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\vsize=25truecm                                                             
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Matalien l\"amp\"otilojen fysiikan teoria (Kyl-0.104)\hfil Autumn 2000\break
Electronic transport in mesoscopic systems\hfil Exercise set 1\break
\parindent=20pt                                                                    
\bigskip 
Exercises E.1.1, E.1.2 from Datta's book.
\bigskip                                                                        
3. As will be discussed later in this course, the effect of interactions can be
represented by replacing the potential $U$ in the Schr\"odinger equation by a self
energy $\Sigma$, which is complex valued. Study the effect of constant $\Sigma$ on a
plane wave $\psi(t=0,x)=\exp(ikx)/\sqrt{L}$ and relate $\Sigma$ to the phase
relaxation time
$\tau_\phi$.
\bigskip                                                                        

4. Show that the two lowest eigenfunctions ($n=0$ and 1) of the harmonic oscillator  
\begin{equation} 
[-{\hbar^2\over 2m}{d^2\over d x^2}
+{1\over 2}m\omega_0^2x^2]\chi(x)=E\chi(x)
\label{e.Gibbs} \end{equation} 
 have the (unnormalized) forms
\begin{equation} 
\chi_0(x)=\exp(-{m\omega_0\over 2\hbar}x^2), \chi_1(x)=x \exp(-{m\omega_0\over
2\hbar}x^2)
\label{e.Gibbs} \end{equation}
and the energies $E=(n+{1\over 2})\hbar\omega_0$.
\bigskip                                                                        

5. Show that in the symmetric gauge ${\bf A}={1\over 2}B(-y\hat{\bf x}+x\hat{\bf
y})$ the lowest Landau level has eigenfunctions (polar coordinates, $l=0$, 1, 2,
$\ldots$, $\omega_c=\vert e\vert B/m$)  
\begin{equation} 
\psi_l(r,\phi)=r^le^{-il\phi}\exp(-{m\omega_c\over 4\hbar}r^2).
\label{e.Gibbs} \end{equation} 
 Describe their differences and similarities to the corresponding solutions
\begin{equation} 
\psi_k(x,y)=e^{ikx}\exp[-{m\omega_c\over 2\hbar}(y-{\hbar k\over m\omega_c})^2].
\label{e.Gibbs} \end{equation}
in the Landau gauge  ${\bf A}=-By\hat{\bf x}$. 
\bigskip                                                                        






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