⃗)ψ(r⃗)=τψ(r⃗) +\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} +

+

Since T̂\hat{T} is unitary, its eigenvalues τ\tau must have the form eiθe^{i \theta}, with θ\theta real. Therefore a translation by a⃗\vec{a} causes a phase shift, for some vector k⃗\vec{k}:

+

ψ(r⃗+a⃗)=T̂(a⃗)ψ(r⃗)=eiθψ(r⃗)=eik⃗⋅a⃗ψ(r⃗) +\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} +

+

Let us now define the following function, keeping our arbitrary choice of k⃗\vec{k}:

+

u(r⃗)=e−ik⃗⋅r⃗ψ(r⃗) +\begin{aligned} + u(\vec{r}) + = e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) +\end{aligned} +

+

As it turns out, this function is guaranteed to be lattice-periodic for any k⃗\vec{k}:

+

u(r⃗+a⃗)=e−ik⃗⋅(r⃗+a⃗)ψ(r⃗+a⃗)=e−ik⃗⋅r⃗e−ik⃗⋅a⃗eik⃗⋅a⃗ψ(r⃗)=e−ik⃗⋅r⃗ψ(r⃗)=u(r⃗) +\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} +

+

Then Bloch’s theorem follows from isolating the definition of u(r⃗)u(\vec{r}) for ψ(r⃗)\psi(\vec{r}).

+ +
+© "Prefetch". Licensed under CC BY-SA 4.0. + + -- cgit v1.2.3