Physics 401 Final

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Questions (3 pts. apiece) Answer questions in complete, well-written sentences WITHIN the spaces provided.

  1. What properties distinguish a metal from other materials?








  2. Describe the model we used to explain $\alpha$ decay.














  3. What is Rutherford scattering?








  4. What are the conditions of orthonormality?








  5. What is the quantum program?














  6. Consider the spectral distribution shown below. What is the mathematical definition of the width of the distribution? Sketch on the plot what this width represents.

    \includegraphics[height=1.5in]{f1.eps}








  7. A ground-state wave function is described by

    \begin{displaymath} \vert\psi(x,t)\rangle = \sum_{n=1}^\infty b_n \vert \phi_n\rangle e^{-i\omega_n t} \end{displaymath}

    where the $\vert\phi_n\rangle$ are the eigenfunctions of the Schroedinger equation with energy states $E_n$. How does the probability for finding the system in a specific state evolve with time? Explain your reasoning.








  8. Is there a classical alternative to resolving the paradox of solar fusion? Explain.











Problems. Clearly show all work for full credit.


1. (8 pts.)

The work function of zinc is 3.6 eV. What is the energy of the most energetic photoelectron emitted by ultraviolet light of wavelength 2400$\rm\AA$?

2. (17 pts)

At $t=0$ it is known that of 800 neutrons in a one-dimensional box of width $10^{-8}~m$, 500 have energy $9E_1$ and 300 have energy $400E_1$. The eigenfunctions and eigenvalues of the one-dimensional particle in a box of width $a$ are

\begin{displaymath} \vert\phi(x)\rangle = \sqrt{\frac{2}{a}}\sin\left ( \frac{n\... ...qquad E_n = n^2 E_1 = n^2 \frac{\hbar^2 \pi^2}{2ma^2} \quad . \end{displaymath}

  1. Construct a state function that has these properties (Coefficients may be complex).

  2. Use the state you have constructed to calculate the density $\phi(x)$ of neutrons per unit length. Note that $\rho(x)$ is a real function.

  3. How many neutrons are in the right half of the box?

3. (17 pts)

Find $\psi(x)$ and $P(E_n)$ at $t=0$ relevant to a one-dimensional box with walls at $(0,a)$ for the following initial state.

\begin{displaymath} \psi(x,0) = A_3 \left ( e^{i\pi(x-a)/a} - 1 \right ) \end{displaymath}

Make sure you get an expression for $\psi(x)$ valid for all eigenstates. The eigenfunctions and eigenvalues of the one-dimensional particle in a box of width $a$ are

\begin{displaymath} \vert\phi(x)\rangle = \sqrt{\frac{2}{a}}\sin\left ( \frac{n\... ...qquad E_n = n^2 E_1 = n^2 \frac{\hbar^2 \pi^2}{2ma^2} \quad . \end{displaymath}

Continue $\rightarrow$

Problems (continued). Clearly show all work for full credit.


4. (17 pts)

In the phenomenon of cold emission, electrons are drawn from a metal at room temperature by an externally supported electric field. The potential well the metal presents to the free electrons before the electric field is turned on is shown in Figure 1a below. After application of the constant electric field $\vec \mathcal{E}$, the potential at the surface slopes down as shown in Figure 1b below, thereby allowing electrons in the Fermi sea to `tunnel' through the potential barrier. If the surface of the metal is taken as the $x=0$ plane, the new potential outside the surface is

\begin{displaymath} V(x) = \Phi + E_f - e\mathcal{E}x \end{displaymath}

where $E_f$ is the Fermi level and $\Phi$ is the work function of the metal.

  1. Use the WKB approximation to calculate the transmission coefficient for cold emission.

  2. Estimate the field strength $\mathcal{E}$ necessary to draw a current a current density out of the surface $J_0 = 10^{-3}~A/cm^2$ for potassium. The value of both $E_f$ and $\Phi$ for potassium is $2.1~ eV$ and the current density $J_{inc}$ in potassium is $2.6\times 10^{10}~ A/cm^2$.




\includegraphics[height=1.0in]{p2a.eps} \includegraphics[height=1.5in]{p2b.eps}
Figure 1a Figure 1b

Continue $\rightarrow$

Problems (continued). Clearly show all work for full credit.


