Physics 310 Test 1

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

  1. Recall your comparison of theory and data in the alpha decay lab. What did the theory get right? What is at least one weakness in the theory?









  2. What is the quantum program?









  3. List a measurement we discussed in class that requires quantum mechanics to explain it and describe how classical physics failed in that explanation.









  4. When we solved the rectangular barrier problem we required the wave function to continuous across the boundary between different potential energy regions. Why?









  5. In solving the carbon-monoxide problem we choose the origin of our coordinate system to be on the center-of-mass of the molecule. Why?

Problems. Clearly show all work for full credit.


1. (20 pts.)

Recall your calculation of the total derivative $d/dy$ for homework. Show

\begin{displaymath} {\partial \theta \over \partial y} = {\sin \phi \cos \theta \over r} \qquad . \end{displaymath}

(Hint: Use the fact that ${\partial z / \partial y} = 0$ where $z = r \cos \theta$ and $\partial r/\partial y = y/r $.)


The transformation from Cartesian coordinates to spherical coordinates is

\begin{eqnarray*} x &= r \sin \theta \cos \phi \\ y &= r \sin \theta \sin \phi \\ z &= r \cos \theta \qquad . \end{eqnarray*}
2. (20 pts.)

A beam of $\rm ^{4}_2He$ ($Z_1 = 2$, $A_1 = 4$), i.e. $\alpha$-particles, of kinetic energy $5.30~ \rm MeV$ and intensity $10^4~ particles/s$, is incident on a gold foil ($Z_2 = 79$, $A_2=197$) of density $19.3~g/cm^3$ and thickness $1.0 \times 10^{-5}~cm$. A detector of area $\rm0.1~ cm^2$ is placed at a distance of $10~ cm$ from the foil. If $\theta$ is the angle between the incident beam and a line from the center of the foil to the center of the detector and the differential cross section for Rutherford scattering at $\theta = 15^\circ$ is $d\sigma/d\Omega = 3.95\times 10^5~fm^2$, then what is the count rate?

3. (30 pts)

A particle in incident on a step barrier of height $V_0$ as shown in the figure. The components of the wave function in different regions are also shown in the figure. What is the reflection coefficient in terms of the wave numbers $k_i$? Leave your answer in symbolic form.

Physics 310 Equations and Constants



\begin{displaymath} R_T(\nu) = {Energy \over time \times area} \quad E = h\nu = ... ...lambda \nu \quad \lambda = {h \over p} \quad p = \hbar k \quad \end{displaymath}


\begin{displaymath} -{\hbar^2 \over 2 m} {\partial^2 \over\partial x^2} \Psi(x,t... ...si(x,t) \quad \hat {p}_x = -i\hbar {\partial \over \partial x} \end{displaymath}


\begin{displaymath} \hat{A~}\vert\phi\rangle = a\vert\phi\rangle \quad \langle\h... ...angle = \int_{-\infty}^{\infty} \psi^* \hat {A~} \psi dx \quad \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} {\rm If}\ f(x) = \sqrt{1 \over 2 \pi \sigma^2} ~ e^{-x^2/2\sigma^2}, \ {\rm then}\ \Delta x = \sigma \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} \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 \... ... {\rm incident\ flux}} \quad {\rm flux} = \vert\psi \vert^2 v \end{displaymath}


\begin{displaymath} V(r) = {Z_1 Z_2 e^2 \over r} \quad \frac{dN_s}{dt} = {d\sigm... ...t ( {\theta \over 2} \right ) } \quad d\Omega = \frac{dA}{r^2} \end{displaymath}


\begin{displaymath} \vec p = m \vec v \quad \vec L = \vec r \times \vec p = I\ve... ...frac{L^2}{2\mu r^2} + V(r) \quad \mu = \frac{m_1 m_2}{m_1+m_2} \end{displaymath}


\begin{displaymath} \hat{H~}\psi = {-\hbar^2 \over 2m} {\nabla^2 \psi} + V\psi... ...nabla} \quad \vert\phi \rangle = R_{nl}(r) Y_l^m(\theta,\phi) \end{displaymath}


\begin{displaymath} d\tau = r^2 d\cos\theta d\phi \quad L_z \vert nlm\rangle = m... ... L^2 \vert nlm\rangle = l (l+1) \hbar^2 \vert nlm\rangle \quad \end{displaymath}



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 ($m_n$) $1.68\times 10^{-27}~kg$ Avogadro's Number ($N_A$) $6.022\times 10^{23}~ mol^{-1} $
$939~MeV/c^2$