Merge branch 'master' of git+ssh://git@git.eluhost.de:522/qm1-script
This commit is contained in:
Daniel Bahrdt
commit
207cd4749b
154
kapII-0.tex
154
kapII-0.tex
@ -41,3 +41,157 @@ Die möglichen $E$-Werte sind die Eigenwerte des $H$-Operators. Diese Form der P
|
||||
\end{equation}
|
||||
\item H-Operator ist durch das Analogon zur klassischen Hamiltonfunktion gegeben
|
||||
\end{enumerate}
|
||||
|
||||
\section{Beispiel 1: $\infty$-Potentialtopf}
|
||||
%\begin{figure}[h]
|
||||
%\includegraphics{pdf/II/00-02-00.pdf}
|
||||
%\end{figure}
|
||||
\begin{equation}
|
||||
V(x) = \left\lbrace \begin{array}{ll} \infty &\text{für } \abs{x} > a\\ 0 &\text{für } \abs{x} < a \end{array} \right.
|
||||
\end{equation}
|
||||
|
||||
\paragraph*{klassisch}
|
||||
$x(t_0), p(t_0) = \sqrt{2m E}$
|
||||
%\begin{figure}[h]
|
||||
%\includegraphics{pdf/II/00-02-01.pdf}
|
||||
%\end{figure}
|
||||
|
||||
\paragraph*{quantal}
|
||||
\subparagraph*{Schritt 1} Stationäre Zustände
|
||||
\begin{equation}
|
||||
\left( -\frac{\hbar^2}{2m}~\partial_x^2 + V(x) \right) \phi(x) \stackrel{!}{=} E \phi(x)
|
||||
\end{equation}
|
||||
mit $V(x) = 0$ für $\abs{x} < a$.\\
|
||||
Randbedingung: $\phi(\pm a) = 0$
|
||||
\begin{equation}
|
||||
\diffPs{x}^2 \phi(x) = -\frac{2 m E}{\hbar} \phi(x)
|
||||
\end{equation}
|
||||
Lösung:
|
||||
\begin{enumerate}
|
||||
\item symmetrisch
|
||||
\begin{equation}
|
||||
\phi(x) = A \cos(kx); ~ k \equiv \sqrt{\frac{2 m E}{\hbar^2}}
|
||||
\end{equation}\\
|
||||
Rand:
|
||||
\begin{equation}
|
||||
\phi(\pm a) = A \cos k a) \stackrel{!}{=} 0
|
||||
\end{equation}\\
|
||||
daraus folgt (mit $n = 0, 2, 4, 6, ...$)
|
||||
\begin{equation}
|
||||
k_n a = \frac{\pi}{2} ( 1 + n )
|
||||
\end{equation}
|
||||
und mit $n = 0, 1, ..., \infty$ ist dann
|
||||
\begin{equation}
|
||||
E_n = \frac{\hbar^2}{2m} \frac{1}{a^2} \left(\frac{\pi}{2} \right)^2 (1 + n)^2
|
||||
\end{equation}
|
||||
\item antisymmetrisch
|
||||
\begin{equation}
|
||||
\phi(x) = A \sin(k x)
|
||||
\end{equation}
|
||||
Rand:
|
||||
\begin{equation}
|
||||
\phi(\pm a) = \pm A \sin(k a) \stackrel{!}{=} 0
|
||||
\end{equation}
|
||||
daraus folgt mit $n = 1, 3, 5, 7, 9, ...$
|
||||
\begin{equation}
|
||||
k_n a = \frac{\pi}{2} (1 + n)\\
|
||||
\end{equation}
|
||||
und mit $n = 0, 1, ..., \infty$ ist dann
|
||||
\begin{equation}
|
||||
E_n = \frac{\hbar^2}{2m} \frac{1}{a^2} \left(\frac{\pi}{2} \right)^2 (1 + n)^2
|
||||
\end{equation}
|
||||
\end{enumerate}
|
||||
\subparagraph*{Fazit}
|
||||
\begin{enumerate}
|
||||
\item Energieeigenwerte sind quantisiert.
|
||||
%\begin{figure}[h]
|
||||
%\includegraphics{pdf/II/00-02-02.pdf}
|
||||
%\end{figure}
|
||||
\item Eigenfunktionen $\phi_n(x)$ bilden ein vollständig normiertes Basissystem.
|
||||
\begin{equation}
|
||||
\phi_n = \frac{1}{\sqrt{a}} \left\lbrace \begin{array}{ll} \cos(k_n x) & n\text{ grade}\\ \sin(k_n x) & \text{sont.} \end{array} \right.
