Physics 401 Test 2

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Questions (5 pts. apiece) Answer questions 1-3 in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions 4-5 circle the correct answer.

  1. Consider a free particle moving in one dimension with an initial wave function $\psi(x,t=0)$ shown in the figure. What happens to the wave function for $t>0$? Your answer should be descriptive and qualitative; not quantitative.

    \includegraphics[height=1.5in]{f1.eps}

  2. What is the quantum program?








  3. The energy states for some molecule are the following: 0.0 eV, 0.4 eV, 1.2 eV, 2.4 eV, 4.0 eV, 6.0 eV, and 8.4 eV. Does this molecule behave like a harmonic oscillator? Explain.








  4. For the one-dimensional harmonic oscillator, the potential energy is $U= (1/2)kx^2$ and the ground state wave function is

    \begin{displaymath} \psi_0 = Ce^{-ax^2} \qquad . \end{displaymath}

    Find the constant $C$. (Note: $\int_{-\infty}^{\infty} {1 \over \sqrt{2 \pi \sigma}} e^{-1/2 x^2/\sigma^2} dx = 1$).

    A. $C = \sqrt{2a/\pi}$ B. $C = \sqrt{a/\pi}$
    C. $C = 1$ D. $C = (2a/\pi)^{1/4}$
    E. $C = a$    

  5. Calculate the coefficient of reflection for a particle of energy $E > V_0$ is incident on a step potential of height $V_0$. Let

    \begin{displaymath} k = \sqrt{2mE}/\hbar \qquad {\rm and} \qquad k^\prime = \sqrt{2m(E - V_0)}/\hbar \quad . \end{displaymath}

    A. $R=0$ D. $R= \left \vert \frac{k-k^\prime}{k+k^\prime} \right \vert$
    B. $R=1$ E. $R= \left \vert \frac{k}{k^\prime} \right \vert$
    C. $R= \left \vert \frac{k-k^\prime}{k+k^\prime} \right \vert^2$    


    \includegraphics[height=1.0in]{stepbarrier.eps}

Problems. Clearly show all work for full credit.


1. (25 pts.)

A free particle of mass $m$ moving in one dimension is known to be in the initial state

\begin{displaymath} \psi(x,0) = Cx(x-a) \qquad 0 < x < a \end{displaymath}

where $C=\sqrt{30/a^5}$. The eigenfunctions and eigenvalues for the free particle are the following.

\begin{displaymath} \vert\phi(x)\rangle = \frac{1}{\sqrt{2\pi}} e^{ikx} \qquad E = \frac{\hbar^2 k^2 }{2m} \end{displaymath}

What is $\psi (x,t)$?

2. (25 pts)

Recall our old friends, Newton's Second Law, $ \vec F = m \vec a$ and Hooke's Law, $\vert \vec F \vert = -K x$ which can be combined to form a differential equation

\begin{displaymath} m {d^2x \over dt^2} = -K x \qquad {\rm or} \qquad {d^2x \over dt^2} = - \omega_0 ^2x \end{displaymath}

where $\omega_0 = \sqrt{K \over m}$. The solutions of this equation have, of course, been known to us since our earliest childhood, but now solve this differential equation using the Method of Frobenius and generate the recursion relationship that relates different coefficients to one another.

3. (25 pts)

An electron beam is incident on a barrier of height $V=8~eV$. At $E=10~eV$, $T=4.0\times 10^{-2}$. What is the width of the barrier? Do these results look reasonable?

