Physics 309 Test 2

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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. The separations between energy states for some molecule are the following: 0.5132 eV, 0.5610 eV, 0.4917 eV, 0.4715 eV, 0.4963 eV, 0.4869 eV, 0.4631 eV, 0.5037 eV, 0.5658 eV, 0.546 eV. The average separation is $0.510\pm 0.036~\rm eV$. Does this molecule behave like a harmonic oscillator? Explain and be quantitative.

   
   
   
   
   
   
   

2. When we solved the harmonic oscillator Schroedinger equation we used the power series method to solve the differential equation. As we developed that solution we applied the technique of truncating the infinite series so it would not go to infinity. Why did we do this?

   
   
   
   
   
   
   

3. A one-dimensional, free particle wave packet $\psi(x)$ can be described with an infinite series of eigenfunctions
   
   $$\psi(x) = \int_{-\infty}^\infty b(k) \frac{e^{ikx}}{\sqrt{2\pi}} dk$$
   
   
   where the $b(k)$ and $k$ are known. How would you modify this expression to study its time development? Express your answer in terms of known quantities like the mass $m$, $k$, $b(k)$, or other physical constants.

   
   
   
   
   
   
   

4. 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?

   
   
   
   
   
   
   

   Do not write below this line.

5. Is there a classical physics explanation for solar fusion to occur even though the average energy of protons in the Sun is well below the Coulomb barrier? What is the explanation?

   
   
   
   
   
   
   
   

Problems. Clearly show all work for full credit.

1. (15 pts.)	A 5.30-MeV $\alpha$ particle is incident on a gold foil. Calculate the distance of closest approach for a head-on collision. The alpha nucleus has two protons (positive charges) and the gold nucleus has 79 protons (positive charges).
2. (25 pts.)	Recall our old friend, Newton's Second Law, $\vec F = m \vec a$ and perhaps a new one in the drag force equation $F_d = -bv$ which can be combined to form a differential equation for an object falling straight down $$m \frac{d^2y}{dt^2} = -b\frac{dy}{dt} - mg \qquad {\rm or} \qquad \frac{d^2y}{dt^2} + \frac{b}{m}\frac{dy}{dt} + g = 0$$ where $b$ is a parameter describing the drag force. Now solve this differential equation using the Method of Frobenius (the power series method) and generate the recursion relationship that relates different coefficients to one another.
3. (30 pts)	Electrons in a beam of density $\rho = 10^{15}~electrons/m$ are accelerated through a potential difference $V_1 = 50~eV$. The resulting current impinges on a potential step of height $V_0 = 20~eV$ as shown in the figure. The solution to the Schroedinger equation in Region 1 in the figure is $\phi_1 = A e^{ik_1 x} + B e^{-i k_1 x}$. The solution to the Schroedinger equation in Region 2 in the figure is $\phi_2 = C e^{ik_2 x} + D e^{-i k_2 x}$. What conditions must these solutions satisfy? Apply them to get conditions on the coefficients $A$, $B$, $C$, and $D$. What fraction of the incident beam is reflected at the barrier?

Physics 309 Equations

$$E = h\nu = \hbar \omega \qquad v_{wave} = \lambda \nu \qquad... ...\vert^2 \qquad \lambda = {h \over p} \qquad p = \hbar k \qquad$$

$$-{\hbar^2 \over 2 m} {\partial^2 \over\partial x^2} \Psi(x,t... ...{A~}\rangle = \int_{-\infty}^{\infty} \psi^* \hat {A~} \psi dx$$

$$\langle\phi_{n'} \vert \phi_n \rangle = \int_{-\infty}^{\i... ... \delta(k - k') \quad e^{i\phi} = \cos\phi + i\sin\phi \quad$$

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

$$\vert\psi (t) \rangle = \sum b_n \vert\phi_n\rangle e^{-i\om... ... (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle^2 \qquad$$

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 ( $\psi_1(a) = \psi_2(a)$ and $\psi^\prime_1(a) = \psi^\prime_2 (a)$) .

$$V_{HO} = {\kappa x^2 \over 2} \quad \omega = 2 \pi \nu = \sq... ...(u) \quad u = \beta x \quad \beta^2 = {m\omega_0 \over \hbar}$$

$$\psi_1 = {\bf t} \psi_3 = {\bf d_{12} p_2 d_{21} p_1^{-1... ..._{x_0}^{x_1} \sqrt {2m(V(x) - E) \over \hbar^2} ~ dx\right ]$$

$$E = {\hbar^2 k^2 \over 2 m} \quad k = \sqrt{2m (E-V) \over \... ... {\rm incident\ flux}} \quad {\rm flux} = \vert\phi \vert^2 v$$

$$V(r) = {Z_1 Z_2 e^2 \over r} \quad \frac{dN}{dt} = {d\sigma ... ...\over 2} \right ) } \quad E = \frac{1}{2}\mu v^2 + V(r) \quad$$

$$\mu = \frac{m_1 m_2}{m_1+m_2} \quad \psi(x) = \sum_{n=1}^\in... ...t ( {m \over 2 \pi k_B T} \right )^{3/2} v^2 e^{-mv^2/2k_B T}$$

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