Sample Test Problems

1. A hydrogen atom is in a uniform electric field in the $z$ direction which turns on abruptly at $t=0$ and decays exponentially as a function of time, $E(t)=E_0 e^{-t/\tau}$. The atom is initially in its ground state. Find the probability for the atom to have made a transition to the $2P$ state as $t\rightarrow\infty$. You need not evaluate the radial part of the integral. What $z$ components of orbital angular momentum are allowed in the $2P$ states generated by this transition?

2. A hydrogen atom is in the $n=5$, $\ ^{3}D_{5\over 2}$ state. To which states is it allowed to decay via electric dipole transitions? What will be the polarization for a photon emitted along the $z$-axis if $m_l$ decreases by one unit in the decay?

3. A hydrogen atom is in a uniform electric field in the $z$ direction which turns on abruptly at $t=0$ and decays exponentially as a function of time, $E(t)=E_0 e^{-t/\tau}$. The atom is initially in its ground state. Find the probability for the atom to have made a transition to the $2P$ state as $t\rightarrow\infty$. You need not evaluate the radial part of the integral. What $z$ components of orbital angular momentum are allowed in the $2P$ states generated by this transition?

4. Derive the selection rules for radiative transitions between hydrogen atom states in the electric dipole approximation. These are rules for the change in $l, m$, and $s$.

5. State the selection rules for radiative transitions between hydrogen atom states in the electric dipole approximation. These are rules for the allowed changes in $l, m$, $s$, and parity. They can be easily derived from the matrix element given on the front of the test. Draw an energy level diagram (up to $n=3$) for hydrogen atoms in a weak $B$ field. Show the allowed E1 transitions from $n=3$ to $n=1$ on that diagram.

6. Calculate the differential cross section, ${d\sigma\over d\Omega}$, for high energy scattering of particles of momentum p, from a spherical shell delta function
   
   $$V({\bf r}) = \lambda \delta (r-r_0)$$
   
   
   Assume that the potential is weak so that perturbation theory can be used. Be sure to write your answer in terms of the scattering angles.
7. Assume that a heavy nucleus attracts $K_0$ mesons with a weak Yakawa potential $V(r)={V_0 \over r}e^{-\alpha r}$. Calculate the differential cross section, ${d\sigma\over d\Omega}$, for scattering high energy $K_0$ mesons (mass $m_K$) from that nucleus. Give your answer in terms of the scattering angle $\theta$.

