Physics 309 Test 1

I pledge that I have given nor received unauthorized assistance during the completion of this work.

Signature:


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

  1. Cite at least one experimental result that shows that particles can behave as waves and interfere.








  2. High-energy neutrons with a short de Broglie wavelength are reflected out the sides of a piece of polycrystalline graphite. Low-energy neutrons with a long de Broglie pass through the graphite. Why?









  3. What is the quantum program?









  4. When we solved the particle-in-a-box problem (i.e., the infinite rectangular well) we required the wave function to be zero at the edges of the well. Why?








  5. Suppose the initial wave function of a particle-in-a-box is

    \begin{displaymath} \vert\psi \rangle = \frac{9\vert\phi_9\rangle + 6\vert\phi_7\rangle +3\vert\phi_5\rangle}{\sqrt{126}} \end{displaymath}

    and a first measurement of the energy is made with the result $E_a = E_7$. A second measurement is made on the same system. What is the probability the result is $E_b = E_5$? Explain.








Problems. Clearly show all work for full credit.


1. (20 pts.)

The number of hairs $N_l$ on a certain rare species can only be the number $3^l$ where $l=0,1,2,3,...$. The probability of finding such an animal with $3^l$ hairs is $e^{-1}/l!$. What is $\langle N^2\rangle$? You should obtain a numerical result.
2. (20 pts.)

The time-dependent Schroedinger equation in one dimension is the following.

 \begin{displaymath} -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t) = i \hbar \frac{\partial \Psi(x,t)}{\partial t} \end{displaymath} (1)

The time-independent form of the Schroedinger equation is
 \begin{displaymath} -\frac{\hbar^2}{2m}\frac{\partial^2 \psi(x)}{\partial x^2} + V(x)\psi(x) = E\psi(x) \end{displaymath} (2)

where $\Psi(x,t) = \psi(x)T(t)$ and $V=V(x)$ depends only on position. Starting with Eq 1 obtain the time-independent form of the Schroedinger equation (Eq 2). What equation does $T(t)$ satisfy? How does the energy $E$ enter the problem?

3. (30 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 $n=3$ 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. What are the probabilities for finding the neutron in the two lowest energy states of the fused system.

Physics 309 Equations and Constants



\begin{displaymath} R_T(\nu) = {Energy \over time \times area} \quad E = h\nu =... ... E\vert^2 \quad K_{max} = h\nu - W \quad K = \frac{p^2}{2m} \end{displaymath}


\begin{displaymath} \lambda = {h \over p} \quad p = \hbar k \quad -{\hbar^2 \o... ...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 ... ...psi (t) \rangle = \sum b_n \vert\phi_n\rangle e^{-i\omega_n t} \end{displaymath}


\begin{displaymath} \Delta p \Delta x \propto \hbar \quad (\Delta x)^2 = \langle... ... P(x) dx \quad \langle f_n \rangle = \sum_{n=0}^\infty f_n P_n \end{displaymath}


\begin{displaymath} {\rm If}\ f(x) = \sqrt{1 \over 2 \pi \sigma^2} ~ e^{-x^2/2\s... ...le = \langle \hat B \phi \vert \phi\rangle~({\rm Hermiticity}) \end{displaymath}


\begin{displaymath} e^{ix} = \cos x + i\sin x \quad \sin x = \frac{e^{ix}-e^{-ix... ...{g(x)} = \lim_{x\rightarrow c} \frac{f^\prime(x)}{g^\prime(x)} \end{displaymath}

The wave function, $\psi(\vec r,t)$, contains all we know of a system and $\vert\psi\vert^2$ is the probability of finding it 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} \frac{df}{du} = \frac{df}{dx}\frac{du}{dx} \quad \frac{d}{d... ...(\cos x) = -\sin x \quad \frac{d}{dx}(e^{ax}) = a e^{ax} \quad \end{displaymath}


\begin{displaymath} \frac{d}{dx}(\ln ax) = \frac{1}{x} \quad \int x^n dx = \frac... ...ax} dx = \frac{e^{ax}}{a} \quad \int \frac{1}{x} = \ln x \quad \end{displaymath}



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