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@@ -6,19 +6,19 @@ entering extended mode **main.tex (./main.tex LaTeX2e <2021-11-15> patch level 1 -L3 programming layer <2021-11-22> (/usr/share/texmf-dist/tex/latex/base/article.cls +L3 programming layer <2022-02-24> (/usr/share/texmf-dist/tex/latex/base/article.cls Document Class: article 2021/10/04 v1.4n Standard LaTeX document class (/usr/share/texmf-dist/tex/latex/base/size10.clo File: size10.clo 2021/10/04 v1.4n Standard LaTeX file (size option) ) -\c@part=\count183 -\c@section=\count184 -\c@subsection=\count185 -\c@subsubsection=\count186 -\c@paragraph=\count187 -\c@subparagraph=\count188 -\c@figure=\count189 -\c@table=\count190 +\c@part=\count185 +\c@section=\count186 +\c@subsection=\count187 +\c@subsubsection=\count188 +\c@paragraph=\count189 +\c@subparagraph=\count190 +\c@figure=\count191 +\c@table=\count192 \abovecaptionskip=\skip47 \belowcaptionskip=\skip48 \bibindent=\dimen138 @@ -30,11 +30,11 @@ Package: keyval 2014/10/28 v1.15 key=value parser (DPC) ) (/usr/share/texmf-dist/tex/generic/iftex/ifvtex.sty Package: ifvtex 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead. (/usr/share/texmf-dist/tex/generic/iftex/iftex.sty -Package: iftex 2020/03/06 v1.0d TeX engine tests +Package: iftex 2022/02/03 v1.0f TeX engine tests )) -\Gm@cnth=\count191 -\Gm@cntv=\count192 -\c@Gm@tempcnt=\count193 +\Gm@cnth=\count193 +\Gm@cntv=\count194 +\c@Gm@tempcnt=\count195 \Gm@bindingoffset=\dimen139 \Gm@wd@mp=\dimen140 \Gm@odd@mp=\dimen141 @@ -61,7 +61,7 @@ Package: titling 2009/09/04 v2.1d maketitle typesetting \thanksmargin=\skip52 \droptitle=\skip53 ) (/usr/share/texmf-dist/tex/latex/hyperref/hyperref.sty -Package: hyperref 2021-06-07 v7.00m Hypertext links for LaTeX +Package: hyperref 2022-02-21 v7.00n Hypertext links for LaTeX (/usr/share/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty Package: ltxcmds 2020-05-10 v1.25 LaTeX kernel commands for general use (HO) ) (/usr/share/texmf-dist/tex/generic/pdftexcmds/pdftexcmds.sty @@ -88,73 +88,71 @@ Package: auxhook 2019-12-17 v1.6 Hooks for auxiliary files (HO) Package: kvoptions 2020-10-07 v3.14 Key value format for package options (HO) ) \@linkdim=\dimen152 -\Hy@linkcounter=\count194 -\Hy@pagecounter=\count195 +\Hy@linkcounter=\count196 +\Hy@pagecounter=\count197 (/usr/share/texmf-dist/tex/latex/hyperref/pd1enc.def -File: pd1enc.def 2021-06-07 v7.00m Hyperref: PDFDocEncoding definition (HO) +File: pd1enc.def 2022-02-21 v7.00n Hyperref: PDFDocEncoding definition (HO) Now handling font encoding PD1 ... ... no UTF-8 mapping file for font encoding PD1 -) (/usr/share/texmf-dist/tex/latex/hyperref/hyperref-langpatches.def -File: hyperref-langpatches.def 2021-06-07 v7.00m Hyperref: patches for babel languages ) (/usr/share/texmf-dist/tex/generic/intcalc/intcalc.sty Package: intcalc 2019/12/15 v1.3 Expandable calculations with integers (HO) ) (/usr/share/texmf-dist/tex/generic/etexcmds/etexcmds.sty Package: etexcmds 2019/12/15 v1.7 Avoid name clashes with e-TeX commands (HO) ) -\Hy@SavedSpaceFactor=\count196 +\Hy@SavedSpaceFactor=\count198 (/usr/share/texmf-dist/tex/latex/hyperref/puenc.def -File: puenc.def 2021-06-07 v7.00m Hyperref: PDF Unicode definition (HO) +File: puenc.def 2022-02-21 v7.00n Hyperref: PDF Unicode definition (HO) Now handling font encoding PU ... ... no UTF-8 mapping file for font encoding PU ) -Package hyperref Info: Hyper figures OFF on input line 4192. -Package hyperref Info: Link nesting OFF on input line 4197. -Package hyperref Info: Hyper index ON on input line 4200. -Package hyperref Info: Plain pages OFF on input line 4207. -Package hyperref Info: Backreferencing OFF on input line 4212. +Package hyperref Info: Hyper figures OFF on input