Solved general equation

ma5/uppg/main.fdb_latexmk

@@ -1,75 +1,86 @@
 # Fdb version 3
-["pdflatex"] 1648131639 "main.tex" "main.pdf" "main" 1648131640
+["pdflatex"] 1648150168 "main.tex" "main.pdf" "main" 1648150169
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ma5/uppg/main.fls

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 Package: url 2013/09/16  ver 3.4  Verb mode for urls, etc.
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 Package: atbegshi-ltx 2021/01/10 v1.0c Emulation of the original atbegshi
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 Package hyperref Info: Driver (autodetected): hpdftex.
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 LaTeX Info: Redefining \frac on input line 234.
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 LaTeX Info: Redefining \overline on input line 399.
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 LaTeX Info: Redefining \ldots on input line 496.
 LaTeX Info: Redefining \dots on input line 499.
 LaTeX Info: Redefining \cdots on input line 620.
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 LaTeX Font Info:    Redeclaring font encoding OMS on input line 744.
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 LaTeX Info: Redefining \[ on input line 2938.
 LaTeX Info: Redefining \] on input line 2939.
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+Output written on main.pdf (3 pages, 168684 bytes).
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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}