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\textbf{\large{Indian Institute of Information Technology, Allahabad}}
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\textbf{Assignment 03}: 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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\textbf{[1]} The mass of a baseball is $100$grams which travelling at the speed of $90$mph. Calculate the uncertainty in the position of a baseball if the velocity is known to $0.1$mph. (\textbf{Hint:} $0.1$mph is the uncertainty in the velocity.) \\
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\textbf{[2]} A pendulum consisting of a 0.01 kg mass is suspended from a string 0.1 m in length. Let the amplitude of its oscillation be such that the string in its extreme positions makes an angle of 0.1 rad with the vertical. The energy of the pendulum decreases due, for instance, to frictional effects. Is the energy decrease observed to be continuous or dis- continuous? ({\bf Ans.} $\Delta E= 10^{-33}$ joule)\\
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\textbf{[3]} Consider the one dimensional wave function,
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$ \psi(x)=A(\frac{x}{x_{0}})^{n}e^{-\frac{x}{x_{0}}} $
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Where $A$, $n$, and $x_{0}$ are constants. Assuming $x\rightarrow\infty$, $V(x)\rightarrow 0$, using Schrodinger equation, find the potential $V(x)$ and energy $E$ for which this wave function is an eigenfunction.\\
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\textbf{[4]} At time $t=0$ a particle in the potential $V(x)=\frac{1}{2}m\omega^{2}x^{2}$ is described by the wave function,
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$ \psi(x,0)=A\Sigma_{n}(\frac{1}{\sqrt{2}})^{n}\psi_{n}(x) $
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Where $\psi_{n}(x)$ are eigenstates of the energy with eigenvalues $E_{n}=(n+\frac{1}{2})\hbar\omega$. You are given that ($\psi_{n}\psi_{n'})=\delta_{nn'}$. Find the normalisation constant $A$. Write an expression for $\psi(x,t)$ for $t>0$. \\
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\textbf{[5]} At time $t=0$, for a free particle of mass $m$ moving in one dimensional space, the normalised wave function of the particle is,
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$ \psi(x,0,\sigma_{x}^{2})=(2\pi\sigma_{x}^{2})^{-\frac{1}{4}}e^{-\frac{x^{2}}{4\sigma_{x}^{2}}} $
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Compute the momentum spread $\sigma_{p}=\sqrt{(<p^{2}>-<p>^{2})}$ associated with this wave function.\\
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\textbf{[6]} A particle of mass $m$ moves in 1-D under the influence of a potential $V(x)$. Suppose it is an energy eigenstate $\psi(x)=(\frac{\gamma^{2}}{\pi})^{\frac{1}{2}}e^{-\frac{\gamma^{2}x^{2}}{2}}$ with energy $E=\frac{\hbar^{2}\gamma^{2}}{2m}$. Find mean position $\&$ mean momentum of the particle.\\
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\textbf{[7]} For the given wave function below,
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$\psi(x)=A\cos(\frac{2\pi x}{L})$ \qquad for $-\frac{L}{4}\leq x \leq \frac{L}{4}$
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Determine the normalisation constant $A$. Calculate the probability that the particle will be found between $x=0$ to $x=\frac{L}{8}$.\\
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\textbf{[8]} For an oscillator, the solution of Schrodinger’s equation is,
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$\psi(x)=Cxe^{-\alpha x^{2}}$
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Normalize the wave-function. Determine $\alpha$ in terms of $m$ and $\omega$. What is the energy of this state? \\
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\textbf{[9]} The wave-function of a particle confined to the $x-axis$ is $\psi=e^{-x}$ for $x>0$ and $\psi=e^{x}$ for $x<0$. Normalize the wave-function and calculate the probability of finding the particle between $x=-1$ to $x=1$.\\
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\textbf{[10]} Normalised wave-function of a particle is,
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\psi(x)=& 2\alpha\sqrt{\alpha}xe^{-\alpha x} \qquad for x>0\\
=&0 \qquad\qquad\qquad\quad for x<0
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Calculate $<x>$ and $<x^{2}>$. Estimate the probability that the particle is found between $x=0$ and $x=\frac{1}{\alpha}$. Consider the limit from $0$ to $\frac{1}{\alpha}$.\\
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\textbf{[11]} Show that the function,
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$\psi(x,t)=\frac{e^{-iE_{1}t}}{\hbar}\psi_{1}(x)+\frac{e^{-iE_{2}t}}{\hbar}\psi_{2}(x)$
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is normalised. Find $<E>$ and $\bigtriangleup E$ for this state.\\
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\textbf{[12]} What is the de Broglie wavelength of a baseball moving at a speed $v = 10 $ m/sec?//

What is the de Broglie wavelength of an electron whose kinetic energy is 100 eV?


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