diff --git a/PS2/doc.aux b/PS2/doc.aux index 3a44f65..d7e9143 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -14,4 +14,4 @@ \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{5}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Sketch of appropriate energy levels\relax }}{7}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).\relax }}{10}{}\protected@file@percent } -\gdef \@abspage@last{12} +\gdef \@abspage@last{13} diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index 52fd9da..8fb6934 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,8 +1,8 @@ # Fdb version 3 -["pdflatex"] 1614655256 "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc" 1614655259 +["pdflatex"] 1614658426 "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc" 1614658632 "C:/Users/speed/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1614620319 11548 693fc704971bedc6e913b37b081cf40e "" "C:/Users/speed/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1612675286 9293900 83195ab665706b6f706823352908aa95 "" - "C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614638559 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" + "C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614658033 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/jknappen/ec/tcrm1000.tfm" 993058908 1436 c7f957a372ef2fbe93c0982f96625e12 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1233951848 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -25,10 +25,18 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr6.tfm" 1136765053 1300 b62933e007d01cfd073f79b963c01526 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr8.tfm" 1136765053 1292 21c1c5bfeaebccffdb478fd231a0997d "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr9.tfm" 1136765053 1292 6b21b9c2c7bebb38aa2273f7ca0fb3af "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmss10.tfm" 1136765053 1316 b636689f1933f24d1294acdf6041daaa "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmss12.tfm" 1136765053 1324 37b971caf729d7edd9cbb9f9b0ea76eb "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmss8.tfm" 1136765053 1296 d77f431d10d47c8ea2cc18cf45346274 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmss9.tfm" 1136765053 1320 49357c421c0d469f88b867dd0c3d10e8 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy10.tfm" 1136765053 1124 6c73e740cf17375f03eec0ee63599741 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy6.tfm" 1136765053 1116 933a60c408fc0a863a92debe84b2d294 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy8.tfm" 1136765053 1120 8b7d695260f3cff42e636090a8002094 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy9.tfm" 1136765053 1116 25a7bf822c58caf309a702ef79f4afbb "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmtt10.tfm" 1136765053 768 1321e9409b4137d6fb428ac9dc956269 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmtt12.tfm" 1136765053 772 9a936b7f5e2ff0557fce0f62822f0bbf "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmtt8.tfm" 1136765053 768 d7b9a2629a0c353102ad947dc9221d49 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmtt9.tfm" 1136765053 764 c98a2af25c99b73a368cf7336e255190 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1247593066 34811 78b52f49e893bcba91bd7581cdc144c0 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1247593066 32080 340ef9bf63678554ee606688e7b5339d "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1247593067 30251 6afa5cb1d0204815a708a080681d4674 "" @@ -86,8 +94,10 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/kvoptions/kvoptions.sty" 1602224496 22521 d2fceb764a442a2001d257ef11db7618 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3backend/l3backend-pdftex.def" 1611916124 27097 58278863d97b10ab86e334b8da33df7a "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3kernel/expl3.sty" 1610202356 6209 031757bc8d0350c53dd99ad8ae4875eb "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3packages/l3keys2e/l3keys2e.sty" 1603800684 4615 0436b95d48df75eb5f264ca6898802f6 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3packages/xparse/xparse-generic.tex" 1603800698 80141 edbf9289c99ff37db17116af7a3a423f "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3packages/xparse/xparse.sty" 1603800698 5905 c6eb253894f4e808af476e034b49df36 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/latex-tools/array.sty" 1602844017 13051 9a7bbb9e485cd81cdcc1d56212b088ff "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/lipsum/lipsum.ltd.tex" 1546855561 99745 c6fa29828cc60471827afe275c8bd77f "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/lipsum/lipsum.sty" 1546855560 18705 8cf65af2c4529eed91b5d364b50d3ada "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/listings/listings.cfg" 