|
||||
\end{equation}
|
||||
\begin{align}
|
||||
\intgr{-\infty}{+\infty}{\phi_m(x) \phi_n(x)}{x} &= \delta_{m,n}\\
|
||||
\sum_{n=0}^{\infty} \phi_n(x) \phi_n(x') &= \delta(x - x')
|
||||
\end{align}
|
||||
d.h. jede Funktion $\psi(x)$ kann entwickelt werden in dieser Basis $\psi(x) = \sum_{n=0}^{\infty} c_n \phi_n(x)$.
|
||||
%\begin{figure}[h]
|
||||
%\includegraphics{pdf/II/00-02-03.pdf}
|
||||
%\caption{Skizze der Eigenfunktionen}
|
||||
%\end{figure}
|
||||
\end{enumerate}
|
||||
\paragraph{Schritt 2} Dynamik\\
|
||||
Sei nun $\psi(x, t)$ beliebig gegeben durch
|
||||
\begin{equation}
|
||||
\psi(x,t) = \sum_{n=0}^{\infty} c_n(t) \phi_n(x)
|
||||
\end{equation}
|
||||
eingesetzt in die Schrödinger Gleichung
|
||||
\begin{align}
|
||||
i \hbar \sum_{n=0}^{\infty} \left( \diffPs{t} c_n(t) \right) \phi_n(x) &= \sum_{n'=0}^{\infty} \left( -\frac{\hbar^2}{2m} \diffPs{x}^2 \right) c_{n'}(t) \phi_{n'}(x)\\
|
||||
i \hbar \sum_{n=0}^{\infty} \diffPs{t} c_n(t) \phi_n(x) &= \sum_{n'=0}^{\infty} c_{n'}(t) E_{n'} \phi_{n'}(x) &\left| \intgr{-a}{+a}{\phi_m(x)}{x} \right.\\
|
||||
i\hbar \diffPs{t} c_n(t) &= E_n c_m(t)
|
||||
\end{align}
|
||||
dann ist
|
||||
\begin{equation}
|
||||
c_m(t) = e^{-\frac{i}{\hbar} E_m t}e_m(0)
|
||||
\end{equation}
|
||||
mit
|
||||
\begin{equation}
|
||||
c_m(0) = \intgr{-a}{+a}{\phi_m(x) \psi(x,t)}{x}
|
||||
\end{equation}
|
||||
und damit
|
||||
\begin{align}
|
||||
\psi(x,t) &= \intgru{\sum_n \phi_n(x) e^{-\frac{i}{\hbar} E_n (t-t_0) \phi_n(x') \psi(x,t)}}{x}\\
|
||||
&\equiv \intgru{U(x,t;x',t_0) \psi(x',t_0)}{x'}
|
||||
\end{align}
|
||||
($U(x,t;x',t_0)$ ... Zeitentwicklungsoperator in Ortsdarstellung)
|
||||
\section{Beispiel 2: $\delta$-Potentialtopf}
|
||||
%\begin{figure}[h]
|
||||
%\includegraphics{pdf/II/00-03-00.pdf}
|
||||
%\end{figure}
|
||||
Mit
|
||||
\begin{equation}
|
||||
V(x) = -\alpha \delta(x)
|
||||
\end{equation}
|
||||
ergeben sich die Stationären Zustände:
|
||||
\begin{align}
|
||||
\left[ -\frac{\hbar^2}{2m} -\alpha \delta(x) \right] \phi(x) &= E \phi(x) &\left| \intgr{-\varepsilon}{+\varepsilon}{}{x} \right.\\
|
||||
-\frac{\hbar^2}{2m} \left[ \phi'(0+\varepsilon) - \phi'(0-\varepsilon) \right] - \alpha \phi(0) &= \underbrace{2 \varepsilon E \phi(0)}_{\rightarrow 0}
|
||||
\end{align}
|
||||
$\phi'(x)$ springt bei der Null, wobei $\phi$ selbst stetig ist.
|
||||
|
||||
\paragraph*{Fall 1} $E < 0$\\
|
||||
$x > 0$:
|
||||
\begin{align}
|
||||
\diffPs{x}^2 \phi(x) &= K^2 \phi(x) &K^2 \equiv \frac{\abs{E} 2m}{\hbar^2}\\[15pt]
|
||||
\phi(x) &= A e^{\pm K x}
|
||||
\end{align}
|
||||
$+K$-Lösung nicht normierbar, also:
|
||||
\begin{equation}
|
||||
\phi(x) = A_+ e^{-K x}
|
||||
\end{equation}\\[15pt]
|
||||
$x > 0$:
|
||||
\begin{equation}
|
||||
\phi(x) = A_- e^{-K \abs{x}}
|
||||
\end{equation}
|
||||
Aus der Stetigkeit von $\phi$ folgt:
|
||||
\begin{equation}
|
||||
A_+ = A_- = A
|
||||
\end{equation}
|
||||
\subparagraph*{Sprungbedingung}
|
||||
\begin{align}
|
||||
\frac{- \hbar^2}{2m} \left( \right) - \alpha A &= 0\\
|
||||
K &= \frac{m \alpha}{\hbar^2}\\
|
||||
\rightarrow E &= -\frac{\hbar^2}{2m} \left( \frac{2m}{\hbar} \right)^2 \alpha^2
|
||||
\end{align}
|
||||
$\rightarrow$ Ein gebundener Zustand.
