Physics 309 Test 2

I pledge that I have given nor received unauthorized assistance during the completion of this work.

Signature:


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

  1. What is the paradox of alpha decay?








  2. When we solved the harmonic oscillator Schroedinger equation with a power series, we plugged the power series ( $ \sum a_i \xi^i$ ) into the differential equation and gathered terms of like power and required the coefficient that multiplied $ \xi^i$ go to zero. Why?









  3. The figure below shows the center-of-mass of the dumbbell at three possible locations a, b, or c with $ m_1 > m_2$ . Which one is closest to the actual center-of-mass? Explain.

    Image cm1(-156,20)$ m_1$ (-23,10)$ m_2$




  4. The figures below show the change in the wave function for an initial square wave in an infinite square well from $ t=0$ to a later time. Why does the square wave change so radically?





  5. Consider the potential barrier shown below. How would you use the transfer-matrix approach to connect the wave function $ \substack{\text{\normalsize $\ \psi_0 $} \\ \sim}$ in region 0 to the wave function $ \substack{\text{\normalsize $\ \psi_4 $} \\ \sim}$ in region 4? Give your answer in the appropriate notation used in class for the discontinuity and propagation matrices. What is the form of the wave number $ k$ in each region?



Problems. Work out your solutions on a separate sheet and clearly show all work for full credit.


1. (20 pts.)

An electron beam (the electron mass is $ m_e$ ) is sent through a rectangular potential barrier like the one in the figure of total length $ b$ . The transmission coefficient exhibits a second maximum at an energy $ E_2$ . What is the barrier height in terms of $ m_e$ , $ b$ , $ E_2$ and any other constants?



From our text we know the following.

$\displaystyle T = \left [ 1 + \frac{1}{4}\frac{V^2}{E(E-V)} \sin^2 2 k_2 a\right ]^{-1} \quad E>V_0 \quad k_2 = \sqrt{\frac{2 m (E-V_0)}{\hbar^2}} $

$\displaystyle T = \left [ 1 + \frac{1}{4}\frac{V^2}{E(E-V)} \sinh^2 2 \kappa_2 a\right ]^{-1} \quad E<V_0 \quad \kappa_2 = \sqrt{\frac{2 m (V_0-E)}{\hbar^2}} $

2. (25 pts)

The energy eigenvalues of a molecule indicate the molecule is a one-dimensional harmonic oscillator. In going from the second excited state to the first excited state, it emits a photon of energy $ \hbar\omega_0$ . Assuming that the oscillating portion of the molecule is a proton of mass $ m_p$ , calculate the probability that a proton in the first excited state is at a distance from the origin that would be forbidden to it by classical mechanics. You may have difficulty performing the integration necessary for the final answer. In that case, express your answer in terms of the unsolved integral, $ \omega_0$ , $ m_p$ , and any other necessary constants. You may find the table of equations helpful.

3. (25 pts.)

Electrons of mass $ m_e$ in a beam of density $ \rho$ are accelerated through a voltage to obtain an energy $ E_{b}$ which `strikes' a potential step of height $ V_0$ where $ E_{b} > V_0$ as shown in the figure. The beam is incident from the left so it's really `falling off' the barrier. Starting from the general solution to the Schroedinger equation in each region in the figure, what are the reflection coefficient and the reflected current (or flux) in terms of the parameters given above? The eigenfunctions in each region are $ e^{\pm ik_{1,2}x}$ .





Do not write below this line. width14cm height1pt depth0pt

Physics 309 Equations and Constants


$\displaystyle E = h\nu = \hbar \omega \qquad v_{wave} = \lambda \nu \qquad I \p... ...t^2 \qquad \lambda = {h \over p} \qquad p = \hbar k \qquad K = \frac{p^2}{2m} $

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

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

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

$\displaystyle \vert\psi (t) \rangle = \sum b_n \vert\phi_n\rangle e^{-i\omega t... ... \over 2} \qquad (\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)$ ) .

$\displaystyle V_{HO} = {\kappa x^2 \over 2} \quad \omega = 2 \pi \nu = \sqrt{\k... ...-\xi^2/2}H_n(\xi) \quad \xi = \beta x \quad \beta^2 = {m\omega_0 \over \hbar} $

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

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

$\displaystyle V(r) = {Z_1 Z_2 e^2 \over r} \quad {d\sigma \over d\Omega} = \l... ...frac{1}{2}\mu v^2 + V(r) \quad \vec R = \frac{\sum_i m_i\vec r_i}{\sum_i m_i} $

$\displaystyle \mu = \frac{m_1 m_2}{m_1+m_2} \quad \psi(x) = \sum_{n=1}^\infty a... ...) = 4 \pi N \left ( {m \over 2 \pi k_B T} \right )^{3/2} v^2 e^{-mv^2/2k_B T} $

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$    

$\displaystyle \frac{df}{du} = \frac{df}{dx}\frac{du}{dx} \quad \frac{d}{dx}(x^... ...uad \frac{d}{dx}(\cos x) = -\sin x \quad \frac{d}{dx}(e^{ax}) = a e^{ax} \quad $

$\displaystyle \frac{d}{dx}(\ln ax) = \frac{1}{x} \quad \int x^n dx = \frac{x^{n... ...frac{1}{\sqrt{x^2 + a^2}} dx = \ln \left [ x + \sqrt{x^2 + a^2} \right ] \quad $

$\displaystyle \int \frac{x}{\sqrt{x^2 + a^2}} dx = \sqrt{x^2 + a^2} \quad \int ... ...t{x^2 + a^2} - \frac{1}{2} a^2 \ln \left [ x + \sqrt{x^2 + a^2} \right ] \quad $

$\displaystyle \int \frac{x^3}{\sqrt{x^2 + a^2}} dx = \frac{1}{3} (-2a^2 + x^2) \sqrt{x^2 + a^2} \quad $

Hermite polynomials ($ H_n(\xi)$ )

$\displaystyle H_0 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{ \sqrt \pi}} \qquad$ $\displaystyle H_4 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{384\sqrt \pi}} (16\xi^4 - 48\xi^2 + 12)$    
$\displaystyle H_1 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{2 \sqrt \pi}} 2\xi \qquad$ $\displaystyle H_5 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{3840\sqrt \pi}} (32\xi^5 - 160\xi^3 + 120\xi)$    
$\displaystyle H_2 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{8 \sqrt \pi}} (4\xi^2 -2) \qquad$ $\displaystyle H_6 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{46080\sqrt \pi}} (64\xi^6 - 480\xi^4 + 720\xi^2 - 120)$    
$\displaystyle H_3 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{48\sqrt \pi}} (8\xi^3 - 12\xi) \qquad$