1585216535 1879 e31effa752c61538383451ae21332364 "" @@ -96,11 +106,15 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/listings/lstmisc.sty" 1585216535 79106 ee25ce086f4a79d8cf73bac6f94c02a5 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/parskip/parskip.sty" 1592833306 4414 25631f9679adf750bac898538ededd61 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/setspace/setspace.sty" 1324294077 22913 a27d7908fc6f0385466454a966a316eb "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/siunitx/siunitx-abbreviations.cfg" 1582616262 4745 793d4d5a808c37a7085f620706b56fe1 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/siunitx/siunitx.sty" 1582616262 273434 c3d90844d64bf82fded1b064359b4e14 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/titlesec/titlesec.sty" 1571240058 48596 2b6a95da931c07a430b1a61904aaa42d "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614655259 1163 a0354534513b31cd9af2bee0ca4d6d1a "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614655255 21406 c34edfd6306342301c6d3fe25c1a3804 "" - "doc.aux" 1614655259 1163 a0354534513b31cd9af2bee0ca4d6d1a "pdflatex" - "doc.tex" 1614655255 21406 c34edfd6306342301c6d3fe25c1a3804 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/translator/translator-basic-dictionary-English.dict" 1596559992 3535 7dc96051305a7e943219126c49c44cd6 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/translator/translator.sty" 1596560046 8765 3368c4a5a4b5466261cafb836195e916 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614658429 1163 a2a422c07ff91b546dabf95ba81d6b33 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614658631 23472 b184c802ebac7b1a4bfe6070d84478f0 "" + "doc.aux" 1614658429 1163 a2a422c07ff91b546dabf95ba81d6b33 "pdflatex" + "doc.tex" 1614658631 23472 b184c802ebac7b1a4bfe6070d84478f0 "" "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" "q1cii.png" 1614640691 20581 6a6b3aa285fd3f5be369532a6d2a033d "" @@ -113,8 +127,8 @@ "q2c_fig2_2s.png" 1614652829 29139 9c3bdf9df7dc973686f5105c04140b36 "" "q2f_fig.png" 1614654314 34695 af1d8fa7a33b7f04767332fb391a90f6 "" (generated) - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" - "doc.pdf" + "doc.aux" "doc.log" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" - "doc.aux" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" + "doc.pdf" diff --git a/PS2/doc.fls b/PS2/doc.fls index bbd8d7b..f82d6f2 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -427,6 +427,56 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3kernel\expl3.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\xparse\xparse.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsmath\amstext.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3kernel\expl3.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\graphics\keyval.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty @@ -450,6 +500,43 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstop INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator-basic-dictionary-English.dict +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator-basic-dictionary-English.dict +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator-basic-dictionary-English.dict +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator-basic-dictionary-English.dict +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsa.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsa.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsa.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsa.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam10.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam7.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam5.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsb.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsb.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsb.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsb.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm10.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm7.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm5.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmss10.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmss8.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmss8.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmtt10.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmtt8.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmtt8.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmr17.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmr12.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmr8.tfm @@ -463,17 +550,15 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmsy6.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmex10.