line 4137. +Package hyperref Info: Link nesting OFF on input line 4142. +Package hyperref Info: Hyper index ON on input line 4145. +Package hyperref Info: Plain pages OFF on input line 4152. +Package hyperref Info: Backreferencing OFF on input line 4157. Package hyperref Info: Implicit mode ON; LaTeX internals redefined. -Package hyperref Info: Bookmarks ON on input line 4445. -\c@Hy@tempcnt=\count197 +Package hyperref Info: Bookmarks ON on input line 4390. +\c@Hy@tempcnt=\count199 (/usr/share/texmf-dist/tex/latex/url/url.sty \Urlmuskip=\muskip16 Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. ) -LaTeX Info: Redefining \url on input line 4804. +LaTeX Info: Redefining \url on input line 4749. \XeTeXLinkMargin=\dimen153 (/usr/share/texmf-dist/tex/generic/bitset/bitset.sty Package: bitset 2019/12/09 v1.3 Handle bit-vector datatype (HO) (/usr/share/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty Package: bigintcalc 2019/12/15 v1.5 Expandable calculations on big integers (HO) )) -\Fld@menulength=\count198 +\Fld@menulength=\count266 \Field@Width=\dimen154 \Fld@charsize=\dimen155 -Package hyperref Info: Hyper figures OFF on input line 6076. -Package hyperref Info: Link nesting OFF on input line 6081. -Package hyperref Info: Hyper index ON on input line 6084. -Package hyperref Info: backreferencing OFF on input line 6091. -Package hyperref Info: Link coloring OFF on input line 6096. -Package hyperref Info: Link coloring with OCG OFF on input line 6101. -Package hyperref Info: PDF/A mode OFF on input line 6106. -LaTeX Info: Redefining \ref on input line 6146. -LaTeX Info: Redefining \pageref on input line 6150. +Package hyperref Info: Hyper figures OFF on input line 6027. +Package hyperref Info: Link nesting OFF on input line 6032. +Package hyperref Info: Hyper index ON on input line 6035. +Package hyperref Info: backreferencing OFF on input line 6042. +Package hyperref Info: Link coloring OFF on input line 6047. +Package hyperref Info: Link coloring with OCG OFF on input line 6052. +Package hyperref Info: PDF/A mode OFF on input line 6057. +LaTeX Info: Redefining \ref on input line 6097. +LaTeX Info: Redefining \pageref on input line 6101. (/usr/share/texmf-dist/tex/latex/base/atbegshi-ltx.sty Package: atbegshi-ltx 2021/01/10 v1.0c Emulation of the original atbegshi package with kernel methods ) -\Hy@abspage=\count199 -\c@Item=\count266 -\c@Hfootnote=\count267 +\Hy@abspage=\count267 +\c@Item=\count268 +\c@Hfootnote=\count269 ) Package hyperref Info: Driver (autodetected): hpdftex. (/usr/share/texmf-dist/tex/latex/hyperref/hpdftex.def -File: hpdftex.def 2021-06-07 v7.00m Hyperref driver for pdfTeX +File: hpdftex.def 2022-02-21 v7.00n Hyperref driver for pdfTeX (/usr/share/texmf-dist/tex/latex/base/atveryend-ltx.sty Package: atveryend-ltx 2020/08/19 v1.0a Emulation of the original atveryend package with kernel methods ) -\Fld@listcount=\count268 -\c@bookmark@seq@number=\count269 +\Fld@listcount=\count270 +\c@bookmark@seq@number=\count271 (/usr/share/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty Package: rerunfilecheck 2019/12/05 v1.9 Rerun checks for auxiliary files (HO) (/usr/share/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty @@ -165,14 +163,14 @@ Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 28 \Hy@SectionHShift=\skip54 ) (/usr/share/texmf-dist/tex/latex/tools/multicol.sty Package: multicol 2021/10/28 v1.9b multicolumn formatting (FMi) -\c@tracingmulticols=\count270 +\c@tracingmulticols=\count272 \mult@box=\box51 \multicol@leftmargin=\dimen156 -\c@unbalance=\count271 -\c@collectmore=\count272 -\doublecol@number=\count273 -\multicoltolerance=\count274 -\multicolpretolerance=\count275 +\c@unbalance=\count273 +\c@collectmore=\count274 +\doublecol@number=\count275 +\multicoltolerance=\count276 +\multicolpretolerance=\count277 \full@width=\dimen157 \page@free=\dimen158 \premulticols=\dimen159 @@ -222,15 +220,15 @@ Package: multicol 2021/10/28 v1.9b multicolumn formatting (FMi) \@tempa=\box91 \@tempa=\box92 \@tempa=\box93 -\c@minrows=\count276 -\c@columnbadness=\count277 -\c@finalcolumnbadness=\count278 +\c@minrows=\count278 +\c@columnbadness=\count279 +\c@finalcolumnbadness=\count280 \last@try=\dimen162 \multicolovershoot=\dimen163 \multicolundershoot=\dimen164 \mult@nat@firstbox=\box94 \colbreak@box=\box95 -\mc@col@check@num=\count279 +\mc@col@check@num=\count281 ) (/usr/share/texmf-dist/tex/latex/amsmath/amsmath.sty Package: amsmath 2021/10/15 v2.17l AMS math features \@mathmargin=\skip57 @@ -248,13 +246,13 @@ Package: amsbsy 1999/11/29 v1.2d Bold Symbols ) (/usr/share/texmf-dist/tex/latex/amsmath/amsopn.sty Package: amsopn 2021/08/26 v2.02 operator names ) -\inf@bad=\count280 +\inf@bad=\count282 LaTeX Info: Redefining \frac on input line 234. -\uproot@=\count281 -\leftroot@=\count282 +\uproot@=\count283 +\leftroot@=\count284 LaTeX Info: Redefining \overline on input line 399. -\classnum@=\count283 -\DOTSCASE@=\count284 +\classnum@=\count285 +\DOTSCASE@=\count286 LaTeX Info: Redefining \ldots on input line 496. LaTeX Info: Redefining \dots on input line 499. LaTeX Info: Redefining \cdots on input line 620. @@ -263,15 +261,15 @@ LaTeX Info: Redefining \cdots on input line 620. \big@size=\dimen167 LaTeX Font Info: Redeclaring font encoding OML on input line 743. LaTeX Font Info: Redeclaring font encoding OMS on input line 744. -\macc@depth=\count285 -\c@MaxMatrixCols=\count286 +\macc@depth=\count287 +\c@MaxMatrixCols=\count288 \dotsspace@=\muskip17 -\c@parentequation=\count287 -\dspbrk@lvl=\count288 +\c@parentequation=\count289 +\dspbrk@lvl=\count290 \tag@help=\toks20 -\row@=\count289 -\column@=\count290 -\maxfields@=\count291 +\row@=\count291 +\column@=\count292 +\maxfields@=\count293 \andhelp@=\toks21 \eqnshift@=\dimen168 \alignsep@=\dimen169 @@ -285,31 +283,38 @@ LaTeX Font Info: Redeclaring font encoding OMS on input line 744. \mathdisplay@stack=\toks23 LaTeX Info: Redefining \[ on input line 2938. LaTeX Info: Redefining \] on input line 2939. +) (/usr/share/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Redeclaring math symbol \hbar on input line 98. +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 106. ) (/usr/share/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def -File: l3backend-pdftex.def 2021-12-14 L3 backend support: PDF output (pdfTeX) -\l__color_backend_stack_int=\count292 +File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX) +\l__color_backend_stack_int=\count294 \l__pdf_internal_box=\box98 ) (./main.aux) \openout1 = `main.aux'. -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 62. -LaTeX Font Info: ... okay on input line 62. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 62. -LaTeX Font Info: ... okay on input line 62. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 62. -LaTeX Font Info: ... okay on input line 62. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 62. -LaTeX Font Info: ... okay on input line 62. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 62. -LaTeX Font Info: ... okay on input line 62. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 62. -LaTeX Font Info: ... okay on input line 62. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 62. -LaTeX Font Info: ... okay on input line 62. -LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 62. -LaTeX Font Info: ... okay on input line 62. -LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 62. -LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 63. +LaTeX Font Info: ... okay on input line 63. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 63. +LaTeX Font Info: ... okay on input line 63. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 63. +LaTeX Font Info: ... okay on input line 63. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 63. +LaTeX Font Info: ... okay on input line 63. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 63. +LaTeX Font Info: ... okay on input line 63. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 63. +LaTeX Font Info: ... okay on input line 63. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 63. +LaTeX Font Info: ... okay on input line 63. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 63. +LaTeX Font Info: ... okay on input line 63. +LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 63. +LaTeX Font Info: ... okay on input line 63. *geometry* driver: auto-detecting *geometry* detected driver: pdftex @@ -345,7 +350,7 @@ LaTeX Font Info: ... okay on input line 62. * \@reversemarginfalse * (1in=72.27pt=25.4mm, 1cm=28.453pt) -Package hyperref Info: Link coloring OFF on input line 62. +Package hyperref Info: Link coloring OFF on input line 63. (/usr/share/texmf-dist/tex/latex/hyperref/nameref.sty Package: nameref 2021-04-02 v2.47 Cross-referencing by name of section (/usr/share/texmf-dist/tex/latex/refcount/refcount.sty @@ -353,34 +358,41 @@ Package: refcount 2019/12/15 v3.6 Data extraction from label references (HO) ) (/usr/share/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty Package: gettitlestring 2019/12/15 v1.6 Cleanup title references (HO) ) -\c@section@level=\count293 +\c@section@level=\count295 ) -LaTeX Info: Redefining \ref on input line 62. -LaTeX Info: Redefining \pageref on input line 62. -LaTeX Info: Redefining \nameref on input line 62. +LaTeX Info: Redefining \ref on input line 63. +LaTeX Info: Redefining \pageref on input line 63. +LaTeX Info: Redefining \nameref on input line 63. (./main.out) (./main.out) \@outlinefile=\write3 \openout3 = `main.out'. - [1 +LaTeX Font Info: Trying to load font information for U+msa on input line 68. + (/usr/share/texmf-dist/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 68. + (/usr/share/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2013/01/14 v3.01 AMS symbols B +) [1 {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] (./main.aux) Package rerunfilecheck Info: File `main.out' has not changed. 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PDF statistics: - 116 PDF objects out of 1000 (max. 8388607) - 84 compressed objects within 1 object stream + 131 PDF objects out of 1000 (max. 8388607) + 93 compressed objects within 1 object stream 15 named destinations out of 1000 (max. 500000) 33 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/ma5/uppg/main.pdf b/ma5/uppg/main.pdf index 901649b..f642136 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 0c1acfb..8b67c93 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 daf912c..e35142f 100644 --- a/ma5/uppg/main.tex +++ b/ma5/uppg/main.tex @@ -6,6 +6,7 @@ \usepackage[hidelinks]{hyperref} \usepackage{multicol} \usepackage{amsmath} +\usepackage{amsfonts} \titleformat{\section} {\Large\bfseries} @@ -94,7 +95,7 @@ Slutligen, eftersom $\shrodprob$ motsvarar sannolikhetstätheten för att partik \item Partikelns fullständiga vågfunktion är egentligen även en funktion utav tiden. För en partikel som befinner sig i ett så kallat energiegentillstånd är den fullständiga vågfunktionen $\fullshrodequ$ Dock innebär den extra faktorn $\shrodtime$ inte någon intressant tidsutveckling av sannolikhetsfördelningen eftersom $|\Psi(x, t)|^2 = |\psi_n(x) \shrodtime|^2 = \shrodprob$. Intressantare blir det om en partikel befinner sig i en