Physics 401 Test 2

I pledge that I have given nor received unauthorized assistance during the completion of this work.

Signature: width10cm height1pt depth0pt


Multiple Choice Questions (5 points apiece).

  1. A deuteron ( $Z=1,\ m = 2\,u$) is incident on a lead nucleus ( $Z=82,\ m=208\,u$) at the Brookhaven National Laboratory. The terminal voltage of the accelerator is $15~MeV$. Find the distance of closest approach in a head-on collision.

    (a) 7.87 fm (d) 10.64 fm
    (b) 15.74 fm (e) 13.20 fm
    (c) 5.32 fm    

  2. In the derivation of the Stefan-Boltzmann law from the Planck distribution, one must evaluate the integral

    \begin{displaymath} \int_0^\infty z^3 dz /(e^z - 1) \end{displaymath}

    where $z=hv/kT$. What is the value of this integral?

    (a) $\pi^4/15$ (d) $\pi^3/15$
    (b) $\pi^3/90$ (e) $\pi^4/90$
    (c) $\pi^2/40$    


  3. A ball with a mass of $0.002~kg$ and a kinetic energy of $10^{-3}~J$ is incident upon a rectangular hill $0.20~m$ in height and $0.02 ~m$ in width as shown. What is the probability the ball will quantum mechanically tunnel through the hill and appear on the other side?

    (a) $e^{-1.3\times 10^{30}}$ (d) $e^{-13.8}$
    (b) $e^{-3.0\times 10^{30}}$ (e) $e^{-7.4\times 10^{28}}$
    (c) $e^{-9.2}$    


    \includegraphics[]{mp3.eps}

  4. One must use the de Broglie wavelength concept to `derive' the Schroedinger equation from the one-dimensional wave equation. What de Broglie wavelength must be used to get the general time independent equation?

    (a) $\lambda = h/\sqrt{2mE}$ (d) $\lambda = h/\sqrt{2m(E+U)}$
    (b) $\lambda = h/\sqrt{2mU}$ (e) $\lambda = h/\sqrt{m(E-U)}$
    (c) $\lambda = h/\sqrt{2m(E-U)}$    

Problems. Clearly show all work for full credit.


1. (20 pts.) A harmonic oscillator with energy $E_n = (n+1/2)\hbar \omega_0$ consists of a mass $m=0.002\, kg$ on a spring oscillating with a frequency of $\nu = 1.0~Hz$. It passes through its equilibrium position with a velocity of $v_e = 12~m/s$. How many quanta are in the system?


2. (20 pts.) Calculate the expectation value of the momentum for a particle in the $n=15$ harmonic oscillator state using the annihilation and creation operators shown below. Clearly show all your steps. Does your result make sense? Explain. You may find the following relationships useful

\begin{displaymath} \hat a = {\beta \over \sqrt 2} \left ( \hat x + {i\hat p \ov... ... \sqrt 2} \left ( \hat x - {i\hat p \over m \omega_0} \right ) \end{displaymath}


\begin{displaymath} \hat a \vert \phi_n\rangle = \sqrt n ~\vert \phi_{n-1}\rangl... ...gger \vert \phi_n \rangle = \sqrt{n+1} ~\vert\phi_{n+1}\rangle \end{displaymath}

where $\beta = \sqrt{m\omega_0 /\hbar}$.

3. (40 pts.) In class we found the general solution to the rectangular barrier problem for the potential shown in the figure below.



\includegraphics[]{pr3.eps}

This general solution in the three regions labelled in the figure is

\begin{eqnarray*} \phi_1 &=& Ae^{ik_1 x} + Be^{-ik_1 x} \\ \phi_2 &=& Ce^{ik_2 x} + De^{-ik_2 x} \\ \phi_3 &=& Fe^{ik_1 x} + Ge^{-ik_1 x} \end{eqnarray*}



where the wave numbers are defined in the following way.

\begin{displaymath} k_1 = \sqrt{ 2mE \over \hbar^2 } \qquad k_2 = \sqrt{ 2m(E-V_0) \over \hbar^2} \end{displaymath}

3. (cont.) We expressed the wave functions in each region in the form of column vectors

\begin{displaymath} \psi_1 = \pmatrix{ A \cr B } \qquad \psi_2 = \pmatrix{ C \cr D } \qquad \psi_3 = \pmatrix{ F \cr G } \end{displaymath}

and the boundary conditions in the form of the matrices

\begin{displaymath} \psi_1 = {\rm\bf d}_{12} {\rm\bf p}_2 {\rm\bf d}_{21} {\rm\bf p}_{1}^{-1}\psi_3 = {\rm\bf t}\psi_3 \end{displaymath}

where $\rm\bf t$ is the transfer matrix, $\rm\bf d_{12}$ and $\rm\bf d_{21}$ are discontinuity matrices, and $\rm\bf p_2$ and $\rm\bf p_1^{\bf -1}$ are the propagation matrices. The discontinuity and propagation matrices are defined in the following way.

\begin{displaymath} {\bf d_{12}} = {1 \over 2} \pmatrix{ 1 + {k_2 \over k_1} & ... ...2} \cr 1 - {k_1 \over k_2} & 1 + {k_1 \over k_2} \cr } \quad \end{displaymath}


\begin{displaymath} {\bf p_{2}} = \pmatrix{ e^{-ik_2 2a} & 0 \cr 0 & e^{ ik_2 ... ...f -1} = \pmatrix{ e^{ ik_1 2a} & 0 \cr 0 & e^{-ik_1 2a} \cr} \end{displaymath}

Consider the elements of $\rm\bf d_{12}$, $\rm\bf d_{21}$, $\rm\bf p_{2}$, and $\rm\bf p_{1}^{-1}$ to be known quantities. In region 3 we set $G=0$ because no waves were incident from the right. The coefficient $A$ represents the incident wave coming from the left. It is our `beam' and so we consider it to be known.

(a)
What is the coefficient $F$ in terms of $A$ and any known matrix element (i.e., a matrix element from $\rm\bf t$, $\rm\bf d_{12}$, $\rm\bf d_{21}$, $\rm\bf p_{1}^{-1}$, and $\rm\bf p_{2}$)?

(b)
Now get an expression for $B$ in terms of $A$ and any known matrix element.

(c)
How is the wave function in region 1, $\psi_1$, related to the wave function in region 2, $\psi_2$?

(d)
Now get a relationship for $C$ in terms of $A$, $B$, $F$, and any matrix element. Don't work too hard at getting things pretty if you are running out of time.

Table of Constants

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