Physics 309 Test 2

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Questions (6 pts. apiece) Answer questions in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions circle the correct answer.

  1. What is the CLASSICAL expectation for the transmission coefficient of a particle of energy $E$ striking a one-dimensional rectangular barrier of height $V_0$? Explain.







  2. Consider a particle in a box. Measurement of the energy reveals the particle is in the $n=3$ state. A follow-up measurement finds the particle is located at the mid-point of the potential well. What result do you expect for a repeat of the energy measurement? Do not calculate anything; answer in words.







  3. What is particle flux?







  4. In solving the three-dimensional Schroedinger equation for the $\rm CO$ rotator problem we encountered the following equation

    \begin{displaymath} -\left \{ {\sin^2\theta \over R} {\partial \ \over \partia... ...t \} = {1 \over \Phi } {\partial^2 \Phi \over \partial \phi^2} \end{displaymath}

    where $R$, $\Theta$, and $\Phi$ are solutions to the radial ($r$), angular ($\theta$), and azimuthal ($\phi$) parts. What does this equation equal? Explain your reasoning.







  5. A particle of mass $m$ follows the Maxwell-Boltzmann distribution at temperature $T$. What is the most probable speed for this particle?

    (a) $\sqrt{2kT/m}$ (c) $\sqrt{8kT/m}$ (e) $\sqrt{3kT/\pi m}$
    (b) $\sqrt{kT/m}$ (d) $\sqrt{3kT/m}$    

Problems. Clearly show all work for full credit.


1. (15 pts.)

A harmonic oscillator consists of a mass $m=0.001~kg$ on a spring. Its frequency is $\nu_0= 1~Hz$ and the mass passes through the equilibrium point with a velocity $v=15~m/s$. What is the order of magnitude of the quantum number associated with the energy of this system?

2. (15 pts)

The inverse propagation matrix in the region where $V=0$ for the rectangular barrier problem is

\begin{displaymath} {\bf p_1} = \pmatrix{ e^{ -i k_1 2 a} & 0 \cr 0 & e^{ i k_1 2 a} \cr } \qquad \end{displaymath}

where $a$ is the half-width of the barrier and $k_1$ is the wave number in the region of the barrier. What is the inverse of ${\bf p_1}$? Show ALL work for full credit.

3. (20 pts)

An electron beam is sent through a rectangular potential barrier of half-width $a=4.5~ \rm\AA$ long. The transmission coefficient exhibits a fourth maximum at an incident electron energy $E = 200~eV$. What is the height of the barrier? The transmission coefficient $T$ is

\begin{displaymath} \frac{1}{T} = 1 + \frac{1}{4} \frac{V_0^2}{E(E-V_0)}\sin^2\left (2k_2 a\right) \qquad E > V_0 \end{displaymath}


\begin{displaymath} \frac{1}{T} = 1 + \frac{1}{4} \frac{V_0^2}{E(E-V_0)}\sinh^2\left (2\kappa a\right) \qquad E < V_0 \end{displaymath}

where $V_o$ is the barrier height, $\kappa = \sqrt{2 m_e (V_0 - E)/\hbar^2} > 0$, $k_2 = \sqrt{2 m_e (E-V_0)/\hbar^2} > 0$, and $m_e$ is the electron mass.

4. (20 pts)

A beam of $\alpha$-particles, of kinetic energy $E=5.30~\rm MeV$ and intensity $\Delta N_{inc}/\Delta t =10^4~ particles/s$, is incident on a foil of density $\rm\rho=8.9~g/cm^3$, atomic weight $A=63.6$, atomic number $Z=29$, and thickness $L=\rm 2.0 \times 10^{-5}~cm$. A detector of area $\rm 1.5~ cm^2$ is placed at a distance of $r=10~ cm$ from the foil. The count rate in the detector is $\rm 10~counts/s$ at $\theta = 10^\circ$. What is the cross section?

Physics 309 Equations



\begin{displaymath} E = h\nu = \hbar \omega \quad v_{wave} = \lambda \nu \quad I... ...ert\vec E\vert^2 \quad \lambda = {h \over p} \quad p = \hbar k \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... ... \delta(k - k') \quad e^{i\phi} = \cos\phi + i\sin\phi \quad \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 ( $\psi_1(a) = \psi_2(a)$ and $\psi^\prime_1(a) = \psi^\prime_2 (a)$) .


\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} \psi_1 = {\bf t} \psi_3 = {\bf d_{12} p_2 d_{21} p_1^{-1... ..._{x_0}^{x_1} \sqrt {2m(V(x) - E) \over \hbar^2} ~ dx\right ] \end{displaymath}


\begin{displaymath} E = {\hbar^2 k^2 \over 2 m} \quad k = \sqrt{2m (E-V) \over \... ...= \vert\phi \vert^2 v \quad V(r) = {Z_1 Z_2 e^2 \over r} \quad \end{displaymath}


\begin{displaymath} \frac{dN_{scat}}{dt} = {d\sigma \over d\Omega} ~d\Omega \fra... ...\frac{1}{2}\mu v^2 + V(r) \quad \mu = \frac{m_1 m_2}{m_1+m_2} \end{displaymath}


\begin{displaymath} \vec L = \vec r \times \vec p \quad \vert\vec L \vert = rp\s... ...t ( {m \over 2 \pi k_B T} \right )^{3/2} v^2 e^{-mv^2/2k_B T} \end{displaymath}

Physics 309 Conversions, and Constants

Avogadro's Number ($N_A$) $6.022\times 10^{23}$ 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 ($m_n$) $1.68\times 10^{-27} kg$ Speed of light ($c$) $2.9979\times 10^8 m/s$
$939 MeV/c^2$