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex8.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsa.fd -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsa.fd -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsa.fd -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsa.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmss12.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmss8.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmss8.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmtt12.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmtt8.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmtt8.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam10.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam10.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam7.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsb.fd -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsb.fd -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsb.fd -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\amsfonts\umsb.fd INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm10.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm10.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm7.tfm @@ -495,14 +580,6 @@ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmr9.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam10.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam7.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam5.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm10.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm7.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm5.tfm INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png @@ -527,6 +604,8 @@ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmmi9.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmsy9.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex9.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmss9.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmtt9.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam10.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm10.tfm INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1civ_fig2.png diff --git a/PS2/doc.log b/PS2/doc.log index 31f4e7b..499f5ee 100644 --- a/PS2/doc.log +++ b/PS2/doc.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 20:20 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 21:13 entering extended mode **d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex (d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex @@ -232,43 +232,99 @@ Package: enumitem 2019/06/20 v3.9 Customized lists \enitdp@description=\count282 ) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/cancel\cancel.sty Package: cancel 2013/04/12 v2.2 Cancel math terms -) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/listings\lstlang1.sty +) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/siunitx\siunitx.sty +Package: siunitx 2020/02/25 v2.8b A comprehensive (SI) units package + (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/latex-tools\array.sty +Package: array 2020/10/01 v2.5c Tabular extension package (FMi) +\col@sep=\dimen175 +\ar@mcellbox=\box56 +\extrarowheight=\dimen176 +\NC@list=\toks27 +\extratabsurround=\skip64 +\backup@length=\skip65 +\ar@cellbox=\box57 +) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/l3packages/l3keys2e\l3keys2e.sty +Package: l3keys2e 2020-10-27 LaTeX2e option processing using LaTeX3 keys +) +\l__siunitx_tmp_box=\box58 +\l__siunitx_tmp_dim=\dimen177 +\l__siunitx_tmp_int=\count283 +\l__siunitx_number_mantissa_length_int=\count284 +\l__siunitx_number_uncert_length_int=\count285 +\l__siunitx_round_int=\count286 +\l__siunitx_process_decimal_int=\count287 +\l__siunitx_process_uncertainty_int=\count288 +\l__siunitx_process_fixed_int=\count289 +\l__siunitx_process_integer_min_int=\count290 +\l__siunitx_process_precision_int=\count291 +\l__siunitx_group_min_int=\count292 +\l__siunitx_angle_marker_box=\box59 +\l__siunitx_angle_unit_box=\box60 +\l__siunitx_angle_marker_dim=\dimen178 +\l__siunitx_angle_unit_dim=\dimen179 +\l__siunitx_unit_int=\count293 +\l__siunitx_unit_denominator_int=\count294 +\l__siunitx_unit_numerator_int=\count295 +\l__siunitx_unit_prefix_int=\count296 +\l__siunitx_unit_prefix_base_int=\count297 +\l__siunitx_unit_prefix_gram_int=\count298 +\l__siunitx_number_product_int=\count299 +\c__siunitx_one_fill_skip=\skip66 +\l__siunitx_table_unit_align_skip=\skip67 +\l__siunitx_table_exponent_dim=\dimen180 +\l__siunitx_table_integer_dim=\dimen181 +\l__siunitx_table_mantissa_dim=\dimen182 +\l__siunitx_table_marker_dim=\dimen183 +\l__siunitx_table_result_dim=\dimen184 +\l__siunitx_table_uncert_dim=\dimen185 +\l__siunitx_table_fill_pre_dim=\dimen186 +\l__siunitx_table_fill_post_dim=\dimen187 +\l__siunitx_table_fill_mid_dim=\dimen188 +\l__siunitx_table_pre_box=\box61 +\l__siunitx_table_post_box=\box62 +\l__siunitx_table_mantissa_box=\box63 +\l__siunitx_table_result_box=\box64 +\l__siunitx_table_number_align_skip=\skip68 +\l__siunitx_table_text_align_skip=\skip69 + (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/translator\translator.sty +Package: translator 2020-08-03 v1.12c Easy translation of strings in LaTeX +)) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/listings\lstlang1.sty File: lstlang1.sty 2020/03/24 1.8d listings language file ) (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) \openout1 = `doc.aux'. -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 73. -LaTeX Font Info: ... okay on input line 73. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 73. -LaTeX Font Info: ... okay on input line 73. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 73. -LaTeX Font Info: ... okay on input line 73. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 73. -LaTeX Font Info: ... okay on input line 73. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 73. -LaTeX Font Info: ... okay on input line 73. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 73. -LaTeX Font Info: ... okay on input line 73. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 73. -LaTeX Font Info: ... okay on input line 73. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 74. +LaTeX Font Info: ... okay on input line 74. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 74. +LaTeX Font Info: ... okay on input line 74. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 74. +LaTeX Font Info: ... okay on input line 74. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 74. +LaTeX Font Info: ... okay on input line 74. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 74. +LaTeX Font Info: ... okay on input line 74. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 74. +LaTeX Font Info: ... okay on input line 74. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 74. +LaTeX Font Info: ... okay on input line 74. (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/context/base/mkii\supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] -\scratchcounter=\count283 -\scratchdimen=\dimen175 -\scratchbox=\box56 -\nofMPsegments=\count284 -\nofMParguments=\count285 -\everyMPshowfont=\toks27 -\MPscratchCnt=\count286 -\MPscratchDim=\dimen176 -\MPnumerator=\count287 -\makeMPintoPDFobject=\count288 -\everyMPtoPDFconversion=\toks28 +\scratchcounter=\count300 +\scratchdimen=\dimen189 +\scratchbox=\box65 +\nofMPsegments=\count301 +\nofMParguments=\count302 +\everyMPshowfont=\toks28 +\MPscratchCnt=\count303 +\MPscratchDim=\dimen190 +\MPnumerator=\count304 +\makeMPintoPDFobject=\count305 +\everyMPtoPDFconversion=\toks29 ) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/epstopdf-pkg\epstopdf-base.sty Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 485. ) -\c@lstlisting=\count289 +\c@lstlisting=\count306 Package caption Info: Begin \AtBeginDocument code. Package caption Info: End \AtBeginDocument code. @@ -306,94 +362,100 @@ Package caption Info: End \AtBeginDocument code. * \@reversemarginfalse * (1in=72.27pt=25.4mm, 1cm=28.453pt) -LaTeX Font Info: Trying to load font information for U+msa on input line 75. -(C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/amsfonts\umsa.fd +(C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/siunitx\siunitx-abbreviations.cfg +File: siunitx-abbreviations.cfg 2017/11/26 v2.7k siunitx: Abbreviated units +) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/translator\translator-basic-dictionary-English.dict +Dictionary: translator-basic-dictionary, Language: English +) +LaTeX Font Info: Trying to load font information for U+msa on input line 74. + (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/amsfonts\umsa.fd File: umsa.fd 2013/01/14 v3.01 AMS symbols A ) -LaTeX Font Info: Trying to load font information for U+msb on input line 75. +LaTeX Font Info: Trying to load font information for U+msb on input line 74. (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/amsfonts\umsb.fd File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) [1 {C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map}] [2] -LaTeX Font Info: Trying to load font information for OMS+cmr on input line 149. +LaTeX Font Info: Trying to load font information for OMS+cmr on input line 150. (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/base\omscmr.fd File: omscmr.fd 2019/12/16 v2.5j Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10> not available -(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 