Physics 309 Final

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

  1. Recall the vibration-rotation spectrum of carbon monoxide shown in the figure. If the $ \rm CO$ molecule absorbed a photon to go from the $ n=0$ to $ n=2$ vibrational state where would it appear on the plot below. Explain your reasoning.


  2. For the $ \rm CO$ vibrator-rotator what is the formula for the peaks above the red line in the figure corresponding to the transition $ n\rightarrow n+1$ , $ \ell\rightarrow \ell+1$ ? Start from the expression for the $ \rm CO$ energy levels and get your result in terms of $ \omega_0$ , $ \mathcal{I}$ , and any other constants.

    Image COspectrum4





  3. Recall again the vibration-rotation spectrum of carbon monoxide shown in the figure above. How would it change (if at all) if you replaced the oxygen ($ \rm O$ ) and carbon ($ \rm C$ ) atoms with sulfur ($ \rm S$ ) and magnesium ($ \rm Mg$ ) atoms each with twice the mass of the oxygen and carbon respectively. Assume the separation of the new atoms is the same as in the $ \rm CO$ molecule. Explain your reasoning.









  4. A harmonic oscillator is in the state $ \vert\psi \rangle = A(2\vert\phi_3\rangle + \vert\phi_4\rangle)$ What is $ A$ ? Show your reasoning.







  5. The claim is made that the free particle eigenfunction $ e^{-ikx}$ corresponds to a wave traveling to the right. Is that true? Explain your reasoning.







  6. Why do we express the wave function in terms of energy eigenstates?







  7. For an electron in the three-dimensional hydrogen atom is the following uncertainty relationship correct?

    $\displaystyle \Delta x \Delta p_y \ge\frac{\hbar}{2} $

    The $ \Delta p_y$ part refers to the uncertainty in the $ y$ component of the electron. Explain your reasoning.







  8. For the step barrier shown in the figure, what is the solution to the Schrödinger equation in each region and what are the boundary conditions those solutions must satisfy. Express your answer in terms of the mass $ m$ , energy $ E$ , potential $ V_0$ , and any other constants.



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  9. For a particle in a box in the $ n=2$ state as shown in the figure, the probability density at certain points is zero. Does this imply the particle cannot move across those points? Explain.


  10. Consider the potential barrier shown below. How would you use the transfer-matrix approach to connect the wave function $ \tilde{\psi_0}$ in region 0 to the wave function $ \tilde{\psi_3}$ in region 3? 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. Clearly show all work for full credit on a separate piece of paper.


1. (10 pts.) A mass $ m_0 = 0.910~ kg$ is oscillating freely on a vertical spring. The period for $ m_0$ is $ T_o = 1.10~ s$ . An unknown mass $ m_1$ replaces $ m_0$ on the same spring and has a period of $ T_1=1.32~s$ . What is the spring constant $ k$ and the unknown mass $ m_1$ ?

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2. (10 pts.)

A thousand quarks are trapped in a one-dimensional box in the range $ 0 < x < a$ . At $ t=0$ each particle is in the state

$\displaystyle \psi (x,0) = A x^2 (x-a) $

where

$\displaystyle A = \sqrt{\frac{105}{a^7}} $

The eigenfunctions and eigenvalues for this particle in a box are

$\displaystyle \vert\phi_n \rangle = \sqrt{\frac{2}{a}} \sin \left ( \frac{n\pi x}{a} \right ) \quad E_n = n^2\frac{\hbar^2 \pi^2}{2ma^2}=n^2E_1 \quad 0 < x < a $

and $ \vert\phi_n\rangle$ is zero outside the box. How many particles are in the interval $ (0,a/3)$ at $ t=0$ ? Get your answer in terms of $ A$ , $ a$ , and any other constants.

3. (10 pts.) In studying rotational motion, we take advantage of the center-of-mass system to make life easier. Consider the two-particle system shown in the figure including the center-of-mass vector $ {\bf R}$ . For convenience we will place our origin at the center-of-mass of the system ( $ {\bf R = 0}$ ). Show the classical mechanical energy of the two-particle system in the center-of-mass frame can be written as

$\displaystyle E_{cm} = {1 \over 2 } \mu v^2 + V(r) \qquad {\rm where} \qquad \mu = {m_1 m_2 \over m_1 + m_2} \qquad {\rm and} \qquad v = {dr \over dt} $

and $ r$ is the relative coordinate between the two particles as shown in the figure. Notice that $ V(r)$ depends only on the relative coordinate.

Image coordinates3

4. (10 pts.)

Consider a time-independent Schrödinger equation, $ \hat H$ , that is composed of two independent parts, so that

$\displaystyle \hat H(x_1,x_2) = \hat H_1(x_1) + \hat H_2(x_2) \quad . $

The two parts of the Hamiltonian have solutions $ \psi_1 (x_1)$ and $ \psi_2(x_2)$ such that


$\displaystyle \hat {H_1} \psi_1(x_1)$ $\displaystyle = E_1\psi_1(x_1)$    
$\displaystyle \hat {H_2} \psi_2(x_2)$ $\displaystyle = E_2\psi_2(x_2) \quad .$    


Show the wave function of the composite system $ \hat {H~}$ is the following.

$\displaystyle \psi(x_1,x_2) = \psi_1(x_1)\psi_2(x_2) $

5. (15 pts.)

