Physics 309 Test 1

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Questions (5 pts. apiece) Answer questions 1-4 in complete, well-written sentences WITHIN the spaces provided. For multiple-choice question 5 circle the correct answer.

  1. Cite two experimental results that show that energy can be quantized.











  2. What is the time-dependent solution $\vert\psi(x,t)\rangle$ for a system with initial wave function $\vert\psi(x,t=0)\rangle$, eigenfunctions $\vert\phi_n\rangle$, and eigenvalues $E_n$? Your answer should be in terms of the knowns $\vert\psi(x,t=0)\rangle$, $\vert\phi_n\rangle$, $E_n$ or other quantities you express in terms of those knowns.








  3. The figure below shows evidence of the molar specific heat freeze out for diatomic molecules such as $\rm N_2$ and $\rm O_2$. At low temperatures like $40~K$ the molar specific heat of these molecules is the same as monatomic ones such as argon. Explain why the molar specific heat rises from $3R/2$ to $5R/2$ (where $R = N_A k_B$, $N_A$ is Avogadro's number, and $k_B$ is Boltzmann's constant) as the temperature rises to $500~K$.

    \includegraphics[height=1.5in]{freezeout2.eps}



  4. Consider a particle-in-a-box in the following initial state

    \begin{displaymath} \vert\psi(,x,t=0)\rangle = \frac{\vert\phi_2\rangle + 3\vert\phi_4\rangle + 2\vert\phi_6\rangle}{\sqrt {14}} \end{displaymath}

    where the $\vert\phi_n\rangle$ are the eigenfunctions for the states with energies $E_n$. What is the probability of measuring $E_6$?










  5. Which of the following is NOT a true statement about quantum mechanics?

    1. The wave function is always a real quantity.

    2. The wave function represents the complete physical state.

    3. The quantities $\psi$, $\psi^\prime$, and $\psi^{\prime \prime}$ are finite, single-valued, and continuous.

    4. For every observable, there is a quantum mechanical operator.

    5. In one dimension, $\int_{-\infty}^{\infty} \psi^* \psi dx =1$ is required.

Problems. Clearly show all work for full credit. Use a separate sheet to show your work.


1. (15 pts)

The work function of zinc is $W=3.6~eV$. What is the energy of the most energetic photoelectrons emitted by ultraviolet light of wavelength $\lambda = 2800~{\rm\AA}$.

2. (25 pts)

What is the uncertainty relation for the product $\Delta x \Delta E$?

3. (35 pts)

Consider a case of one dimensional nuclear `fusion'. A neutron is in the potential well of a nucleus that we will approximate with an infinite square well with walls at $x=0$ and $x=a$. The eigenfunctions and eigenvalues are

\begin{eqnarray*} E_n = {n^2 \hbar^2 \pi^2 \over 2 m a^2} \qquad \phi_n & = & \... ... a \\ & = & 0 \hspace{3.5cm} x < 0 ~ {\rm and} ~ x > a \qquad. \end{eqnarray*}

The neutron is in the $n=4$ state when it fuses with another nucleus that is the same size, instantly putting the neutron in a new infinite square well with walls at $x=0$ and $x=2a$. (1) What are the new eigenfunctions and eigenvalues of the fused system? (2) Calculate the probabilities for finding the neutron in the two lowest energy states of the fused system.


Physics 309 Equations, Conversions, and Constants


\begin{displaymath} R_T(\nu) = {Energy \over time \times area} \quad E = h\nu = ... ...nu \quad I \propto \vert\vec E\vert^2 \quad K_{max} = h\nu - W \end{displaymath}


\begin{displaymath} \lambda = {h \over p} \quad p = \hbar k \quad -{\hbar^2 \ove... ...i(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... ... dx \quad [ \hat A, \hat B ~ ] = \hat A \hat B - \hat B \hat A \end{displaymath}


\begin{displaymath} \langle\phi_{n'} \vert \phi_n \rangle = \int_{-\infty}^{\i... ...\infty}^{\infty} \phi(k^\prime)^* \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... ...r 2 \pi \sigma^2} ~ e^{-x^2/2\sigma^2}, \ \Delta x = \sigma \end{displaymath}


\begin{displaymath} \Delta p \Delta x \ge {\hbar \over 2} \quad (\Delta x)^2 = \... ...\ \Delta A \Delta B \ge \frac{\vert\langle C \rangle \vert}{2} \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.



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 ($h$) $6.621 \times 10^{-34} J-s$ Neutron mass ($m_n$) $1.68\times 10^{-27} kg$
  $4.1357\times 10^{-15} eV-s$   $939 MeV/c^2$
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 ($u$) $1.66\times 10^{-27} kg$
  $938 MeV/c^2$   $931.5 MeV/c^2$