Physics 310 Test 2

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Questions (6 pts. apiece) Answer questions in complete, well-written sentences WITHIN the spaces provided.

1. The hydrogen atom Zeeman effect shows that hydrogen atoms act like magnets. What is the source of that magnetism?

   
   
   
   
   
   
   
   

2. The electric and magnetic fields in electromagnetic waves oscillate sinusoidally which average out to zero. Nevertheless, they transmit energy. How can that be?

   
   
   
   
   
   
   
   

3. Recall the carbon-monoxide spectrum we studied (see the figure). How would the spectrum change if the distance between the carbon and oxygen atoms increased? Explain.

   

   
   
   
   
   
   

4. In solving the radial part of the hydrogen atom we explored the solution of the Schroedinger equation at large values of $r$, the radial coordinate. Why?

   
   
   
   
   
   
   
   

5. To explain the carbon-monoxide spectrum and the Zeeman effect we used the selection rules $\Delta l = \pm 1$ and $\Delta m = 0, \pm 1$. What quantity did we calculate to explain these selection rules? Why?

   
   
   
   
   
   
   
   

Problems. Answer the questions below on a separate sheet of paper. Clearly show all work for full credit.

1. (10 pts.)	In the Bohr model of the hydrogen atom, the electron is in a circular orbit at a radius of $a_0 = 0.53~{\rm\AA}$. What is the expression for the electron's magnetic moment due to its circular orbit in terms of the electron velocity $v_e$, $a_0$, and any other necessary constants.
2. (10 pts.)	In a region of free space, the electric field of an electromagnetic wave is $\vec E = (40.0 \hat {i\,} + 16.0 \hat {j~} + 32.0 \hat k)~ N/C$ and the magnetic field is $\vec B= (0.100 \hat {i\,} + 0.040 \hat {j~} - 0.145 \hat k)~ \mu T$. (1) Are the fields perpendicular to each other? (2) What is their Poynting vector?
3. (20 pts.)	In our theory of radiation emission from hydrogen the matrix element of the $z$-component of the dipole has the following integral representation. $$\langle n \vert z \vert n^\prime\rangle = \int_{4\pi} \left ... ...ta d\Omega \int_0^\infty R_{nl}^* R_{n^\prime l^\prime} r^3 dr$$ Using this result and the equations listed below, generate selection rules. Clearly show your reasoning and identify any equations you use from the list below. $$Y_l^m (\theta,\phi) = \left [ \frac{2l + 1}{4\pi} \frac{(l-m)!}{(l+m)!} \right ]^{1/2} P_l^m(\cos\theta)e^{im\phi}$$ (1) $$\int_{-1}^1 d\cos\theta \int_0^{2\pi} d\phi Y_l^m (Y_{l^\prime}^{m^\prime})^* = \delta_{m m^\prime} \delta_{l l^\prime}$$ (2) $$(2l+1)\mu P_l^m(\mu) = (l-m+1)P_{l+1}^m (\mu) + (l+m)P_{l-1}^m(\mu)$$ (3) $$(2l+1)\sqrt{1-\mu^2} P_l^m(\mu) = P_{l-1}^{m+1} (\mu) - P_{l+1}^{m+1}(\mu)$$ (4) $$\begin{array}{lcl} \sqrt{1-\mu^2} P_l^{m+1}(\mu) &= & (l-m)\... ...&= & -(l+m+1) P_{l}^m (\mu) + (l-m+1)P_{l+1}^m(\mu) \end{array}$$ (5) $$\begin{array}{lcll} \int_{-1}^1 P_l^m(\mu) P_k^m(\mu) d \mu ... ...1} \frac{(l-m)!}{(l+m)!} &l=k \\ &= & 0 &l\neq k \end{array}$$ (6)

More Problems. Answer the questions below on a separate sheet of paper. Clearly show all work for full credit.

4. (30 pts)

A $\rm CO$ molecule behaves as a rigid rotator and at $t=0$ is in the state $$\psi = \frac{3 Y_3^3 + 4 Y_3^2 + 5 Y_4^3}{N_0}$$ What is the value of the normalization constant $N_0$? What values of $L$ and $L_z$ will measurement find? What are the probabilities for the measurements in the previous question? What is $\langle L_z \rangle$ for the molecule at $t=0$?

cmrConstants and Conversion Factors

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 ($N_A$)	$6.022\times 10^{23}~ mol^{-1}$
	$939~MeV/c^2$
Permeability of free space ($\mu_0$)	$4\pi\times 10^{-7}~T-m/A$

cmrPhysics 310 Study Sheet Test 2

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

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

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

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

$$\vert\psi (t) \rangle = \sum b_n \vert\phi_n\rangle e^{-i\om... ... (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle^2 \qquad$$

cmrThe 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 $\psi^\prime_1(a) = \psi^\prime_2 (a)$) .

$$\vec F_{coul} = \frac{Z_1 Z_2 e^2}{r^2}\hat r \quad V(r) = {... ...1}{2}\mu v^2 + V(r) \quad \mu = \frac{m_1 m_2}{m_1+m_2} \quad$$

$$\psi(x) = \sum_{n=1}^\infty a_n x^n \quad \vert nlm\rangle =... ...theta)e^{i m_l \phi} \quad E_n = -\frac{\mu e^4}{2\hbar^2 n^2}$$

$$R_{nl} (r) = \sqrt{\left ( \frac{2}{na_0} \right )^3 \frac{(... ...)!~ \mathcal{L}_{n-l-1}^{2l+1} \left ( \frac{2r}{na_0}\right )$$

$$\vec L = \vec r \times \vec p = I\vec \omega \quad I = \int ... ...\tau = \vec \mu \times \vec B \quad V = -\vec \mu \cdot \vec B$$

$$E_{Zee} = E_n + \frac{eB}{2m_e} m_l B \quad \vec p = m \vec ... ...{2\mu} + \frac{L^2}{2\mu r^2} + V(r) \quad a_c = \frac{v^2}{r}$$

$$d\tau = r^2 dr d\cos\theta d\phi \quad d\Omega = d\cos\theta... ... L^2 \vert nlm\rangle = l (l+1) \hbar^2 \vert nlm\rangle \quad$$

$$\vec S = \frac{1}{\mu_0} \vec E \times \vec B \quad \frac{E... ...t\vec d\vert^2 \quad \phi(x,t) = \phi(x)e^{iE_n t/\hbar} \quad$$

$$\frac{\lambda}{T} = \lambda f = v_{wave} \quad \omega = \frac{2\pi}{T} \quad k = \frac{2\pi}{\lambda}$$

$$\vec A \times \vec B = (A_y B_z - A_z B_y) \hat {i\thinspace} - (A_x B_z - A_z B_x) \hat {j~} + A_x B_y - A_y B_x) \hat k$$

$$\vec A \cdot \vec B = A_x B_x + A_y B_y + A_z B_z$$