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Solved general equation

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E. Almqvist 3 years ago
parent 1a6de05f09
commit 05047ef105
6 changed files with 267 additions and 185 deletions
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  1. 153
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  2. 32
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  3. 220
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  4. BIN
      ma5/uppg/main.pdf
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      ma5/uppg/main.synctex.gz
  6. 47
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153
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# Fdb version 3
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@ -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}

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