$\hbar$ = $1.05 \times 10^{-27}$ erg sec	$c$ = $3.00 \times 10^{10}$ cm/sec	$e = 1.602 \times 10^{-19}$ coulomb
$1 eV = 1.602 \times 10^{-12}$ erg	$\alpha = {e^2\over \hbar c} = 1/137$	$\hbar c$ = 1973 eV Å= 197.3 MeV F
1 Å= $1.0 \times 10^{-8}$ cm	1 Fermi = $1.0 \times 10^{-13}$ cm	$a_0 = {\hbar\over \alpha m_e c} = 0.529 \times 10^{-8}$ cm
$m_p$ = 938.3 MeV/ $c^2$	$m_n$ = 939.6 MeV/ $c^2$	$m_e$ = $9.11 \times 10^{-28}$ g = 0.511 MeV/ $c^2$
$k_B = 1.38 \times 10^{-16}$ erg $/^{\circ}$K	$g_e = 2 + {\alpha\over\pi}$	$g_p = 5.6$
$\mu_{Bohr} = {e\hbar\over 2m_e c}$ = $0.579 \times 10^{-8}$ eV/gauss		$\int\limits_{-\infty}^\infty dx\; f(x)\; \delta(g(x))=[{1\over \vert{dg\over dx}\vert}f(x)]_{g(x)=0}$
$\int\limits_{-\infty}^\infty dx\; f(x)\; \delta(x-a)=f(a)$	$\int\limits_{-\infty}^{\infty}\; dx\; e^{-ax^2} = \sqrt{\pi\over a}$	$\int\limits_0^\infty dr\; r^n e^{-ar}={n!\over a^{n+1}}$
$e^A=\sum\limits_{n=0}^\infty {A^n\over n!}$	$\sin\theta= \sum\limits_{n=1,3,5...}^\infty {\theta^n\over n!}(-1)^{{n-1\over 2}}$	$\cos\theta= \sum\limits_{n=0,2,4...}^\infty {\theta^n\over n!}(-1)^{{n\over 2}}$
GENERAL WAVE MECHANICS
$\Delta p\; \Delta x \geq {\hbar\over 2}$	$\Delta A\; \Delta B \geq \langle{1\over 2} [A,B]\rangle$	$\Delta A=\sqrt{\langle A^2\rangle-\langle A\rangle^2}$
$\lambda = h/p$	$E=h\nu=\hbar\omega$	$E=\sqrt{m^2c^4+p^2c^2}$
$\langle\phi\vert\psi\rangle=\int\limits_{-\infty}^\infty dx \phi^*(x)\psi(x)$	$\langle u_i\vert u_j\rangle=\delta_{ij}$	$\sum\limits_i\vert u_i\rangle\langle u_i\vert=1$
$\langle\phi\vert A\vert\psi\rangle=\langle\psi\vert A\vert\... ...^*= \langle\phi\vert A\psi\rangle=\langle A^\dagger\phi\vert\psi\rangle$
$\psi (x) = {1\over\sqrt{2\pi \hbar}} \int\limits_{-\infty}^{\infty}dp\; \phi (p)\; e^{ipx/\hbar}$		$\phi (p) = {1\over\sqrt{2\pi \hbar}} \int\limits_{-\infty}^{\infty}dx\; \psi (x)\; e^{-ipx/\hbar}$
$p_{op} = {\hbar\over i}{\partial\over \partial x}$	$E_{op} = i\hbar {\partial\over \partial t}$	$x_{op} = i\hbar {\partial\over \partial p}$
Schr $\ddot{\rm o}$dinger: $\lbrack{1\over 2m}(\vec{p}+{e\over c}\vec{A})^2 + V(\vec{r})\rbrack \psi(\vec{r}) = E \psi(\vec{r})$		${d\langle A\rangle \over dt}= \langle {\partial A\over \partial t}\rangle +{i\over \hbar}\langle [H,A]\rangle$
$[p_x,x] = {\hbar\over i}$	$[L_x,L_y] = i \hbar L_z$	$[L^2,L_z] = 0$
$\psi_i=\langle u_i\vert\psi\rangle$	$A_{ij}=\langle u_i\vert A\vert u_j\rangle$
ANGULAR MOMENTUM
$L^2 Y_{\ell m} = \ell (\ell +1)\hbar^2 Y_{\ell m}$	$L_z Y_{\ell m} = m\hbar Y_{\ell m}$	$-\ell\leq m \leq \ell$
$L_\pm = L_x \pm iL_y$	$L_\pm Y_{\ell m} = \hbar \sqrt{\ell (\ell +1) - m(m\pm 1)}\; Y_{\ell ,m\pm 1}$
$Y_{00}={1\over\sqrt{4\pi}}$	$Y_{11} = -\sqrt{3\over 8\pi}\; e^{i\phi}\; \sin\theta$	$Y_{10} = \sqrt{3\over 4\pi}\; \cos\theta$
$Y_{22} = \sqrt{15\over 32\pi}\; e^{2i\phi}\; \sin^2\theta$	$Y_{21} = -\sqrt{15\over 8\pi}\; e^{i\phi}\; \sin\theta \cos\theta$	$Y_{20} = \sqrt{5\over 16\pi}\; (3 \cos^2\theta-1)$
$Y_{\ell\ell} = e^{i\ell\phi}\; \sin^\ell \theta$	$Y_{\ell m}(\pi -\theta ,\phi +\pi) = (-1)^\ell \; Y_{\ell m}(\theta ,\phi )$
$H = H_0 - \vec{\mu} \cdot \vec{B}$	$\vec{\mu} = {e\over 2mc}\vec{L}$	$\vec{\mu} = {ge\over 2mc}\vec{S}$
$S_i = {\hbar\over 2} \sigma_i$
$\sigma_x = \pmatrix{0 &1\cr 1 &0\cr}$	$\sigma_y = \pmatrix{0 &-i\cr i &0\cr}$	$\sigma_z = \pmatrix{1 &0\cr 0 &-1\cr}$
$S_x=\hbar\pmatrix{0&{1\over \sqrt{2}}&0 \cr {1\over \sqrt{2}}&0&{1\over \sqrt{2}} \cr 0&{1\over \sqrt{2}}&0}$	$S_y=\hbar\pmatrix{0&{-i\over \sqrt{2}}&0 \cr {i\over \sqrt{2}}&0&{-i\over \sqrt{2}} \cr 0&{i\over \sqrt{2}}&0}$	$S_z=\hbar\pmatrix{1&0&0\cr 0&0&0\cr 0&0&-1}$

