Physics 401 Test 1

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Multiple Choice Questions (5 points apiece).

  1. Light of wavelength $\rm 4500 \AA$ is incident on a Na surface for which the threshold wavelength of the photoelectrons is $\rm 5420 \AA$. Calculate the work function of sodium.

    (a) 2.76 eV (d) 4.76 eV
    (b) 2.29 eV (e) 1.00 eV
    (c) 0.47 eV    

  2. Consider $N$ noninteracting bosons in an infinite potential box of width $a$. What is the ground state energy?


    \includegraphics[]{f1.eps}


    (a) $\hbar^2 \pi^2 N/ma^2$ (d) $\hbar^2 \pi^2 /ma^2$
    (b) $\hbar^2 \pi^2 N/2ma^2$ (e) $\hbar^2 \pi^2 N/2ma^2$
    (c) $\hbar^2 \pi^2 /ma^2$    


  3. The lowest nucleon resonance state is the $\Delta$ which has a mass of $1232~MeV/c^2$ and a width of $120~MeV$. Calculate the lifetime of this $I=3/2$ nucleon state.

    (a) $5.5\times 10^{-24}~s$ (d) $6.9\times 10^{-9}~s$
    (b) $1.2\times 10^{-19}~s$ (e) $8.4\times 10^{-17}~s$
    (c) $3.3\times 10^{-23}~s$    

  4. In quantum mechanics, one may picture a wave function in either momentum space (the spectral distribution) or configuration space (the spatial distribution). If the wave function in configuration space is $\psi (x) = N/(x^2 + \alpha^2)$, then calculate the wave function in momentum space (aside from a multiplicative constant).

    (a) $e^{-\alpha^2 k^2 /\hbar^2}$ (d) $e^{-\alpha \vert k\vert}$
    (b) $\cos (p x/\hbar)$ (e) $e^{ikx}$
    (e) $\sin (p x/\hbar)$    

Problems. Clearly show all work for full credit.


1. (20 pts.)

A criterion which discerns if a given configuration is classical or quantum mechanical may be stated in terms of the de Broglie wavelength $\lambda$. If $L$ is a length characteristic of the configuration, then one has the following criteria.

\begin{eqnarray*} \lambda & \ll L \quad : &\quad {\rm Classical} \\ \lambda & \... ...l{\textstyle >}{\sim} L \quad : & \quad {\rm Quantum Mechanical} \end{eqnarray*}



A rubidium atom of mass $m = 102~u$ is in a magnetic trap with an active region about $1~\mu m$ (a $\mu m $ is $10^{-6}m$) across. The active region has been cooled to a temperature $T = 1.7\times 10^{-7}~K$ corresponding to an average energy of $2.2\times 10^{-11}~eV$. Is the configuration classical or quantum mechanical? Why?

2. (20 pts.)

Consider the functions below defined over the interval $(-a/2,+a/2)$.

\begin{displaymath} \vert \phi_k \rangle = {1 \over \sqrt a}~ e^{ikx} \end{displaymath}


a. What is the inner product between two elements of this function space in terms of $k$, $k^\prime$, $a$, and any other necessary parameters?
b. Do these functions comprise an orthonormal set in the limit $a\rightarrow \infty$?

3. (40 pts.)

Consider a case of one dimensional nuclear `fusion'. A neutron is in the potential well of a nucleus that we will approximate with an infinite square well with walls at $x=0$ and $x=L$. The eigenfunctions and eigenvalues are

\begin{eqnarray*} E_n = {n^2 \hbar^2 \pi^2 \over 2 m L^2} \qquad \vert \phi_n \... ... L \\ & = & 0 \hspace{3.5cm} x < 0 ~ {\rm and} ~ x > L \qquad. \end{eqnarray*}



The neutron is in the $n=3$ state when it fuses with another nucleus that is twice as large, instantly putting the neutron in a new infinite square well with walls at $x=0$ and $x=3L$.


a. What are the new eigenfunctions and eigenvalues of the fused system?
b. In the fused system the $n^\prime = 9$ state of the neutron has the same energy as the $n=3$ neutron state in the original system. Calculate the probability for finding the neutron in the $n^\prime = 10$ (not 9) final state.
c. Your answer in part b should be greater than one. This suggests the system could have more energy after the `fusion' than before in violation of the conservation of energy. How would you save the conservation of energy from being thrown on the trash heap of physics?

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 $
    $1970~eV-{\rm\AA}$
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 $u$ $1.66\times 10^{-27}~kg$
    $931.5~MeV/c^2$
fermi $fm$ $10^{-15}~m$
angstrom $\rm\AA$ $10^{-10}~m$
electron-volt $eV$ $1.6\times 10^{-19}~J$
mega-electron-volt $MeV$ $10^6~eV$
giga-electron-volt $GeV$ $10^9~eV$
electron-charge squared $e^2$ $\hbar c / 137$