5. (17 pts)

In solving the Schroedinger equation for the harmonic oscillator potential we rewrote the Schroedinger equation in the form

\begin{displaymath} {d^2 \phi \over du^2} + \left ( {\alpha \over \beta^2} - u^2 \right )\phi = 0 \end{displaymath}

where $u = \beta x$, $\alpha = 2mE/\hbar^2$ and $\beta = \sqrt{m \omega_0 /\hbar}$. We then showed the asymptotic solution is

\begin{displaymath} \vert\phi_{asymp}\rangle = A_{asymp}e^{-u^2/2} + B_{asymp}e^{u^2/2} \quad . \end{displaymath}

We then made the guess that the initial wave function will be of the form

\begin{displaymath} \vert\phi(u)\rangle = A e^{-u^2/2} H(u) \end{displaymath}

where $u = \beta x$, $H(u)$ is some, as-yet-to-be-determined function, and $A$ is a normalization constant. This guess was made in the hope of ensuring the finiteness of the wave function far outside the range of the potential. Starting from this form of the wave function and the Schroedinger equation, show the new differential equation we must solve is

\begin{displaymath} {d^2 H(u) \over du^2} - 2 u {d H(u) \over du} + \left ( {\alpha \over \beta^2} - 1 \right ) H(u) = 0 \end{displaymath}

where

\begin{displaymath} \alpha = {2 m E \over \hbar^2} \qquad {\rm and} \qquad \beta... ... \pi m \nu_0 \over \hbar} = {m \omega_0 \over \hbar} \qquad . \end{displaymath}

Equations

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.


\begin{displaymath} R_T(\nu) = {Energy \over time \times area} \quad E = h\nu = \hbar \omega \quad v_{wave} = \lambda \nu \end{displaymath}


\begin{displaymath} \vec F_e = q \vec E \quad I \propto \vert\vec E\vert^2 \quad... ...quad p = \hbar k \quad \Psi( x,t) = e^{\pm i(kx \pm \omega t)} \end{displaymath}


\begin{displaymath} -{\hbar^2 \over 2 m} \nabla^2 \Psi(x,t) + V(x) \Psi(\vec r,t... ...~}\rangle = \int_{-\infty}^{\infty} \psi^* \hat {A~} \psi dx \end{displaymath}


\begin{displaymath} \langle\phi_{n'} \vert \phi_n \rangle = \int_{-\infty}^{\inf... ...int_{-\infty}^{\infty} \phi_{k'}^* \phi_k dx = \delta(k - k') \end{displaymath}


\begin{displaymath} \vert\psi\rangle = \sum b_n \vert\phi_n\rangle \quad b_n = \... ...\vert\phi (x,t)\rangle = \vert\phi(x,0)\rangle e^{-i \omega t} \end{displaymath}


\begin{displaymath} \vert\phi_n\rangle = \sqrt{2 \over L} \sin {n \pi x \over L}... ...A,\ {\rm then}\ \lim_{x \rightarrow a} {f(x) \over g(x)} = A \end{displaymath}


\begin{displaymath} (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle^2 \quad... ...gma^2} ~ e^{-x^2/2\sigma^2}, \ {\rm then\ } \Delta x = \sigma \end{displaymath}


\begin{displaymath} V_{HO} = {\kappa x^2 \over 2} \quad \omega = 2 \pi \nu = \sq... ...ad \vert\phi_n\rangle = A_ne^{-u^2/2}H_n(u) \quad u = \beta x \end{displaymath}


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


\begin{displaymath} \psi(x) = \sum_{n=1}^\infty a_n x^n \quad n(v) = 4 \pi N \left ( {m \over 2 \pi k_B T} \right )^{3/2} v^2 e^{-mv^2/2k_B T} \end{displaymath}


\begin{displaymath} 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 \end{displaymath}


\begin{displaymath} \overline K = {3\over 2} kT \quad \zeta_1 = {\bf t}\zeta_3 =... ..._{x_0}^{x_1} \sqrt {2m(V(x) - E) \over \hbar^2} ~ dx\right ] \end{displaymath}


\begin{displaymath} \frac{dN}{dt} = {d\sigma \over d\Omega} ~d\Omega I n_{tgt} \... ... H \vert \psi \rangle \over \langle \psi \vert \psi \rangle } \end{displaymath}


\begin{displaymath} e^{i\phi} = \cos\phi + i\sin\phi \quad \psi_1(a) = \psi_2(a... ...ad {\rm and} \quad \psi^\prime_1(a) = \psi^\prime_2 (a) \quad \end{displaymath}

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 ($m_n$) $1.68\times 10^{-27}~kg$ Avogadro's Number $6.023\times10^{23}$
$939~MeV/c^2$ ($N_A$)

\includegraphics[width=5.5in]{periodic_chart2.eps}