|
||||
%\begin{figure}[h]
|
||||
%\includegraphics{pdf/II/00-03-01.pdf}
|
||||
%\end{figure}
|
||||
\subparagraph*{Normierung}
|
||||
\begin{equation}
|
||||
\phi_0(x) = \frac{1}{\sqrt{K}} e^{-K \abs{x}}
|
||||
\end{equation}
|
||||
|
||||
\paragraph*{Fall 2} $E > 0$: Streuzustände (nicht normierbar)
|
||||
|
||||
@ -121,7 +121,7 @@ Heisenberg:
|
||||
für Gauss'sches Wellenpacket ist Gleichheit erreicht.
|
||||
\paragraph*{Dynamik}
|
||||
\begin{align}
|
||||
\psi(x,t) &= \intgr{-\infty}{+infty}{U(x,t; x',t_0)}{x'} &\left| t_0 = 0; ~ U(x,t; x',t_0) = \dirac{x}{U(t,t_0}{x'} \right.\\
|
||||
\psi(x,t) &= \intgr{-\infty}{+infty}{U(x,t; x',t_0)}{x'} &\left| \begin{array}{l} t_0 = 0;\\ U(x,t; x',t_0) = \dirac{x}{U(t,t_0}{x'} \end{array} \right.\\
|
||||
&= \left( \sqrt{\pi} \left( \Delta + \frac{i \hbar t}{m \Delta} \right) \right)^{-\frac{1}{2}} e^\frac{-\left(x - \frac{p_0 t}{m} \right)^2}{2 \Delta^2 \left( 1 + i \hbar \frac{t}{m \Delta^2} \right)} e^{\frac{i p_0}{\hbar} \left( x - \frac{p_0 t}{m} \right)}
|
||||
\end{align}
|
||||
\begin{equation}
|
||||
|
||||
75
kapII-3.tex
75
kapII-3.tex
@ -25,3 +25,78 @@ der Wahscheinlichkeitsstromdichte (``Kontinuitätsgleichung''; gilt für jede Er
|
||||
\end{align}
|
||||
|
||||
\section{Streuung an der Potentialstufe}
|
||||
%\begin{figure}[h]
|
||||
%\includegraphics{pdf/II/03-02-00.pdf}
|
||||
%\end{figure}
|
||||
\paragraph*{klassisch}
|
||||
\subparagraph*{Fall 1} $E > V_0$
|
||||
\begin{align}
|
||||
x < 0:~ & p(x < 0) = \sqrt{2m E}\\
|
||||
x > 0:~ & p(x > 0) = \sqrt{2m (E - V_0)}
|
||||
\end{align}
|
||||
Teilchen passiert die Potentialstufe, verliert Impuls
|
||||
\subparagraph*{Fall 2} $E < V_0$
|
||||
\begin{equation}
|
||||
p(x < 0) = \sqrt{2m E}
|
||||
\end{equation}
|
||||
Teilchen wird reflektiert
|
||||
|
||||
\paragraph*{quantal}
|
||||
\subparagraph*{Fall 1} $E > 0$\\
|
||||
stationäre SG:
|
||||
\begin{equation}
|
||||
\left( -\frac{\hbar^2}{2m} \diffPs{x}^2 V(x) \right) \phi(x) = E \phi(x)
|
||||
\end{equation}
|
||||
links: $x < 0$
|
||||
\begin{equation}
|
||||
\diffPs{x}^2 \phi(x) = -k^2 \phi(x)
|
||||
\end{equation}
|
||||
mit
|
||||
\begin{equation}
|
||||
k = \sqrt{\frac{2m}{\hbar^2}}
|
||||
\end{equation}
|
||||
Lösung:
|
||||
\begin{equation}
|
||||
\phi(x) = A e^{i k x} + B e^{-i k x}
|
||||
\end{equation}
|
||||
rechts: $x > 0$
|
||||
\begin{equation}
|
||||
\left( -\frac{\hbar^2}{2m} \diffPs{x}^2 + V_0 \right) \phi(x) = E \phi(x)
|
||||
\end{equation}
|
||||
Lösung:
|
||||
\begin{equation}
|
||||
\phi(x) = C e^{i q x} + D e^{-i q x}
|
||||
\end{equation}
|
||||
mit
|
||||
\begin{equation}
|
||||
