Physics 309 Test 2

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Questions (6 pts. apiece) Answer questions in complete, well-written sentences WITHIN the spaces provided.

  1. Consider the set of data in the table. A theoretical prediction requires these data to be constant. In words, describe a procedure to determine quantitatively if the data agreed with that prediction. Explain your reasoning.

    0.59395 0.635293
    0.523303 0.342906
    0.638051 0.42535
    0.4888 0.759856
    0.381186 0.562729

  2. What is the paradox of the alpha decay of $\rm ^{238}U$?







  3. Let the potential energy of the NaCl molecule be described by

    \begin{displaymath} V(r) = {-e^2 \over r} + ke^{-r/r_0} \end{displaymath}

    where $r$ is the internuclear separation. If the equilibrium separation is $r^* = 2.50~{\rm\AA}$ and the dissociation energy is $V(r^*) = -3.60~eV$, then find the constants $r_0$ and $k$.

    1. (A.) $r_0 = 2.50~ {\rm\AA}, k = 3.60~eV$

    2. (B.) $r_0 = 1.25~ {\rm\AA}, k = 7.20~eV$

    3. (C.) $r_0 = 0.94~ {\rm\AA}, k = 30.94~eV$

    4. (D.) $r_0 = 1.88~ {\rm\AA}, k = 15.47~eV$

    5. (E.) $r_0 = 2.50~ {\rm\AA}, k = 5.76~eV$


    \includegraphics[height=0.75in]{NaCl.eps}

  4. 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(V_0 - E)}/\hbar \quad . \end{displaymath}

    Find the transmission coefficient.

    1. 1

    2. 0

    3. $k^2 / {k^\prime}^2$

    4. $4k^2 / (k^2 + {k^\prime}^2)$

    5. $k / {k^\prime}$


    \includegraphics[height=1.25in]{stepbarrier.eps}

  5. A deuteron ($Z_1=1$, $m_1 = 2~u$) is incident on a lead nucleus ($Z_2=82$, $m_2=208~u$) at the Brookhaven National Laboratory. The terminal voltage of the accelerator is $15~\rm MeV$. 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    

Problems. Clearly show all work for full credit.


1. (20 pts.)

A mass $m$ is oscillating freely on a vertical spring. When $m = 0.810~ kg$, the period is $0.910~ s$. An unknown mass on the same spring has a period of $1.16~s$. What is the spring constant $k$ and the unknown mass?

2. (20 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 are well known, but now solve this differential equation using the Method of Frobenius (i.e. the power series method) and obtain the recursion relationship.

3. (30 pts)

Find $\psi(x,t)$ and $P(E_n)$ at $t>0$, relevant to a particle in a one-dimensional box with walls at $(0,a)$ for the initial state

\begin{displaymath} \psi(x,0) = A x^2 (x-a) \end{displaymath}

where $A=\sqrt{105/a^7}$. The eigenfunctions and eigenvalues for this particle in a box are the following.

\begin{displaymath} \vert\phi_n \rangle = \sqrt{\frac{2}{a}} \sin \left ( \frac{... ...a} \right ) \qquad E_n = n^2\frac{\hbar^2 \pi^2}{2ma^2}=n^2E_1 \end{displaymath}

Your answers should be in terms of $x$, $t$, $n$, $a$, and $E_1$ and any other necessary constants.

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Physics 309 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 ($m_n$) $1.68\times 10^{-27}~kg$    
  $939~MeV/c^2$    

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Physics 309 Equations


\begin{displaymath} E = h\nu = \hbar \omega \qquad v_{wave} = \lambda \nu \qquad... ...\vert^2 \qquad \lambda = {h \over p} \qquad p = \hbar k \qquad \end{displaymath}


\begin{displaymath} -{\hbar^2 \over 2 m} {\partial^2 \over\partial x^2} \Psi(x,t... ...{A~}\rangle = \int_{-\infty}^{\infty} \psi^* \hat {A~} \psi dx \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} \vert\psi (t) \rangle = \sum b_n \vert\phi_n\rangle e^{-i\om... ... (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle^2 \qquad \end{displaymath}

The wave function, $\Psi(\vec r,t)$, contains all we know of a system and its square is the probability of finding the system in the region $\vec r$ to $\vec r + d\vec r$. The wave function and its derivative are (1) finite, (2) continuous, and (3) single-valued.


\begin{displaymath} V_{HO} = {\kappa x^2 \over 2} \quad \omega = 2 \pi \nu = \sq... ...ad \vert\phi_n\rangle = A_ne^{-u^2/2}H_n(u) \quad u = \beta x \end{displaymath}


\begin{displaymath} \beta^2 = {m\omega_0 \over \hbar} \quad \left ( \matrix{ \h... ...dagger \vert\phi_n\rangle = \sqrt{n+1} \vert\phi_{n+1} \rangle \end{displaymath}


\begin{displaymath} \langle K \rangle = {3\over 2} kT \qquad \psi_1 = {\bf t} ... ...\over \vert t_{11}\vert^2} \qquad V(r) = {Z_1 Z_2 e^2 \over r} \end{displaymath}


\begin{displaymath} \psi(x) = \sum_{n=1}^\infty a_n x^n \quad n(v) = 4 \pi N \left ( {m \over 2 \pi k_B T} \right )^{3/2} v^2 e^{-mv^2/2k_B T} \end{displaymath}


\begin{displaymath} E = {\hbar^2 k^2 \over 2 m} \quad k = \sqrt{2m (E-V) \over \... ... {\rm incident\ flux}} \quad {\rm flux} = \vert\phi \vert^2 v \end{displaymath}


\begin{displaymath} \overline K = {3\over 2} kT \quad \zeta_1 = {\bf t}\zeta_3 =... ...}\zeta_3 \quad T = {1 \over \vert t_{11}\vert^2} \quad R+T = 1 \end{displaymath}


\begin{displaymath} T_{WKB} = \exp\left [ -2 \int_{x_0}^{x_1} \sqrt {2m(V(x) - E) \over \hbar^2} ~ dx\right ] \end{displaymath}


\begin{displaymath} \frac{dN}{dt} = {d\sigma \over d\Omega} ~d\Omega I n_{tgt} \... ... H \vert \psi \rangle \over \langle \psi \vert \psi \rangle } \end{displaymath}


\begin{displaymath} e^{i\phi} = \cos\phi + i\sin\phi \quad \psi_1(a) = \psi_2(a... ...hen }\frac{f(x_0)}{g(x_0)}=\frac{f^\prime(x_0)}{g^\prime(x_0)} \end{displaymath}