Physics 309 Test 1

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Questions (6 pts. apiece) Answer questions 1-3 in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions 4-5 circle the correct answer.

1. In solving the particle-in-a-box we assumed the wave function was zero at the edges of the box ($x=0$ and $x=4$). Why?

   
   
   
   
   
   
   
   
   
   
   

2. Consider a particle-in-a-box in the following initial state
   
   $$\vert\psi(,x,t=0)\rangle = \frac{\vert\phi_1\rangle + \vert\phi_2\rangle + \vert\phi_3\rangle}{\sqrt 3}$$
   
   
   where the $\vert\phi_n\rangle$ are the eigenfunctions for the states with energies $E_n$. A measurement of the energy of the state yields a value of $E_2$. What possible values can a second measurement of the energy yield? Why? Assume there are no changes in the system between the measurements.

   
   
   
   
   
   
   
   
   
   
   

3. Using the previous question as the starting point: after the second energy measurement in Question 2, the position of the particle is measured yielding a value $x_1$. The energy is subsequently measured again after the $x$ measurement. What possible values can this last measurement of the energy yield? Why? Assume there are no changes in the system between the measurements.

   
   
   
   
   
   
   
   
   
   
   

   
   
   
   
   
   
   
   

4. The lowest nucleon resonance state is the $\Delta$ which has a mass of $1232~\rm MeV/c^2$ and a width of $120~\rm MeV$. Calculate the lifetime of this $I=3/2$ nucleon state.

   (a)	$5.5\times 10^{-24}~s$	(d)	$6.9\times 10^{-9}~s$
   (b)	$1.2\times 10^{-19}~s$	(e)	$8.4\times 10^{-17}~s$
   (c)	$3.3\times 10^{-23}~s$

   
   
   

5. Determine the speed of the photoelectrons ejected from a metal surface. The threshold wavelength is 2638 angstroms and the wavelength of incident light is 1600 angstroms.

   (a)	$5.2 \times 10^5 m/s$	(d)	$1.66 \times 10^6 m/s$
   (b)	$2.6 \times 10^5 m/s$	(e)	$1.04 \times 10^5 m/s$
   (c)	$2.08 \times 10^6 m/s$

   
   

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

1. (25 pts)

One thousand neutrons are in a one-dimensional box with walls at $x=0$ and $x=a$. At $t=0$, the state of each particle is $$\psi(x,0) = A x^2(x-a)$$ where $A=\sqrt{105}~ a^{-7/2}$. The eigenfunctions and eigenvalues are the following. $$E_n = {n^2 \hbar^2 \pi^2 \over 2 m_p L^2 } \qquad \vert\phi_... ...t ( { n \pi x \over a} \right ) \qquad 0 \ge x \ge a \qquad .$$ How many particles have energy $E_4$?

2. (45 pts)

A particle beam has a continuous wave function that can be described by $$\psi(x,t) = e^{i(k_0x-\omega t)} \qquad .$$ This equation describes a wave train moving in the positive $x$ direction. A beam `pulse' of length $L$ is produced by sending the beam through a `chopper' that opens long enough to let part of the original beam through and then closes again, cutting off the remainder. The wave function of the pulse at time $t=0$ is: $$\begin{eqnarray*} \Psi (x,t) & = & {1 \over \sqrt{L}} e^{ik_0 x } \qquad \vert x... ...\le L/2 \\ & = & 0 \hspace{2.35cm} \vert x\vert > L/2 \qquad . \end{eqnarray*}$$ The eigenfunctions are $\vert\phi(k)\rangle = e^{ikx}/\sqrt{2\pi}$. What is the spectral distribution (i.e. the spectrum of wave numbers) necessary to produce such a wave packet? Calculate the variance $(\Delta x)^2$ of the initial wave packet. At what values of the momenta will no particles be found at $t=0$?

Physics 309 Equations, Conversions, and Constants

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

$$\lambda = {h \over p} \quad p = \hbar k \quad -{\hbar^2 \ove... ...i(x,t) \quad \hat {p~}_x = -i\hbar {\partial \over \partial x}$$

$$\hat{A~}\vert\phi\rangle = a\vert\phi\rangle \quad \langle\h... ...gle \quad [ \hat A, \hat B ~ ] = \hat A \hat B - \hat B \hat A$$

$$\langle\phi_{n'} \vert \phi_n \rangle = \int_{-\infty}^{\i... ...nt_{-\infty}^{\infty} \phi_{k'}^* \phi_k~ dx = \delta(k - k')$$

$$\vert\psi\rangle = \sum b_n \vert\phi_n\rangle \rightarrow ... ...angle dk \rightarrow b(k) = \langle\phi(k) \vert \psi \rangle$$

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

$$\Delta p \Delta x \ge {\hbar \over 2} \quad (\Delta x)^2 = \... ...ma^2} ~ e^{-x^2/2\sigma^2}, \ {\rm then}\ \Delta x = \sigma$$

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 ($h$)	$6.621 \times 10^{-34}~J-s$	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 ($m_n$)	$1.68\times 10^{-27}~kg$
	$939~MeV/c^2$