Another thousand quarks are trapped in a same-sized-as-above, one-dimensional box. This time at $ t=0$ each particle is in the following (different than before) state

$\displaystyle \psi (x,0)$ $\displaystyle = A (a - x) \qquad$   $\displaystyle a/2 < x < a$    
  $\displaystyle = 0$   $\displaystyle {\textrm otherwise}$    


where $ A = 2\sqrt 6 / a^{3/2}$ . The eigenfunctions and eigenvalues for this particle in a box are the same as in Problem 2. How many particles have energy $ E_6$ at $ t=0$ ? You should get a number for this.

6. (15 pts.)

A pion is in a harmonic potential (i.e., it feels a Hooke's-Law-like force) and has the initial wave function

$\displaystyle \vert\Psi(x,0)\rangle = e^{-\xi^2/2} \left [\frac{ H_0(\xi) - \sqrt 2 H_2(\xi) - H_3(\xi) + H_5(\xi) + H_6(\xi)}{\sqrt 6} \right ] $

where $ H_n(\xi)$ are the Hermite polynomials and $ \xi=\beta x$ where $ \beta = \sqrt{m\omega_0 /\hbar}$ . The eigenfunctions and eigenvalues of the particle are

$\displaystyle \vert\phi_n \rangle = H_n(\xi) e^{-\xi^2/2} \qquad E_n = (n+ {1 \over 2}) \hbar \omega_0 \qquad . $

  1. What is the average value of the energy for this state in terms of $ \omega_0$ and any other constants?

  2. What are the probabilities of finding the pion in the $ n=0$ , $ n=2$ , and $ n=4$ states?

  3. What is $ \vert\Psi(x,t)\rangle$ for $ t > 0$ ? Express your answer in terms of $ \omega_0$ and any other necessary constants.


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Physics 309 Equations


$\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 (x,t) \rangle = \sum b_n \vert\phi_n\rangle e^{-i\omega... ... \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}} \psi_3 \qquad T = {1 \over \vert t_{11}\vert^2} \qquad R+T = 1 \quad $


$\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 E = \frac{1}{2}\mu v^2 + V(r) ... ...m} = \frac{\sum_i m_i\vec r_i}{\sum_i m_i} \quad \mu = \frac{m_1 m_2}{m_1+m_2} $


$\displaystyle \psi(x) = \sum_{n=1}^\infty a_n x^n \quad \langle K \rangle = {3\... ...v^2/2k_B T} \quad \vec L = \vec r \times \vec p = \mathcal{I}\vec \omega \quad $


$\displaystyle \mathcal{I} = \sum_i m_i r_1^2 = \int r^2 dm \quad KE_{rot} = \fr... ... Z_2 e^2}{r} \quad ME = \frac{p_r^2}{2\mu} + \frac{L^2}{2\mu r^2} + V(r) \quad $


$\displaystyle L_z \vert nlm\rangle = m \hbar \vert nlm\rangle \quad L^2 \vert nlm\rangle = \ell (\ell+1) \hbar^2 \vert nlm\rangle \quad $


$\displaystyle {\bf d_{ij}} = \frac{1}{2} \left (\begin{array}{cc} 1+ \frac{k_j... ...begin{array}{cc} e^{ik_i2a} & 0 \\ 0 & e^{-ik_i2a} \end{array} \right ) \quad $


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$



Integrals and Derivatives

$\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... ... dx = \frac{2 x \sin (a x)}{a^2}-\frac{\left(a^2 x^2-2\right) \cos (a x)}{a^3} $


$\displaystyle \int x \sin(ax) dx = \frac{\sin(ax)}{a^2} - \frac{x\cos(ax)}{a} \... ...x^2-2\right) \sin (a x)}{a^4}-\frac{x \left(a^2 x^2-6\right) \cos (a x)}{a^3} $



Hermite polynomials ($ H_n(\xi)$ )

$\displaystyle H_0 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{ \sqrt \pi}} \qquad$ $\displaystyle H_5 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{3840\sqrt \pi}} (32\xi^5 - 160\xi^3 + 120\xi)$    
$\displaystyle H_1 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{2 \sqrt \pi}} 2\xi \qquad$ $\displaystyle H_6 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{46080\sqrt \pi}} (64\xi^6 - 480\xi^4 + 720\xi^2 - 120)$    
$\displaystyle H_2 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{8 \sqrt \pi}} (4\xi^2 -2) \qquad$ $\displaystyle H_7 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{645120\sqrt \pi}} (128\xi^7 - 1344\xi^5 + 3360\xi^3 - 1680\xi)$    
$\displaystyle H_3 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{48\sqrt \pi}} (8\xi^3 - 12\xi) \qquad$ $\displaystyle H_8 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{10321920\sqrt \pi}} (256\xi^8 - 3584\xi^6 + 13440\xi^4 - 13440\xi^2 + 1680) \nonumber$    
$\displaystyle H_4 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{384\sqrt \pi}} (16\xi^4 - 48\xi^2 + 12)$ $\displaystyle H_9 (\xi)$ $\displaystyle = \frac{1}{ \sqrt{185794560\sqrt \pi}} (512\xi^9 - 9216\xi^7 + 48384\xi^5 - 80640\xi^3 + 30240\xi) \nonumber$    
     

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