Physics 309 Final

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Questions (3 pts. apiece) Answer questions in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions circle the correct answer.

1. What is an absorption spectrum?

   
   
   
   
   
   
   

2. In our model for the CO molecule what is the energy? Describe the components of your answer.

   
   
   
   
   
   
   
   

3. Recall how the we explained the vibration-rotation spectrum of the carbon monoxide molecule (see figure). Suppose that when the molecule absorbed a photon it was constrained to change the value of the angular momentum quantum number by $\Delta l = \pm 1$ units AND $\Delta l = 0$. How would the spectrum change? Explain.

   

   
   
   
   

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

   
   
   
   
   
   
   
   

5. Which of the following is NOT a true statement about quantum physics?

   A. The wave function is always a real quantity.

   B. The wave function represents the complete physical state.

   C. $\Psi$, $\Psi^\prime$, and $\Psi^{\prime \prime}$ are finite, single-valued and continuous.

   D. For every observable there is a quantum mechanical operator.

6. What is the value of the commutator $[\hat{H~},x]$ for the quantum mechanical Hamiltonian $\hat{H~} = p^2/2m$?

   A.	$\hbar p/im$	D.	$2\hbar p/im$
   B.	$3\hbar p^2/im$	E.	$4\hbar p^2/im$
   C.	$\hbar p/2im$

   
   
   

7. Two equal masses $m_1 = m_2 = m$ are connected by a spring having Hooke's constant $k$. If the equilibrium separation is $l_0$ and the spring rests on a frictionless horizontal surface, then what is the angular frequency $\omega_0$?

   A.	$\sqrt{k/m}$	D.	$2 \sqrt{k/m}$
   B.	$\sqrt{2k/m}$	E.	$\sqrt{g/l_0}$
   C.	$\sqrt{3k/m}$

   
   
   

8. The figure shows the energy levels in eV for five different infinite potential wells trapping a single electron in each. The electron in well $A$ is excited to the fourth state at 25 eV and then de-excites by emitting one of more photons corresponding to a single long jump or several smaller jumps. There are no restrictions on the states that can be occupied. What photon emission energies of the de-excitation of the electron in well $A$ match a photon absorption transition from the ground state for the other four wells ($B$-$E$)? Give the corresponding quantum numbers for the transitions in each well.

   
   
   

   

Problems. Clearly show all work for full credit.

1. (10 pts.)	What is the uncertainty relationship for $\Delta x\Delta E$? Assume the problem is one dimensional.
2. (15 pts)	Make the substitution $z=\cos \theta$ in the equation $$- {1 \over \sin \theta} {d \over d\theta} \left ( \sin \the... ...d\theta} \right ) + {m^2 \over \sin^2\theta} \Theta = A \Theta$$ and show that $z$ must satisfy Legendre's differential equation $$(1-z^2) {d^2\Theta \over dz ^2} - 2z{d\Theta \over dz } + \left (A - {m^2 \over 1 - z^2} \right ) \Theta = 0 \qquad .$$
3. (17 pts)	The measured moment of inertia of the CO molecule is $I=8.83\times 10^9~\rm eV\AA^2/c^2$. Using the definition of $I$ and the fact that we are using the center-of-mass frame to extract $I$, calculate the separation of the two atoms in the molecule. The mass of carbon is $m_C = 12 ~u$ and the mass of oxygen is $m_O = 16~u$.
4. (17 pts.)	A $D_2$ molecule is a rigid rotator at $T = 30~K$. At $t=0$ it is in the state $$\psi = \frac{3Y_1^1 + 5Y_6^1 + 2Y_6^4}{\sqrt{38}}$$ a. What values of $L$ and $L_z$ will measurement find and what are their probabilities? b. What is $\psi(\theta,\phi,t)$ in terms of the moment of inertia $I$ and any other constants? c. What is $\langle E \rangle$ for the molecule at $t>0$ in terms of the moment of inertia $I$ and any other constants?

Continue $\rightarrow$

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

5. (17 pts)

A molecule behaves like a one-dimensional harmonic oscillator. In going from the third excited state to the second excited state, it emits a photon of energy $E_\gamma = h\nu ~ = ~ 0.1 ~ eV$. Assume the oscillating portion of the molecule is a proton ( $m_p = 938~MeV/c^2$). a. What is the maximum distance $x_t$ from the origin the proton will reach for a classical oscillator? This is the turning point. Get your expression in terms of the total classical energy of the oscillator. b. Now modify the result above for $x_t$ for a quantum mechanical oscillator in the $n=2$ state. Get your answer in terms of $m_p$, $E_\gamma$, and any other necessary constants. c. Now calculate the probability that a proton in the second excited state is at a distance from the origin that would be forbidden to it by classical mechanics. Get your answer in terms of $x_t$ and $\beta$ which is defined below along with the wave function for the $n=2$ state. d. Obtain a numerical value for $x_t$. Wave function and constants for the $n=2$ harmonic oscillator state. $$\vert\phi_2\rangle = \frac{1}{\sqrt{8 \sqrt \pi}}\left ( 4\x... ...i = \beta x \quad \beta = \sqrt{\frac{m\omega_0}{\hbar}}\qquad$$

Physics 309 Equations

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

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

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

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

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

$$\Delta \langle \hat A \rangle \Delta \langle \hat B \rangle ... ... (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle^2 \qquad$$

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.

$$V_{HO} = {\kappa x^2 \over 2} \quad \omega_0 = 2 \pi \nu = \... ...quad E_n = (n + {1\over 2})\hbar \omega_0 = \hbar \omega \quad$$

$$\vert\phi_n\rangle = A_ne^{-u^2/2}H_n(u) \quad u = \beta x \quad \beta^2 = {m\omega_0 \over \hbar} \quad$$

$$\left ( \matrix{ \hat a~ \cr \hat a^\dagger \cr} \right ) ... ...dagger \vert\phi_n\rangle = \sqrt{n+1} \vert\phi_{n+1} \rangle$$

$$\langle K \rangle = {3\over 2} kT \qquad \psi_1 = {\bf t} ... ...\over \vert t_{11}\vert^2} \qquad V(r) = {Z_1 Z_2 e^2 \over r}$$

$$\psi(x) = \sum_{n=1}^\infty a_n x^n \quad n(v) = 4 \pi N \le... ...\hbar^2 k^2 \over 2 m} \quad k = \sqrt{2m (E-V) \over \hbar^2}$$

$$T = {{\rm transmitted\ flux} \over {\rm incident\ flux}} \qu... ... {\rm incident\ flux}} \quad {\rm flux} = \vert\phi \vert^2 v$$

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

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

$$L = mvr \quad \vec L = \vec r \times \vec p \quad \vec p = m... ... {\vert nlm\rangle} = l (l+1) \hbar^2 {\vert nlm\rangle} \quad$$

$$\vert\phi\rangle = \delta(r)(r) Y_l^m(\theta,\phi) \quad KE_{rot} = \frac{L^2}{2I} \quad E_l = \frac{l(l+1)\hbar^2}{2I}$$

$$I = \sum m_i r_i^2 \quad \vec r_{cm} = \frac{\sum m_i\vec r_i}{\sum m_i} \quad \mu = \frac{m_1 m_2}{m_1 + m_2}$$

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

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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	$6.023\times10^{23}$
	$939~MeV/c^2$	($N_A$)