q = \sqrt{\frac{2m (E - V_0)}{\hbar^2}}
|
||||
\end{equation}
|
||||
Randbedinung bei $x = 0$
|
||||
\begin{align}
|
||||
\phi(-\varepsilon) &= \phi(+\varepsilon)\\
|
||||
\diffPs{x} \phi(-\varepsilon) &= \diffPs{x} \phi(+\varepsilon)\\
|
||||
\rightarrow A + B &= C + D\\
|
||||
i k (A - B) &= i q (C - D)\\[15pt]
|
||||
\inlinematrix{1 & 1 \\ i k & -i k} \inlinematrix{A \\ B} &= \inlinematrix{1 & 1 \\ i q & -i q} \inlinematrix{C \\ D}\\
|
||||
\inlinematrix{A \\ B} &= \frac{1}{2k} \inlinematrix{k+q & k-q \\ k-q & k+q} \inlinematrix{C \\ D}
|
||||
\end{align}
|
||||
$\rightarrow$ Randbedingung einer von links laufenden Welle\\
|
||||
$\Rightarrow$ keine Komponente einer von rechts einlaufenden Welle für $x > 0$ erlaubt!\\
|
||||
$\Rightarrow$ $D \equiv 0$\\
|
||||
o.B.d.A.: $A = 1$
|
||||
\begin{align}
|
||||
A &= \frac{k + q}{2k} C ~ \rightarrow C = \frac{2k}{k+q}\\
|
||||
B &= \frac{k - q}{2k} C ~ \rightarrow B = \frac{k - q}{k + q}
|
||||
\end{align}
|
||||
Strom links: $x < 0$
|
||||
\begin{align}
|
||||
j(x < 0) &= \frac{\hbar}{m} \im{\diffPs{x}\phi ~ \phi^*}\\
|
||||
&= \frac{\hbar}{m} \im{i k \left(A e^{i k x} - B e^{-i k x} \right) \left(A^* e^{- i k x} + B^* e^{i k x}\right)}\\
|
||||
&= \frac{\hbar}{m} \im{ik \left( A A^* - B B^*\right) + ik \left( A B^* e^{2 i k x} - A^* B e^{-2 i k x} \right)}\\
|
||||
&= \frac{\hbar}{m} k \left( 1 - \left( \frac{k - q}{k + q} \right)^2 \right) \equiv j_I - j_R
|
||||
\end{align}
|
||||
mit
|
||||
\begin{align}
|
||||
j_I &= \frac{\hbar k}{m} &\text{einfallend}\\
|
||||
j_R &= \frac{\hbar}{m} k \left( \frac{k - q}{k + q} \right)^2 \equiv R j_I &\text{reflectiert}
|
||||
\end{align}
|
||||
|
||||
2
math.tex
2
math.tex
@ -33,7 +33,7 @@
|
||||
\newcommand{\intgr}[4]{\int_{#1}^{#2} #3 ~\text{d}#4}
|
||||
\newcommand{\intgru}[2]{\int #1 ~\text{d}#2}
|
||||
|
||||
\newcommand{\bbracket}[1]{\left #1 \right}
|
||||
\newcommand{\sbk}[1]{\left( #1 \right)}
|
||||
|
||||
\newcommand{\levicivita}[1]{\varepsilon_{#1}}
|
||||
\newcommand{\krondelta}[1]{\delta_{#1}}
|
||||
2
ueb6.tex
2
ueb6.tex
@ -73,7 +73,7 @@ Da $\inlinematrix{l \\ o \\ m} S_x \inlinematrix{n \\ o \\ p} = 0}$ folgt $S_x =
|
||||
\end{align}
|
||||
(1)
|
||||
\begin{align}
|
||||
\braket{\chi_k}{\phi_0} &= \frac{1}{\sqrt{6}} \sum{n=0}{5}{exp(-\i n \delta_{keine ahnung}\bracket{\phi_n}{\phi_0}} \\ \\
|
||||
\braket{\chi_k}{\phi_0} &= \frac{1}{\sqrt{6}} \sum{n=0}{5}{exp(-\i n \delta_{keine ahnung}\braket{\phi_n}{\phi_0}} \\ \\
|
||||
&= \frac{1}{\sqrt{6}} e^{-\i 0 \delta_{keine ahnung}} \\
|
||||
&= \frac{1}{\sqrt{6}}
|
||||
\end{align}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user