Physics 401 Final

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Questions (3 pts. apiece) Answer questions 1-6 in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions 7-8 circle the correct answer.

  1. Explain conduction in metals.








  2. What is the definition of the solid angle $d\Omega$? Be sure to explain the components of your definition.








  3. 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?








  4. What is the Fermi energy $E_f$ in a solid?








  5. Recall that for a bound, one-dimensional system system at $t=0$, $ \vert\psi\rangle = \sum_{n=0}^\infty b_n \vert\phi_n\rangle$. What is $b_n$ and how is it related to a measurement of the momentum?








  6. What is the paradox of solar fusion?








  7. Use the Fermi gas model for electrons in a metal to determine the Fermi momentum $k_F=p_f/\hbar$ for electrons with a density $n_e = 1.23\times 10^{29}~m^{-3}$.


    A. $\rm 3.08~ \AA^{-1}$
    B. $\rm 1.94~ \AA^{-1}$
    C. $\rm 1.22~ \AA^{-1}$
    D. $\rm 1.54~ \AA^{-1}$
    E. $\rm0.77~ \AA^{-1}$


    \includegraphics[height=1.250in]{fermi.eps}

  8. One must use the DeBroglie wavelength concept to `derive' the Schroedinger equation from the one dimensional wave equation. What DeBroglie wavelength must be used to get the general time independent equation?

    A. $\lambda = h/\sqrt{2mE}$ D. $\lambda = h/\sqrt{2m(E+U)}$
    B. $\lambda = h/\sqrt{2mU}$ E. $\lambda = h/\sqrt{m(E-U)}$
    C. $\lambda = h/\sqrt{2m(E-U)}$    




Problems. Clearly show all work for full credit.


1. (12 pts.)

A beam of $\alpha$-particles, of kinetic energy 6.30 MeV and intensity $10^4$ particles/sec, is incident on a gold foil of density $19.3~g/cm^3$ and thickness $1.5 \times 10^{-5}$ cm. A detector of area $\rm 1.0~ cm^2$ is placed at a distance of $r=10~cm$ 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, use the Rutherford scattering differential cross section to find the count rate at $\theta = 12^\circ$. The atomic number of gold is $Z=79$ and the mass number is $A=192$.

Problems (continued). Clearly show all work for full credit.


2. (12 pts)

An electron moves in a three-dimensional cube of side $a$. Show the eigenfunctions and eigenvalues are

\begin{displaymath} \vert\phi \rangle = \left ( \frac{2}{a} \right )^{3/2} \si... ...uad E_n = (n_x^2 + n_y^2 + n_z^2) \frac{\hbar^2 \pi^2}{2ma^2} \end{displaymath}

where $n_x$, $n_y$, and $n_z$ are integers and $m$ is the mass. The eigenfunctions and eigenvalues of the one-dimensional particle in a box of width $a$ are

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

3. (12 pts)

A harmonic oscillator consists of a mass $m=0.002~kg$ on a spring. Its frequency is $f=3.0~Hz$ and the mass passes through the equilibrium position with a velocity $v=0.15~m/s$. What is the average spacing between zeros of an eigenstate with this energy?

4. (12 pts)

If $\hat A$, $\hat B$, and $\hat C$ are three distinct operators, show that $[\hat A \hat B,\hat C] = \hat A [ \hat B,\hat C] + [\hat A,\hat C] \hat B$.

5. (14 pts)

At $t=0$ it is known that of 2000 neutrons in a one-dimensional box of width $10^{-8}~m$, 300 have energy $9E_1$ and 1700 have energy $196E_1$. See Problem 2 for the eigenfunctions and eigenvalues of the one-dimensional particle in a box.

a.
Construct a state function that has these properties.

b.
Use the state you have constructed to calculate the density $\rho (x)$ of neutrons per unit length in the box.

c.
How many neutrons are in the `right' half of the box?

6. (14 pts)

A particle moving in one dimension has the wave function

\begin{displaymath} \psi(x,t) = A \exp[i(ax - bt)] \end{displaymath}

where $A$, $a$ and $b$ are constants.

a.
What is the potential field $V(x)$ in which the particle is moving?

b.
If the momentum is measured, what is the result in terms of $a$ and $b$?

c.
If the energy is measured, what is the result in terms of $a$ and $b$?

Equations


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} 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} \vec F_e = q \vec E \quad I \propto \vert\vec E\vert^2 \quad... ...~}\rangle = \int_{-\infty}^{\infty} \psi^* \hat {A~} \psi dx \end{displaymath}


\begin{displaymath} -{\hbar^2 \over 2 m} \nabla^2 \Psi(x,t) + V(x) \Psi(\vec r,t... ...nabla \quad \hat{A~}\vert\phi\rangle = a\vert\phi\rangle \quad \end{displaymath}


\begin{displaymath} \langle\phi_{n'} \vert \phi_n \rangle = \int_{-\infty}^{\inf... ...int_{-\infty}^{\infty} \phi_{k'}^* \phi_k dx = \delta(k - k') \end{displaymath}


\begin{displaymath} \vert\psi\rangle = \sum b_n \vert\phi_n\rangle \quad b_n = \... ...\vert\phi (x,t)\rangle = \vert\phi(x,0)\rangle e^{-i \omega t} \end{displaymath}


\begin{displaymath} \vert\phi_n\rangle = \sqrt{2 \over L} \sin {n \pi x \over L}... ...A,\ {\rm then}\ \lim_{x \rightarrow a} {f(x) \over g(x)} = A \end{displaymath}


\begin{displaymath} (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle^2 \quad... ...gma^2} ~ e^{-x^2/2\sigma^2}, \ {\rm then\ } \Delta x = \sigma \end{displaymath}


\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(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 =... ...p^{-1}_1}\zeta_3 \quad T = {1 \over \vert t_{11}\vert^2} \quad \end{displaymath}


\begin{displaymath} R+T = 1 \quad 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} \... ...ver r} \quad n_e = \frac{(2m_e)^{3/2}}{3\hbar^3\pi^2}E_F^{3/2} \end{displaymath}


\begin{displaymath} e^{i\phi} = \cos\phi + i\sin\phi \quad \psi_1(a) = \psi_2(a... ...ad {\rm and} \quad \psi^\prime_1(a) = \psi^\prime_2 (a) \quad \end{displaymath}


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 ($m_n$) $1.68\times 10^{-27}~kg$ Avogadro's Number $6.023\times10^{23}$
  $939~MeV/c^2$ ($N_A$)  

\includegraphics[width=6.5in]{periodic_chart2.eps}