superposition av energiegentillstånd, tex: $$\Psi(x, t) = A(\psi_1(x)e^{-i \frac{E_1}{\hbar}t} + \psi_2(x) e^{-i \frac{E_2}{\hbar}t})$$ För denna vågfunktion, \emph{bestäm konstanten $A$ sådan att}: $$\int_0^L |\Psi(x, t)|^2 dx = 1.0$$ \emph{Undersök sedan hur sannolikheten att befinna sig i den vänstra delen $0 < x < \frac{L}{2}$, respektive högra $\frac{L}{2} < x < L$ delen av lådan}. Hitta alltså ett uttryck för: $$P(V, t) = \int_0^{\frac{L}{2}} \fullshrodprob dx$$ $$P(H, t) = \int_{\frac{L}{2}}^L \fullshrodprob dx$$ - \item Gör sedan samma sak för superpositionen av energiegentillstånden 1 och 3: $$\Psi(x, t) = A(\psi_1(x)e^{-i \frac{E_1}{\hbar}t} + \psi_3(x)e^{-i \frac{E_3}{\hbar}t})$$ \emph{På vilket sätt skiljer de sig? Kan du förklara varför?} + \item Gör sedan samma sak för superpositionen av energiegentillstånden 1 och 3: $$\Psi(x, t) = A\left(\psi_1(x)e^{-i \frac{E_1}{\hbar}t} + \psi_3(x)e^{-i \frac{E_3}{\hbar}t}\right)$$ \emph{På vilket sätt skiljer de sig? Kan du förklara varför?} \end{enumerate} \newpage @@ -105,23 +106,57 @@ Slutligen, eftersom $\shrodprob$ motsvarar sannolikhetstätheten för att partik Enligt Schrödingers ekvation får vi: $\shrodequ$ där $\hbar = \frac{h}{2\pi}$ vilket vi kan substituera i ekvationen och vi får följande: $$ -\shrodequ, \quad \left[\hbar / \frac{h}{2\pi}\right] + \shrodequ, \quad \left[\hbar / \frac{h}{2\pi}\right] $$ $$ -E_n\psi_n(x) = - \frac{\left(\frac{h}{2\pi}\right)^2}{2m} \frac{d^2 \psi_n}{dx^2} = - \frac{h^2}{8 \pi^2 m} \left(\frac{d^2 \psi_n}{dx^2}\right) + E_n\psi_n(x) = - \frac{\left(\frac{h}{2\pi}\right)^2}{2m} \frac{d^2 \psi_n}{dx^2} = - \frac{h^2}{8 \pi^2 m} \left(\frac{d^2 \psi_n}{dx^2}\right) $$ där $h$ är Plancks konstant och $m$ är partikelns massa. Väljer därmed att förenkla uttrycket genom att byta ut konstanterna till en variabel (givet att $k = \frac{h^2}{8 \pi ^2 m}$): $$ -E_n\psi_n(x) = - \frac{h^2}{8 \pi ^2 m} \left(\frac{d^2 \psi_n}{dx^2}\right), \quad \left[\frac{h^2}{8 \pi ^2 m}/k\right] + E_n\psi_n(x) = - \frac{h^2}{8 \pi ^2 m} \left(\frac{d^2 \psi_n}{dx^2}\right), \quad \left[\frac{h^2}{8 \pi ^2 m}/k\right] $$ $$ -E_n\psi_n(x) = -k\left(\frac{d^2 \psi_n}{dx^2}\right), \quad / E_n + E_n\psi_n(x) = -k\left(\frac{d^2 \psi_n}{dx^2}\right), \quad + HL $$ $$ -\psi_n(x) = -\frac{k}{E_n} \psi_n''(x) + E_n\psi_n(x) + k\left(\frac{d^2 \psi_n}{dx^2}\right) = 0 $$ +Väljer att skriva om differentialekvationen utan Leibnizs notation och vi får: +$$ + E_n\psi_n + k \psi_n'' = 0, \quad /E_n +$$ +$$ + \psi_n'' + \frac{E_n}{k}\psi_n = 0 +$$ + +Vet att differentialekvationer av andra ordningen har lösningen $y=e^{\lambda x}$ och vi kan därmed beräkna $\lambda$ för vår differentialekvation genom den karakteristiska ekvationen: +$$ + \lambda^2 + a\lambda + b = 0 +$$ + +där $a$ och $b$ är koefficienterna framför respektive "funktion". I vårt fall är $a=0$ och $b = \frac{E_n}{k}$ och vi får därmed den karakteristiska ekvationen: +$$ + \lambda^2 + \frac{E_n}{k} = 0, \quad PQ +$$ +$$ + \lambda = \pm \sqrt{-\frac{E_n}{k}} +$$ +$$ + \lambda = \pm \sqrt{\frac{E_n}{k}}i +$$ + +Då rötterna för den karakteristiska ekvationen är complexa ($\in \mathbb{C}$) får vi den \emph{allmäna lösningen}: +$$ + \psi_n(x) = e^{ax}\left(C \cos bx + D \sin bx\right) \quad | \quad C,D \in \mathbb{R}, \quad \lambda = a + bi +$$ +$$ + \psi_n(x) = e^{0}\left( C \cos \pm\sqrt{\frac{E_n}{k}}x + D \sin \pm\sqrt{\frac{E_n}{k}}x \right) = C \cos \left(\pm\sqrt{\frac{E_n}{k}}x\right) + D \sin \left(\pm\sqrt{\frac{E_n}{k}}x\right) +$$ + +----------------------------------------- \\ +% Last adjustments Schrödinger ekvationen lyder också att vågfunktionen skall följa både ekvation \ref{shrodequ_con1} och \ref{shrodequ_con2} vilket ger $$ \begin{cases}