Physics 309 Test 1

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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. How would you define the width of a distribution?







  2. What is orthonormality for a bound system like the particle in a box?







  3. What was wrong with the Rayleigh-Jeans calculation of the blackbody radiation? In other words, what was the difference between the calculation of the intensity of blackbody radiation versus the frequency of the light (or the wavelength) and the measured result?







  4. Consider a particle in a box confined to the region from $x=0$ to $x=a$ with initial wave function $\vert\psi(x,0)\rangle$. The eigenfunctions and eigenvalues are

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

    where $m$ is the mass of the particle. The coefficients of the Fourier series for $\psi(x,0)\rangle$ are $b_n=\langle\phi_n \vert \psi(x,0)\rangle$. What is the expression for the time evolution of the wave function $\vert\psi(x,t)\rangle$ for all $t$ in terms of $E_n$, $m$, $a$, $b_n$, and any constants.







    Do not write below this line.

  5. Suppose you know the eigenfunctions for a particular form of the Schroedinger equation (e.g., the particle in a box), the initial wave packet $\vert\psi(x,t=0)\rangle$, and the spectral distribution, the distribution of $\vert b_n\vert^2$. How do you construct an uncertainty relationship? Don't calculate anything here, just state the procedure.








Problems. Clearly show all work for full credit.


1. (20 pts.)

What is the de Broglie wavelength of a $1.5~MeV$ neutron? Give your answer in fermis $fm$.

2. (20 pts.)

What is the expectation value of the momentum $\langle p \rangle$ for a particle in the following state.

\begin{displaymath} \psi(x,t) = A e^{-(x/a)^3}e^{-i\omega t} \cos kx \end{displaymath}

3. (30 pts)

A pulse neutrons of length $L$ contains $N$ particles. At $t=0$, each neutron is in the following state.

$\psi(x,0) =$ $\frac{1}{\sqrt{L}} e^{ik_0 x} \quad$ $-\frac{L}{2} \leq x \leq \frac{L}{2}$
$ =$ $0$ otherwise

The eigenfunctions and eigenvalues for the neutrons are the following.

\begin{displaymath} \vert\phi_k(x)\rangle = \frac{e^{ikx}}{\sqrt{2\pi}} \qquad E = \frac{\hbar^2k^2}{2m} \end{displaymath}

At $t=0$, how many neutrons have momentum in the range $0\leq \hbar k \leq \hbar k_0$? Leave your answer in symbolic form.

Physics 309 Equations



\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... ... 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... ...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} \vert\psi (t) \rangle = \sum b_n \vert\phi_n\rangle e^{-i\om... ... \rangle = \int b(k) \vert\phi(k)\rangle e^{-i\omega(k) t} dk \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}


\begin{displaymath} {\rm If}\ f(x) = \sqrt{1 \over 2 \pi \sigma^2} ~ e^{-x^2/2\s... ...gle \hat B \phi \vert \phi\rangle~({\rm Hermitian\ operators}) \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 ($m_n$) $1.68\times 10^{-27}~kg$    
  $939~MeV/c^2$