From 7a3bb3e3b3606bc88506c248f1c29c30c7b22678 Mon Sep 17 00:00:00 2001 From: aj Date: Mon, 11 Nov 2019 23:13:08 +0000 Subject: [PATCH] add probability plots and maths --- coursework.lyx | 204 ++++++++++++++++++++++++++++++- coursework.pdf | Bin 382359 -> 501255 bytes probability-plot-with-bounds.png | Bin 0 -> 79331 bytes probability-plot-with-n-2.png | Bin 0 -> 81627 bytes 4 files changed, 200 insertions(+), 4 deletions(-) create mode 100644 probability-plot-with-bounds.png create mode 100644 probability-plot-with-n-2.png diff --git a/coursework.lyx b/coursework.lyx index 46dacb9..8db1733 100644 --- a/coursework.lyx +++ b/coursework.lyx @@ -259,12 +259,12 @@ name "fig:Well-Band-structure" \end_layout \begin_layout Standard -None of the binary III-V Indium based alloys have bulk band gaps in a suitable +None of the binary III-V indium based alloys have bulk band gaps in a suitable range, as such ternary alloys were investigated. \end_layout \begin_layout Standard -Indium Gallium Arsenide (In +indium gallium arsenide (In \begin_inset script subscript \begin_layout Plain Layout @@ -288,7 +288,7 @@ Ga \end_inset -As) as a well material with Indium Phosphide (InP) as a barrier material +As) as a well material with indium phosphide (InP) as a barrier material would provide a suitable combination assuming that a composition ratio \begin_inset Formula $x$ @@ -751,7 +751,7 @@ Having found two materials that are lattice matched with a suitable band \emph on \begin_inset Formula \begin{equation} -E_{n}=\frac{n^{2}\pi^{2}\text{ħ}^{2}}{2mL^{2}}\label{eq:Energy-levels} +E_{n}=\frac{n^{2}\pi^{2}\mathcal{\text{\emph{ħ}}}^{2}}{2mL^{2}}\label{eq:Energy-levels} \end{equation} \end_inset @@ -1383,6 +1383,57 @@ name "fig:Probability-plot" \end_inset +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename probability-plot-with-n-2.png + lyxscale 30 + width 100col% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Probability plot for electron in 1 +\begin_inset script superscript + +\begin_layout Plain Layout +st +\end_layout + +\end_inset + + excited state +\begin_inset CommandInset label +LatexCommand label +name "fig:Probability-plot-n-2" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + \end_layout \begin_layout Section @@ -1432,6 +1483,151 @@ Where is an arbitrary distance across the well. \end_layout +\begin_layout Standard +For an interval across the well, this becomes: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P\left(a\leq x\leq b\right)=\frac{1}{L}\left(\left(b-a\right)-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi b}{L}\right)-\sin\left(\frac{2n\pi a}{L}\right)\right)\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center +\begin_inset Graphics + filename probability-plot-with-bounds.png + lyxscale 30 + width 100col% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Probability plot for electron in ground state with distance intervals +\begin_inset CommandInset label +LatexCommand label +name "fig:Probability-plot-with-bounds" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +\begin_inset Formula $2\unit{nm}$ +\end_inset + + to +\begin_inset Formula $4\unit{nm}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)=\frac{1}{L}\left(2\unit{nm}-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi\cdotp\left(4\unit{nm}\right)}{L}\right)-\sin\left(\frac{2n\pi\cdotp\left(2\unit{nm}\right)}{L}\right)\right)\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)=\frac{1}{14.87\unit{nm}}\left(2\unit{nm}-\frac{14.87\unit{nm}}{2\pi}\left(\sin\left(\frac{2\pi\cdotp\left(4\unit{nm}\right)}{14.87\unit{nm}}\right)-\sin\left(\frac{2\pi\cdotp\left(2\unit{nm}\right)}{14.87\unit{nm}}\right)\right)\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P\left(2\unit{nm}\leq x\leq4\unit{nm}\right)\thickapprox0.132 +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +\begin_inset Formula $6\unit{nm}$ +\end_inset + + to +\begin_inset Formula $8\unit{nm}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)=\frac{1}{L}\left(2\unit{nm}-\frac{L}{2n\pi}\left(\sin\left(\frac{2n\pi\cdotp\left(8\unit{nm}\right)}{L}\right)-\sin\left(\frac{2n\pi\cdotp\left(6\unit{nm}\right)}{L}\right)\right)\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)=\frac{1}{14.87\unit{nm}}\left(2\unit{nm}-\frac{14.87\unit{nm}}{2\pi}\left(\sin\left(\frac{2\pi\cdotp\left(8\unit{nm}\right)}{14.87\unit{nm}}\right)-\sin\left(\frac{2\pi\cdotp\left(6\unit{nm}\right)}{14.87\unit{nm}}\right)\right)\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P\left(6\unit{nm}\leq x\leq8\unit{nm}\right)\thickapprox0.132 +\] + +\end_inset + + +\end_layout + \begin_layout Standard \begin_inset Newpage pagebreak \end_inset diff --git a/coursework.pdf b/coursework.pdf index 56ded819a25a0a25324ab25ba294296b42989b4e..d74ead95d9226ccc6ea816ae4f95f08983690972 100644 GIT binary patch delta 147148 zcma&Oby!yG_5~^-AtE519}UvoAt@!@4U*E`ETvNfMN&!;>F)0C?(Xh}yHNK&XP@7` z_qqNfkLdc|HRqUPj5%H&W3sUWdx$=gL$kBR6vDz%1L?S(_Sv7LsiZ#iUrlxCHC5B%j=r$cr+fgvkG8)gRjSu@Az`;q zv=RUw>&sB3^^2kGFy1^h$KgpmG^iGdaxH4N$oS#VRAl`z5Zd{pBtf9LAt&z`i}+0P 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