Physics 401 Test 2

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Questions (5 pts. apiece) Answer questions in complete, well-written sentences WITHIN the spaces provided.

1. Why does the sun shine? Your answer should be descriptive and qualitative; not quantitative.

   
   
   
   
   
   
   
   
   
   

2. A one-dimensional, free particle wave packet $\psi(x)$ can be described with an infinite series of eigenfunctions
   

   $$\psi(x) = \int_{-\infty}^\infty b(k) \frac{e^{ikx}}{\sqrt{2\pi}} dk$$ (1)

   
   
   where the $a_n$ 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 $m$, $k$, $a_n$, or other physical constants.

   
   
   
   
   
   
   
   
   
   

3. We have spent most of the semester developing the quantum program. Is there any evidence this method is correct? Explain.

   
   
   
   
   
   
   
   
   
   

4. In solving the Schroedinger equation for the harmonic oscillator potential, we were forced to truncate the infinite sum. What forced us to do this?

   
   
   
   
   
   
   
   

5. The particle-in-a-box problem predicted the energies would be
   

   $$E_n = n^2 \frac{\hbar^2}{2ma^2}$$ (2)

   
   
   where $m$ is the mass and $a$ is the width of the box. Suppose you had some data on the energy levels of some system. How would you show that your data was described by this model (or not)?

   
   
   
   
   
   
   
   
   
   
   

Problems. Clearly show all work for full credit.

1. (20 pts.)

Fusion reactions in the Sun are responsible for the production of solar energy. One of these reactions is the fusion of a proton with a deuteron (a proton and neutron bound together). $$p + d \rightarrow ^3\hspace{-0.065in}He + \gamma$$ The deuteron has a radius of about $1.2 fm$ and the proton has a radius of about $1 fm$ where $1~ fm = 10^{-15}~m$. Calculate the inter-nuclear distance when the proton and the deuteron are just touching and call this distance $r^\prime$. Estimate the Coulomb barrier, $V_C$, between the proton and the deuteron.

2. (25 pts)

Consider the following initial state for a particle in a box with walls at $(0,a)$. $$\psi(x,0) = \sqrt{\frac{8}{a}} \sin \left ( \frac{\pi x }{L} \right ) \cos \left ( \frac{\pi x }{L} \right )$$ The eigenfunctions and eigenvalues are $$\vert\phi_n(x,t) \rangle = \sqrt{\frac{2}{a}} \sin \left (\f... ...}{a} \right ) \qquad E_n = n^2 \frac{\hbar^2}{2ma^2} = n^2 E_1$$ What is $\psi (x,t)$ and $P(E_n)$ at $t>0$ in terms of $a$, $E_1$, and any other necessary constants?

3. (30 pts)

For a harmonic oscillator state in the superposition state at $t=0$ $$\psi(x,t=0) = \frac{\phi_1(x) + \phi_2 (x)}{\sqrt{2}}$$ what is $\langle x \rangle$ for a proton and for an energy-level spacing $\Delta E = 10 ~MeV$?

Physics 401 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 ($m_n$)	$1.68\times 10^{-27} kg$
	$939 MeV/c^2$

Physics 401 Equations

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

$$-{\hbar^2 \over 2 m} {\partial^2 \over\partial x^2} \Psi(x,t... ...(x,t) \qquad \hat {p~}_x = -i\hbar {\partial \over \partial x}$$

$$\hat{A~}\vert\phi\rangle = a\vert\phi\rangle \quad \langle\hat {A~}\rangle = \int_{-\infty}^{\infty} \psi^* \hat {A~} \psi dx$$

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

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

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

$$\Delta p \Delta x \ge {\hbar \over 2} \qquad (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle^2 \qquad$$

Series solution to differential equations: $\phi(x) = \sum_{n=0}^\infty a_n x^n$

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.

$$V(x) = {1 \over 2} K x^2 \quad F = ma = -Kx \quad x(t) = A \... ...ad \omega_0 = \sqrt{\frac{K}{m}} \quad A = \sqrt{\frac{2E}{K}}$$

$$k = {2\pi \over \lambda} \quad \omega_0 = {2\pi \over T} \qu... ...hbar\omega_0 \quad \vert \phi_n(x) \rangle = H_n(x) e^{-x^2/2}$$

$$\beta^2 = {m\omega_0 \over \hbar} \quad \left ( \matrix{ \h... ...dagger \vert\phi_n\rangle = \sqrt{n+1} \vert\phi_{n+1} \rangle$$

$$V_C(r) = \frac{Z_1 Z_2 e^2}{r} \quad e^2 = \frac{\hbar c }{... ...lected\ flux} \over {\rm incident\ flux}} \quad R+T = 1 \qquad$$

$$\langle K \rangle = {3\over 2} kT \qquad \psi_1 = {\bf t} ... ...1^{-1}} \psi_3 \qquad T = {1 \over \vert t_{11}\vert^2} \qquad$$