Physics 401 Final Exam


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Clearly show all reasoning for full credit. Use a separate sheet to show your work. See the last page for a table of constants and conversion factors.

1. 20 pts.

One thousand neutrons are in a one-dimensional box with walls at $x=0$ and $x=L$. The eigenfunctions and eigenvalues are

\begin{displaymath} \vert\phi_n \rangle = \sqrt {2 \over L} \sin\left ( {n \pi... ...ver L} \right ) \qquad E_n = n^2 {\hbar^2 \pi^2 \over 2 m L^2} \end{displaymath}

At $t=0$, the state of each particle is

\begin{displaymath} \psi(x,0) = Ax(x-L) \end{displaymath}

  1. What is the value of $A$?

  2. How many particles have energy $E_7$ at $t=0$? You should get a final, numerical answer.

2. 25 pts.

For the scattering configuration shown in the figure below, given that $V_0=2E$, at what value of $x$ is the probability density in Region 2 one-quarter the density of particles in the incident beam? You're answer should be in terms of the constants of the problem (i.e., $V_0$, the wave numbers, etc.).





\includegraphics[]{step.eps}

3. 25 pts.

In the phenomenon of cold emission, electrons are drawn from a metal (at room temperature) by an externally supported electric field. The potential well that the metal presents to the free electrons before the electric field is turned on is depicted in the left-hand side of the figure below. After application of the constant electric field $\mathcal{E}$, the potential at the surface slopes down as shown in the right-hand side of the figure below, thereby allowing electrons from the Fermi sea to ``tunnel'' through the potential barrier. If the surface of the metal is taken as the $x=0$ plane, the new potential outside the surface is

\begin{displaymath} V(x) = \Phi + E_F - e\mathcal{E} x \end{displaymath}

where $E_F$ is the Fermi level and $\Phi$ is the work function of the metal.

  1. Use the WKB approximation to calculate the transmission coefficient for cold emission.

  2. Estimate the field strength $\mathcal{E}$, in $V/cm$ (or $J/Coulomb-cm$), necessary to draw a current density of $1~mA/cm^2$ from a potassium surface. You need to use the expression

    \begin{displaymath} J_{inc} = env \end{displaymath}

    where $n$ is the electron density and $v$ is the speed of the electrons at the top of the Fermi sea. The expression for $E_F$ is

    \begin{displaymath} E_F = \left ( {h^2 \over 2m} \right ) \left ( {3 n \over 8 \pi} \right )^{2/3} \end{displaymath}

    and has a value $E_F = 2.1~eV$ for potassium. The work function $\Phi$ for potassium is also $2.1~eV$.



    \includegraphics[]{emission.eps}

4. 30 pts.

Consider a one-dimensional harmonic oscillator.

\begin{displaymath} \hat H = - {\hbar^2 \over 2m} {d^2 \over dx^2} + {1 \over 2} m \omega_0^2 x^2 \end{displaymath}

We found after much labor that the ground state wave function and energy in this case are

\begin{displaymath} \vert\phi_0(x)\rangle = \pi^{-1/4} e^{- {m \omega_0 \over 2\hbar} x^2} \qquad E_0 = {1 \over 2} \hbar \omega_0 \end{displaymath}

From experience with other problems we expect the wave function to be a declining exponential of the form $e^{-\alpha x^2}$ where $\alpha$ is some positive parameter. Assume the solution to the harmonic oscillator has the following form.

\begin{displaymath} \vert\psi_\alpha (x) \rangle = e^{-\alpha x^2} \end{displaymath}

  1. Use the variational method to find a wave function that minimizes $\langle E \rangle$.

  2. Normalize the result from the previous part.

  3. What is the ground state energy? In other words, what is the minimum value of $\langle E \rangle$?

  4. How do your approximate results compare with the exact expressions listed above?

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 $e^2$ $\hbar c /137$
energy $1~eV$ $1.6\times 10^{-19}~J$
length $\rm 1~ \AA$ $10^{-10}~m$
  $1~fm$ $10^{-15}~m$