mirror of https://github.com/E-Almqvist/hsf
parent
61f0f65ea9
commit
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\documentclass{article} |
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|
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\usepackage[margin=2cm]{geometry} |
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\usepackage{titlesec} |
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\usepackage{titling} |
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\usepackage[hidelinks]{hyperref} |
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\usepackage{multicol} |
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|
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\titleformat{\section} |
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{\Large\bfseries} |
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{} |
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{0em} |
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{}[\titlerule] |
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\titleformat{\subsection} |
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{\large\bfseries} |
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{} |
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{0em} |
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{} |
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\titlespacing{\subsection} |
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{0em}{0em}{.4em} |
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\titleformat{\subsubsection}[runin] |
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{\bfseries} |
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{} |
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{0em} |
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{}[:] |
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\titlespacing{\subsubsection} |
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{0em}{0em}{1em} |
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\renewcommand{\maketitle}{ |
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\begin{center} |
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{\huge\bfseries\thetitle}\\ |
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\vspace{1em} |
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{\Large\theauthor} \\ |
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\vspace{1em} |
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elalmqvist@gmail.com --- \url{https://wych.dev} |
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\end{center} |
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} |
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|
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% Wave function |
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\newcommand{\wavefun}{\psi_n(x)} |
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|
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% Schrödingers equation |
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\newcommand{\shrodequ}{E_n \psi_n(x) = - \frac{\hbar}{2m} \frac{d^2 \psi_n}{dx^2}} |
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|
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% Probability density for the particle |
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\newcommand{\shrodprob}{|\psi_n(x)|^2} |
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|
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% Time factor thing |
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\newcommand{\shrodtime}{e^{-i \frac{E_n}{\hbar} t}} |
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|
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% Full Schrödinger equation |
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\newcommand{\fullshrodequ}{\Psi_n(x, t) = \psi_n(x) \shrodtime} |
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|
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% Full Schrödinger equation prob density |
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\newcommand{\fullshrodprob}{|\Psi_n(x,t)|^2} |
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|
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|
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\begin{document} |
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|
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\title{Schrödinger ekvationen (partikel i låda)} |
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\author{Elias Almqvist} |
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|
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\maketitle |
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\newpage |
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|
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\section{Uppgiftbeskrivning (taget från dokumentet)} |
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En partikel i en låda är en utav de första tillämpningarna man stöter på när man lär sig om kvantfysik. Man betraktar då en partikel (t.ex. en elektron) som befinner sig i en låda med oändligt höga väggar. |
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För detta undersöker man partikelns vågfunktion $\wavefun$. Vågfunktionen är i allmänhet en komplex funktion, |
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dvs den har både en realdel och en imaginärdel. Vågfunktionens absolutbelopp i kvadrat, $\shrodprob$, represen- |
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terar täthetsfunktionen för att partikeln skall befinna sig vid läge $x$ i lådan. Om partikeln befinner sig i ett så |
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kallat energiegentillstånd så uppfyller den den tidsoberoende Schrödinger ekvationen: |
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\begin{equation} |
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\shrodequ |
||||
\end{equation} |
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|
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där $E_n$ är partikelns energi, $\hbar = \frac{h}{2\pi}$ och $m$ är partikelns massa. |
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Att lådans väggar är oändligt höga innebär att vågfunktionen också behöver uppfylla randvillkoren: |
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$$ |
||||
\psi_n(0) = \psi_n(L) = 0 \quad \& \quad \psi_n'(0) = \psi_n'(L) = 0 |
||||
$$ |
||||
Slutligen, eftersom $\shrodprob$ motsvarar sannolikhetstätheten för att partikeln skall befinna sig vid position $x$, så måste det gälla att: |
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$$ |
||||
\int_0^L \shrodprob dx\ = 1.0 |
||||
$$ |
||||
|
||||
\subsection{Uppgifter} |
||||
\begin{enumerate} |
||||
\item Hitta de olika möjliga värden på $E_n$, och hitta motsvarande vågfunktioner $\wavefun$. Visa grafer över motsvarande sannolikhetsfördelningar för att partikeln skall befinna sig vid olika positioner $x$. |
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|
||||
\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?} |
||||
\end{enumerate} |
||||
|
||||
\newpage |
||||
|
||||
\section{Uppgiftlösningar} |
||||
\subsection{1} |
||||
|
||||
|
||||
\end{document} |
||||
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Reference in new issue