diff --git a/ma5/uppg/main.fdb_latexmk b/ma5/uppg/main.fdb_latexmk index d9b4b88..0b44c67 100644 --- a/ma5/uppg/main.fdb_latexmk +++ b/ma5/uppg/main.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1648345203 "main.tex" "main.pdf" "main" 1648345203 +["pdflatex"] 1648410637 "main.tex" "main.pdf" "main" 1648410638 "/usr/share/texmf-dist/fonts/map/fontname/texfonts.map" 1647844622 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/share/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1647844622 1004 54797486969f23fa377b128694d548df "" "/usr/share/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1647844622 916 f87d7c45f9c908e672703b83b72241a3 "" @@ -86,9 +86,9 @@ "/usr/share/texmf-dist/web2c/texmf.cnf" 1647844622 39911 2da6c67557ec033436fe5418a70a8a61 "" "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1648036410 4278762 af3ac1370db81a450c1118eabf1fe718 "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1648036393 2924504 371aaf66e6dc336df8f5856c9444aca8 "" - "main.aux" 1648345203 2317 eb7ea0ad69ba39707b99e3ab67505cde "pdflatex" - "main.out" 1648345203 1159 2635baec54a16692a01d1db6cb342d5e "pdflatex" - "main.tex" 1648345194 12682 eee475df4c55e8b4eade04905b343359 "" + "main.aux" 1648410638 2317 eb7ea0ad69ba39707b99e3ab67505cde "pdflatex" + "main.out" 1648410638 1159 2635baec54a16692a01d1db6cb342d5e "pdflatex" + "main.tex" 1648410636 13932 ba98d7bd049011d605ce0643529cdf46 "" (generated) "main.aux" "main.log" diff --git a/ma5/uppg/main.log b/ma5/uppg/main.log index 595fd12..28654ad 100644 --- a/ma5/uppg/main.log +++ b/ma5/uppg/main.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.23) 27 MAR 2022 03:40 +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.23) 27 MAR 2022 21:50 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -391,7 +391,7 @@ Here is how much of TeX's memory you used: 1141 hyphenation exceptions out of 8191 60i,7n,63p,809b,330s stack positions out of 5000i,500n,10000p,200000b,80000s -Output written on main.pdf (7 pages, 206528 bytes). +Output written on main.pdf (7 pages, 209097 bytes). PDF statistics: 192 PDF objects out of 1000 (max. 8388607) 145 compressed objects within 2 object streams diff --git a/ma5/uppg/main.pdf b/ma5/uppg/main.pdf index 56ecda7..bb18ff1 100644 Binary files a/ma5/uppg/main.pdf and b/ma5/uppg/main.pdf differ diff --git a/ma5/uppg/main.synctex.gz b/ma5/uppg/main.synctex.gz index bfc3e68..f3f9f62 100644 Binary files a/ma5/uppg/main.synctex.gz and b/ma5/uppg/main.synctex.gz differ diff --git a/ma5/uppg/main.tex b/ma5/uppg/main.tex index a018e38..20dbde7 100644 --- a/ma5/uppg/main.tex +++ b/ma5/uppg/main.tex @@ -348,8 +348,40 @@ Se relevanta grafbilder i \emph{imgs/}. \subsection{3} +Vi vet sedan tidigare $\psi_n(x)$: $$ \psi_n(x) = \sqrt{ \frac{2}{L} } \sin \left( \sqrt{\frac{8 \pi^2 m E_n}{h^2}} x \right) $$ +och i uppgiften får vi att: +$$ +\Psi_n(x, t) = \psi_n(x) \cdot e^{-i \frac{E_n}{\hbar}t } +$$ +$$ +\implies \Psi_n(x, t) = \sqrt{ \frac{2}{L} } \sin \left( \sqrt{\frac{8 \pi^2 m E_n}{h^2}} x \right) \cdot e^{-i \frac{E_n}{\hbar}t } = +$$ +$$ += \sqrt{ \frac{2}{L} } \sin \left( \sqrt{\frac{8 \pi^2 m E_n}{h^2}} x \right) \cdot e^{-i \frac{E_n}{ \frac{h}{2\pi} }t } += \sqrt{ \frac{2}{L} } \sin \left( \sqrt{\frac{8 \pi^2 m E_n}{h^2}} x \right) \cdot e^{-i \frac{2 E_n \pi}{h}t } +$$ +$$ +\therefore \Psi_n(x,t) = \sqrt{ \frac{2}{L} } \left(e^{-i \frac{2 E_n \pi}{h}t }\right) \sin \left( \sqrt{\frac{8 \pi^2 m E_n}{h^2}} x \right) +$$ + +Vi behöver nu bara normalisera integralen sådan att den alltid blir $1$ för följande: +$$ +\Psi_{1,2}(x, t) = A \left( \psi_1(x)e^{-i \frac{2\pi E_1}{h}t } + \psi_2(x)e^{-i \frac{2\pi E_2}{h}t } \right) +$$ +$$ +\int_0^L |\Psi_{1,2}(x, t)|^2 dx\ = 1 +$$ + +Eftersom vi integrerar med respekt till $x$ tyder det på att $t$ är en konstant och vi kan därmed skriva om tidsfaktorn som $z_i$: +$$ +\int_0^L |\Psi_{1,2}(x, t)|^2 dx\ = \int_0^L \left(A |\psi_1(x)z_1 + \psi_2(x)z_2| \right)^2 dx\ +$$ +$$ += A^2 \int_0^L |\psi_1(x)z_1 + \psi_2(x)z_2|^2 dx\ = 1.0 \quad | \quad z_i \in \mathbb{C} +$$ + \end{document}