Physics 402 Test 1

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Multiple Choice Questions (Circle your choice, 4 points apiece).

  1. Consider the hydrogen molecule $\rm H_2$ as a rigid diatomic rotor of separation $r=1.0~{\rm\AA}$ between two protons. Calculate the energy of the $l=3$ level in the rotational spectrum. ( $m_p = 938.280\times 10^6~eV/c^2$, $\hbar = 1973.5~eV-{\rm\AA}/c$).

    (a) 0.10 eV (d) 0.005 eV
    (b) 0.05 eV (e) 0.10 eV
    (c) 0.15 eV    

  2. Calculate the $Q$-value or energy release in MeV for the nuclear reaction $^{27}\! Al(d,p)^{28}\! Al$ given that $m(^{27}\! Al) = 26.98154$, $m(d) = 2.01473$, $m(p) = 1.00794$, and $m(^{28}\! Al)=27.98154$, all in $u$.

    (a) -6.32 MeV (d) 6.83 MeV
    (b) 0.0 MeV (e) -6.83 MeV
    (c) 6.32 MeV    


  3. A simple wave function for the deuteron is given by $\phi(r) = A\sin (k(r-s))/r$ for $a < r < a+b$ and $\phi(r) = B e^{-kr}/r$ for $r > a+b$. Find the probability that the neutron and the proton are between $r=a$ and $r=a+b$ in separation.

    (a) $\int_a^{a+b} \phi^* \phi dr$ (d) $B^2(e^{-k(a+b)} - e^{-ka})$
    (b) $A^2(b/2 - \sin(2kb)/4k)$ (e) 1.0, since the proton and neutron are bound together
    (c) $A^2 b/2$    




  4. The $n=2$, $l=1$ hydrogen atom radial wave function is

    \begin{displaymath} R_{21}(r) = N r e^{-Zr/2a_0} \qquad . \end{displaymath}

    What is the correct normalization factor $N$?

    (a) $Z/\sqrt 3 a_0$ (d) $(Z/2a_0)^3 Z^2/3a_0^2$
    (b) $(Z/2a_0)^{3/2}$ (e) $(Z/2a_0)^3$
    (c) $(Z/2a_0)^{3/2} Z/\sqrt 3 a_0$    

Problems. Clearly show all work for full credit.


1. (20 pts.) A $D_2$ molecule can be treated as a rigid rotator. Consider a case where the molecule is at a temperature of $30~K$ at $t=0$ and in a state

\begin{displaymath} \psi(\theta,\phi,0) = {3 Y_5^4 + 7 Y_5^5 + 2 Y_4^4 \over \sqrt{62} } \end{displaymath}

(a)
What are the possible values and associated probabilities for a measurement of $L^2$ and $L_z$?

(b)
What is $\langle E \rangle$ in terms of $\hbar$, the moment of inertia $I$, and any other constants.


2. (30 pts.) Consider the hydrogen atom eigenfunctions. With $C_0 = 1$ in the recurrence relationship

\begin{displaymath} C_{i+1} = {(i+l+1) - \lambda \over (i+1)(i+2l+2) } C_i \end{displaymath}

obtain $C_1$ and $C_2$ for $l=0$. Then use the following expressions

\begin{displaymath} u_{nl}(\rho) = e^{-\rho/2} \rho^{l+1} F_{nl}(\rho) = A_{nl} e^{-\rho/2} \rho^{l+1} \sum_{i=0}^{n-l-1}C_i \rho^i \end{displaymath}


\begin{displaymath} \rho = 2\kappa_n r \qquad \kappa_n = {Z \over a_0 n} \qquad u(r) = rR(r) \end{displaymath}

to obtain $u_{30}$ in terms of $A_{nl}$ and $\rho$ only. Next, generate an expression for $\phi_{300}$. Do not worry about the normalization constant $A_{nl}$ for $\phi_{300}$; just leave it as a constant.

3. (30 pts.) To derive the Coulomb part of the potential of an atomic nucleus we assume the nuclear charge is uniformly distributed throughout the volume of the nucleus. We can apply Gauss' Law and obtain

\begin{displaymath} \vert\vec E_{in} \vert = {Z_1 e \over R^3} r \qquad \vert\vec E_{out}\vert = {Z_1 e \over r^2} \end{displaymath}

where $\vec E_{in}$ and $\vec E_{out}$ are the electric fields inside and outside the nucleus respectively and $Z_1 e $ is the charge of the nucleus. Consider a point charge $Z_2 e$ brought towards the first nucleus so one can calculate the potential energy $V$ using the following definition.

\begin{displaymath} V = Z_2 e \int_{\infty}^r E dr^\prime \end{displaymath}

Derive the form of the Coulomb part of the potential inside and outside the nucleus.

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 $e^2$ $\hbar c /137$