Physics 401 Equations


\begin{displaymath} E = h\nu = \hbar \omega \quad v_{wave} = \lambda \nu \quad I... ...{max} = h\nu - W \quad \lambda = {h \over p} \quad p = \hbar k \end{displaymath}


\begin{displaymath} -{\hbar^2 \over 2 m} {\partial^2 \over\partial x^2} \Psi(x,t... ...-i\hbar {\partial \over \partial x} \quad T(t) = e^{i\omega t} \end{displaymath}


\begin{displaymath} \hat{A~}\vert\phi\rangle = a\vert\phi\rangle \quad \langle\h... ... dx \quad [ \hat A, \hat B ~ ] = \hat A \hat B - \hat B \hat A \end{displaymath}


\begin{displaymath} \langle\phi_{n'} \vert \phi_n \rangle = \int_{-\infty}^{\i... ...nt_{-\infty}^{\infty} \phi_{k'}^* \phi_k~ dx = \delta(k - k') \end{displaymath}


\begin{displaymath} \vert\psi\rangle = \sum b_n \vert\phi_n\rangle \rightarrow ... ...angle dk \rightarrow b(k) = \langle\phi(k) \vert \psi \rangle \end{displaymath}


\begin{displaymath} \Delta p \Delta x \ge {\hbar \over 2} \quad (\Delta x)^2 = \... ...ma^2} ~ e^{-x^2/2\sigma^2}, \ {\rm then}\ \Delta x = \sigma \end{displaymath}


\begin{displaymath} e^{ix} = \cos x + i\sin x \quad \end{displaymath}


\begin{displaymath} T = {{\rm transmitted\ flux} \over {\rm incident\ flux}} \qu... ...}} \quad R+T = 1 \quad \langle K \rangle = {3\over 2} kT \quad \end{displaymath}


\begin{displaymath} \psi_1 = {\bf t} \psi_3 = {\bf d_{12} p_2 d_{21} p_1^{-1... ... \vert t_{11}\vert^2} \quad V(r) = {Z_1 Z_2 e^2 \over r} \quad \end{displaymath}


\begin{displaymath} \frac{1}{T} = 1 + \frac{1}{4} \frac{V^2}{E(E-V)} \sin^2 (2k_... ...\frac{1}{4} \frac{V^2}{E(E-V)} \sinh^2 (2\kappa a) \quad E < V \end{displaymath}


\begin{displaymath} k_2 = \frac{ \sqrt{2m(E-V)} }{\hbar} \qquad \kappa = \frac{ \sqrt{2m(V-E)} }{\hbar} \end{displaymath}


\begin{displaymath} \hat a = \frac{\beta}{\sqrt{2}} \left ( \hat x + \frac{i\hat... ...r \vert\phi_n \rangle = \sqrt{n+1}\vert\phi_{n-1}\rangle \quad \end{displaymath}


\begin{displaymath} \beta = \sqrt{\frac{m\omega_0}{\hbar}} \end{displaymath}

Physics 401 Conversions, and Constants

Speed of light ($c$) $2.9979\times 10^8 ~m/s$ fermi ($fm$) $10^{-15}~m$
Boltzmann constant ($k_B$) $1.381\times 10^{-23}~J/K$ angstrom ($\rm\AA$) $10^{-10}~m$
  $8.62\times 10^{-5}~eV/k$ electron-volt ($eV$) $1.6\times 10^{-19}~J$
Planck constant ($h$) $6.621 \times 10^{-34}~J-s$ MeV $10^6~eV$
  $4.1357\times 10^{-15}~eV-s$ GeV $10^9~eV$
Planck constant ($\hbar$) $1.0546\times 10^{-34}~J-s$ Electron charge ($e$) $1.6\times 10^{-19}~C$
  $6.5821\times 10^{-16}~eV-s$ $e^2$ $\hbar c / 137$
Planck constant ($\hbar c $) $197~MeV-fm $ Electron mass ($m_e$) $9.11\times 10^{-31}~kg$
  $1970~eV-{\rm\AA}$   $0.511~MeV/c^2$
Proton mass ($m_p$) $1.67\times 10^{-27}kg$ atomic mass unit ($u$) $1.66\times 10^{-27}~kg$
  $938~MeV/c^2$   $931.5~MeV/c^2$
Neutron mass ($m_n$) $1.68\times 10^{-27}~kg$    
  $939~MeV/c^2$