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\textbf{\large{Indian Institute of Information Technology, Allahabad}}
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\textbf{Assignment 04}: Engineering Physics [Section: C]
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\textbf{Instructor}: Dr. Srijit Bhattacharjee \qquad\qquad\quad \textbf{Dated}: Sept. 10, 2018
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\begin{enumerate}

\item A particle of mass $m$ is in a cubic box of size $a$. The potential inside the box $(0\leq x<a,0\leq y<a,0\leq z<a)$ is zero and infinite outside.If the particle is in an eigenstate of energy $E=\frac{14\pi^{2}\hbar^{2}}{2ma^{2}}$. Determine the wave-function. 

%\item[\textbf{2}.] Determine the commutation relation:\quad $[x,p^{n}_{x}]$
{\bf Ans.}$ E_{n_{x},n_{y},n_{z}}=(n_{x}^{2}+n_{y}^{2}+n_{z}^{2})\frac{\pi^{2}\hbar^{2}}{2ma^{2}}=E=\frac{14\pi^{2}\hbar^{2}}{2ma^{2}} $\\
$n_{x}^{2}+n_{y}^{2}+n_{z}^{2}=14 \quad \Rightarrow \quad n_{x}=1, n_{y}=2, n_{z}=3$\\
$ \psi(x,y,z)=(\frac{2}{a})^{3/2}\sin\frac{\pi x}{a}\sin\frac{2\pi y}{a}\sin\frac{3\pi z}{a} $

\item The wave function for a free particle is given as-
\begin{align*}
\psi(x,0)=& A \quad ; \quad if \quad -a<x<a \nonumber\\
=& 0 \quad ; \quad otherwise
\end{align*}
Where $a$ and $A$ are positive real constants. Find $\psi(x,t)$.

{\bf Ans.} Normalise the wave function, $ \int_{-\infty}^{\infty}|\psi(x,0)|^{2}dx=1 \Rightarrow A=\frac{1}{\sqrt{2a}} $\\
Now, \qquad $ \phi(k)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{2a}}\int_{-a}^{a}e^{-ikx}dx \Rightarrow \frac{1}{\sqrt{\pi a}}\frac{sin(ka)}{k} $\\
Therefore, \qquad $ \psi(x,t)=\frac{1}{\sqrt{2\pi^{2}a}}\int_{-\infty}^{\infty}\frac{\sin(ka)}{k}e^{i(kx-\frac{\hbar k^{2}}{2m}t)}dk $.

\item Show that $[Ae^{ikx}+Be^{-ikx}]$ and $[C\cos(kx)+D\sin(kx)]$ are equivalent ways of writing the same function of $x$ and determine the constants $C$ and $D$ in terms of $A$ and $B$ and vice versa.

{\bf Ans.} \begin{align*}
Ae^{ikx}+Be^{-ikx} =& A(\cos(kx)+i\sin(kx))+B(\cos(kx)-i\sin(kx))\\
=& (A+B)\cos(kx)+i(A-B)\sin(kx)\\
=& C\cos(kx)+D\sin(kx)
\end{align*}
with $C=A+B$ and $D=i(A-B)$\\
Now using, 
\begin{align*}
C\cos(kx)+D\sin(kx) =& \frac{1}{2}(C-iD)e^{ikx}+\frac{1}{2}(C+iD)e^{-ikx}\\
=& Ae^{ikx}+Be^{-ikx}
\end{align*}
with $A=\frac{1}{2}(C-iD)$ and $B=\frac{1}{2}(C+iD)$



\item For the given wave function $\psi(x,t)=Ae^{i(kx-\frac{\hbar k^{2}}{2m}t)}$, find the probability current $J$ for the this free particle wave function.

{\bf Ans.} $ J = \frac{i\hbar}{2m}\Big(\psi\frac{\partial\psi^{*}}{\partial x}-\psi^{*}\frac{\partial\psi}{\partial x}\Big)\Rightarrow \frac{\hbar k}{m}|A|^{2} $.

\item For a given short-range potential, $V(x)=-V_{0}\delta(x)$, Obtain the binding energy of the particle of mass $m$ in one dimension.

{\bf Ans.} The Schrodinger equation,\quad $ \psi^{''}(x)+\frac{2m}{\hbar^{2}}(E-V)\psi(x)=0 \quad ; \quad (E<0) $\\
Set, \quad $ U_{0}=\frac{2mV_{0}}{\hbar^{2}} \quad ; \quad k=\frac{\sqrt{2m|E|}}{\hbar} $\\
Therefore, expression becomes,\qquad $\psi^{''}(x)-k^{2}\psi(x)+U_{0}\delta(x)\psi(x)=0  $\\
At $x\neq 0$, Schrodinger eq. has solutions, 
$$ \psi(x)\sim e^{-kx} \quad for \quad x<0 \qquad and \qquad \psi(x)\sim e^{kx} \quad for \quad x>0 $$
Using the boundary condition the energy is $-E=\frac{\hbar^{2}k^{2}}{2m}=\frac{mV_{0}^{2}}{2\hbar^{2}}$

\item Find the reflection and transmission coefficients for the one dimensional potential step if the particles incident from the right.
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{\bf Hint.} Use the boundary condition for the wave function and the first derivative of the wave function are continuous at $x=0$.

\item For rectangular potential barrier problem discussed in the class, find out the transmission coefficient for the case when energy of the particle $E$ is equal to the potential height $V_0$. What is the transmission and reflection co-efficients for $E>V_0$? Show that you can get the case $E<V_0$ by changing $k=ik$. Check the transmission co-efficient for $V_0=0$.  At what values of the barrier width one expects perfect transmission? (Transmission co-efficient $T=\frac{1}{1+\frac{1}{4}\left(\frac{k_1}{k_2}+\frac{k_2}{k_1}\right)^2\sin^2hka}.$ $k_1^2=\frac{2mE}{\hbar^2}, ~k_2^2=\frac{2m(V_0-E)}{\hbar^2}$.

\item Plot the real part of wave function of rectangular barrier problem for the case when $E<V_0$. Try to interpret the behaviour. 

\item Lets consider a wave $f(x,t)=\cos (kx-\omega t)$. Show that it's crest moves with a velocity equal to $\omega/k$. Now take another wave $\cos [(k+\delta k)x -(\omega+\delta \omega)t]$, sum of these two waves will have a rapidly oscillating component and a slowly oscillating outer component. Find out the sum. Show that  the envelop's phase velocity is equal to $\frac{\delta \omega}{\delta k}$. Show that $v_{group}=v_{phase}-\lambda \frac{d v_{phase}}{d \lambda}$. 

\item A solid metal contains $4.0\times 10^{27}$ per unit volume. Assuming that the solid metal acts as a finite potential well i.e. $U=0$ inside the metal, Find the Fermi energy and velocity of the electrons in the metal.

\item Using equation $\psi(x)=A\sin(kx)+B\cos(kx)$ for $0<x<a$ and $A\sin(ka)=[e^{iKa}-\cos(ka)]B$, show that the wave function for a particle in the periodic delta function potential can be written in the form,
$$ \psi(x)=C[\sin(kx)+e^{-iKa}\sin k(a-x)] \quad ; \quad (0\leq x\leq a) $$
{\bf Please note there is a sign mistake in the lecture note of 4 October. The relative sign between the $cos z$ and $sin z/z$ is $'+'$.}
\end{enumerate}

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