149. +(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 150. File: q1a_1and50.png Graphic file (type png) -Package pdftex.def Info: q1a_1and50.png used on input line 157. +Package pdftex.def Info: q1a_1and50.png used on input line 158. (pdftex.def) Requested size: 202.3529pt x 151.76526pt. File: q1a_eigenvals.png Graphic file (type png) -Package pdftex.def Info: q1a_eigenvals.png used on input line 163. +Package pdftex.def Info: q1a_eigenvals.png used on input line 164. (pdftex.def) Requested size: 202.3529pt x 151.76526pt. [3 ] File: q1cii.png Graphic file (type png) -Package pdftex.def Info: q1cii.png used on input line 278. +Package pdftex.def Info: q1cii.png used on input line 279. (pdftex.def) Requested size: 505.89pt x 379.42523pt. [4] File: q1civ_fig1.png Graphic file (type png) -Package pdftex.def Info: q1civ_fig1.png used on input line 306. +Package pdftex.def Info: q1civ_fig1.png used on input line 307. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. File: q1civ_fig2.png Graphic file (type png) -Package pdftex.def Info: q1civ_fig2.png used on input line 311. +Package pdftex.def Info: q1civ_fig2.png used on input line 312. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. [5 ] File: q1di_fig1.png Graphic file (type png) -Package pdftex.def Info: q1di_fig1.png used on input line 340. +Package pdftex.def Info: q1di_fig1.png used on input line 341. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. File: q1di_fig2.png Graphic file (type png) -Package pdftex.def Info: q1di_fig2.png used on input line 345. +Package pdftex.def Info: q1di_fig2.png used on input line 346. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. File: q1div.jpg Graphic file (type jpg) -Package pdftex.def Info: q1div.jpg used on input line 360. +Package pdftex.def Info: q1div.jpg used on input line 361. (pdftex.def) Requested size: 505.89pt x 379.41656pt. [6 ] [7 ] [8] File: q2c_fig1_1s.png Graphic file (type png) -Package pdftex.def Info: q2c_fig1_1s.png used on input line 460. +Package pdftex.def Info: q2c_fig1_1s.png used on input line 461. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. File: q2c_fig2_2s.png Graphic file (type png) -Package pdftex.def Info: q2c_fig2_2s.png used on input line 465. +Package pdftex.def Info: q2c_fig2_2s.png used on input line 466. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. - + [9 ] + File: q2f_fig.png Graphic file (type png) -Package pdftex.def Info: q2f_fig.png used on input line 480. +Package pdftex.def Info: q2f_fig.png used on input line 501. (pdftex.def) Requested size: 505.89pt x 379.42523pt. - [9 ] [10 ] [11] [12] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) + [10 ] [11] [12] [13] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) Here is how much of TeX's memory you used: - 8404 strings out of 479618 - 126248 string characters out of 2865027 - 981750 words of memory out of 3000000 - 25459 multiletter control sequences out of 15000+200000 - 412866 words of font info for 64 fonts, out of 3000000 for 9000 + 10815 strings out of 479618 + 194800 string characters out of 2865027 + 1030669 words of memory out of 3000000 + 27840 multiletter control sequences out of 15000+200000 + 416160 words of font info for 78 fonts, out of 3000000 for 9000 1141 hyphenation exceptions out of 8191 - 68i,12n,75p,1278b,2408s stack positions out of 5000i,500n,10000p,200000b,50000s + 68i,19n,75p,1278b,2408s stack positions out of 5000i,500n,10000p,200000b,50000s -Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (12 pages, 3711631 bytes). +Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (13 pages, 3715189 bytes). PDF statistics: - 130 PDF objects out of 1000 (max. 8388607) + 134 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) 56 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index 7b37162..2c4d1f3 100644 Binary files a/PS2/doc.pdf and b/PS2/doc.pdf differ diff --git a/PS2/doc.synctex.gz b/PS2/doc.synctex.gz index f554f76..41d2003 100644 Binary files a/PS2/doc.synctex.gz and b/PS2/doc.synctex.gz differ diff --git a/PS2/doc.tex b/PS2/doc.tex index fc13037..2f51ed3 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -16,6 +16,7 @@ \usepackage{lipsum} \usepackage{enumitem} \usepackage{cancel} +\usepackage{siunitx} \titleformat{\subsection}[runin]{\normalfont \large \bfseries} {\thesubsection}{1em}{} @@ -284,7 +285,7 @@ legend({'Numerical','Analytical'},'Location','northwest'); \item %1ciii - Applying the approximation $cos(\Theta) = 1 - \frac{\Theta^2}{2}$ for small $\Theta$, + Applying the approximation $cos(\theta) = 1 - \frac{\theta^2}{2}$ for small $\theta$, on equation (\ref{eq:numerical_E}), we get the following expression \begin{equation*} @@ -469,7 +470,27 @@ axis([0 1e-9 0 0.04]); \item %2d For the 1s level, \(E = -13.4978\) eV. \item %2e - \lipsum[1] + Beginning with equation (11) from the assignment, with \(l_o=0\): + \begin{align*} + \left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2} - \frac{q^2}{4\pi\epsilon_0r}\right]f(r) &= Ef(r)\\ + -\frac{\hbar^2}{2m}\frac{d^2}{dr^2}\left(\frac{2r}{a_0^{3/2}}e^{-r/a_0}\right) - \frac{q^2}{4\pi\epsilon_0r}f(r) &= Ef(r)\\ + -\frac{\hbar^2}{2m}\frac{2}{a_0^{3/2}}\frac{d}{dr}\left(e^{-r/a_0} - \frac{r}{a_0}e^{-r/a_0}\right) - \frac{q^2}{4\pi\epsilon_0r}f(r) &= Ef(r)\\ + -\frac{\hbar^2}{2m}\frac{2}{a_0^{3/2}}\left(-\frac{1}{a_0}e^{-r/a_0} - \frac{1}{a_0}e^{-r/a_0} + \frac{r}{a_0^2}e^{-r/a_0}\right) - \frac{q^2}{4\pi\epsilon_0r}f(r) &= Ef(r)\\ + -\frac{\hbar^2}{2m}\left(-\frac{2}{a_0r} + \frac{1}{a_0^2}\right)f(r) - \frac{q^2}{4\pi\epsilon_0r}f(r) &= Ef(r)\\ + -\frac{\hbar^2}{2m}\left(-\frac{2}{a_0r} + \frac{1}{a_0^2}\right) - \frac{q^2}{4\pi\epsilon_0r} &= E. + \end{align*} + Recalling that \(a_0 = 4\pi\epsilon_0\hbar^2/mq^2\), we can eliminate \(r\): + \begin{align*} + \cancel{\frac{\hbar^2}{2m}}\frac{\cancel{2m}q^2}{4\pi\epsilon_0\cancel{\hbar^2}}\frac{1}{r} - \frac{\hbar^2}{2m}\frac{1}{a_0^2} - \frac{q^2}{4\pi\epsilon_0}\frac{1}{r} &= E\\ + \cancel{\frac{q^2}{4\pi\epsilon_0}\frac{1}{r}} - \frac{\hbar^2}{2m}\frac{1}{a_0^2} - \cancel{\frac{q^2}{4\pi\epsilon_0}\frac{1}{r}} &= E. + \end{align*} + We can then solve for \(E\): + \begin{equation*} + E\:\mathrm{[eV]} = -\frac{1}{q}\cdot\frac{\hbar^2}{2ma_0^2} = -\frac{1}{q}\cdot\frac{(\SI{1.054e-34}{\joule\cdot\second})^2}{2(\SI{9.110e-31}{\kilogram})(\SI{0.0529}{\nm})^2} + = - 13.6\:\mathrm{[eV]} + \end{equation*} + This is very similar to the result in (d). + \item %2f In the figure below we can see that the numerical and analytical results agree up to scaling by \(a\). The scale difference is expected, as discussed in Problem 1. From (d), we also expect agreement in the curve shapes because the numerical and analytical energies for the 1s @@ -490,8 +511,13 @@ axis([0 1e-9 0 0.04]); \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] \item %3a - \lipsum[1] + + \begin{equation} + \theta (x') = \sqrt{\frac{2}{L}} sin \left( \frac{\pi (x' + L/2)}{L} \right) = \sqrt{\frac{2}{L}} cos \left( \frac{\pi x'}{L} \right) + \end{equation} + \item %3b + \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %3bi \lipsum[1] diff --git a/PS2/q4a.m b/PS2/q4a.m new file mode 100644 index 0000000..7917caf --- /dev/null +++ b/PS2/q4a.m @@ -0,0 +1,74 @@ +clear all; + +% Physical constants in MKS units, including free-electron mass m + +hbar = 1.054e-34; +q = 1.602e-19; +m = 9.110e-31; + +% SHO frequency + +w0 = (0.2879*q/hbar); + +% Lattice data: lattice constant "a"; SHO distance parameter beta; +% a discretized spatial grid x; total number of lattice points N; +% Hamiltonian parameter t0, as introduced in class, but now +% divided by q so that it has units of eV; total length of the lattice, +% L = a*(N+1) + +% YOU MUST ENTER AN APPROPRIATE VALUE OF "a" + +a = ; +beta = sqrt(m*w0/hbar); +x = [-6/beta:a:6/beta]; +N = length(x); +t0 = (hbar^2)/(2*m*a^2)/q; +L = a*(N+1); + +% Set up the Hamiltonian matrix, in pieces, by constructing square +% matrices containing information for the main +% diagonal, lower diagonal, and upper diagonal, and then adding them +% together; you may need to look up the diag command using help +% to see how this works; U is the potential energy + +U = (1/q)*(0.5*m*w0^2*x.^2); +main_D = diag(2*t0*ones(1,N)+U,0); +lower_D = diag(-t0*ones(1,N-1),-1); +upper_D = diag(-t0*ones(1,N-1),+1); + +H = main_D + lower_D + upper_D; + +% Find the eigenvectors and eigenvalues of H; V will contain the +% eigenvectors in its columns, and D will contain the corresponding +% eigenvalues on its main diagonal + +[V,D] = eig(H); + +% Find the numerical eigenvalues + +E_numerical = diag(D,0); + +% Find the eigenvectors corresponding to sorted eigenvalues 1 and 50; +% we can get these from the matrix V, provided we reference the +% appropriate UNSORTED eigenvalue number + +phi_numerical_1 = V(:,1); +phi_numerical_2 = V(:,2); + +% Find the position-probability densities + +P_numerical_1 = phi_numerical_1 .* conj(phi_numerical_1); +P_numerical_2 = phi_numerical_2 .* conj(phi_numerical_2); + +% Plot out the relevant results + +figure(1); h = plot(x,P_numerical_1,'k-'); grid on; +set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); +xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]'); +legend('Eigenvector 1'); + +figure(2); h = plot(x,P_numerical_2,'k-'); grid on; +set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); +xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]'); +legend('Eigenvector 2'); +