ADDITION OF ANGULAR MOMENTUM
$\vec{J} = \vec{L}+\vec{S}$	$\vert\ell -s\vert \leq j \leq \ell +s$	$\vec{L\cdot S}={1\over 2}(J^2-L^2-S^2)$
$\psi_{jm_j\ell s}=\sum\limits_{m_\ell m_s}C(jm_j;\ell m_\el... ...gle j m_j \ell s\vert\ell m_\ell s m_s\rangle Y_{\ell m_\ell}\chi_{sm_s}$
$\psi_{j,m_j}=\psi_{\ell +{1\over 2},m+{1\over 2}}= \sqrt{\e... ...ell +1}Y_{\ell m}\chi_+ +\sqrt{\ell -m\over 2\ell +1}Y_{\ell ,m+1}\chi_-$		for $s={1\over 2}$ and any $\ell$
$\psi_{j,m_j}=\psi_{\ell -{1\over 2},m+{1\over 2}}= \sqrt{\e... ... +1}Y_{\ell m}\chi_+ -\sqrt{\ell +m+1\over 2\ell +1 }Y_{\ell ,m+1}\chi_-$		for $s={1\over 2}$ and any $\ell$
HYDROGEN ATOM
$H = {p^2\over 2\mu}-{Ze^2\over r}$	$\psi_{n\ell m} = R_{n\ell}(r) Y_{\ell m}(\theta,\phi)$	$E_n = -{Z^2 \alpha^2 \mu c^2\over 2 n^2}\quad n=1,2,3...$
$R_{10} = 2({Z\over a_0})^{3\over 2}\; e^{-Zr\over a_0}$	$R_{20} = 2({Z\over 2a_0})^{3\over 2} (1-{Zr\over 2a_0})\;e^{-Zr\over 2a_0}$	$R_{21} = {1\over\sqrt 3}({Z\over 2a_0})^{3\over 2} ({Zr\over a_0})\;e^{-Zr\over 2a_0}$
$R_{n,n-1}\; \propto\; r^{n-1}\; e^{-Zr/na_0}$	$\mu = {m_1 m_2\over m_1 + m_2}$	$\langle\psi_{n\ell m}\vert{e^2\over r}\vert\psi_{n\ell m}\rangle={Ze^2\over n^2a_0}={Z\alpha^2mc^2\over n^2}$
$H_1=-{p^4\over 8m^3c^2}$	$H_2={e^2\over 2m^2c^2r^3}\vec{S}\cdot\vec{L}$	$\Delta E_{12}=-{1\over 2n^3}\alpha^4mc^2({1\over j+{1\over 2}}-{3\over 4n})$
$H_3={-e^2g_p\over 3mM_pc^2}\vec{S}\cdot\vec{I}\delta({r})$	$\Delta E_{3}={2g_pm\alpha^4mc^2\over 3M_pn^3}(f(f+1)-I(I+1)-{3\over 4})$
$H_B={eB\over 2mc}(L_z+2S_z)$	$\Delta E_B={e\hbar B\over 2mc}(1\pm {1\over 2\ell +1})m_j$ for $j=\ell\pm{1\over 2}$
HARMONIC OSCILLATOR
$H = {p^2\over 2m}+{1\over2}m\omega^2x^2 = \hbar\omega A^\dagger A + {1\over 2}\hbar\omega$		$E_n = (n + {1\over 2}) \hbar \omega \quad n=0,1,2...$
$A = (\sqrt{m\omega\over 2\hbar} x + i {p\over\sqrt{2 m \hbar\omega}})$	$A^\dagger = (\sqrt{m\omega\over 2\hbar} x - i {p\over\sqrt{2 m \hbar\omega}})$	$[A,A^\dagger] = 1$
$A^\dagger\; \vert n\rangle = \sqrt{(n+1)}\; \vert n+1\rangle$	$A\; \vert n\rangle = \sqrt{(n)}\; \vert n-1\rangle$	$u_0 (x) = ({m\omega\over \hbar \pi})^{1\over 4}\; e^{-m\omega x^2/2\hbar}$
PERTURBATION THEORY AND RADIATIVE DECAYS
$E_n^{(1)}=\langle \phi_n\vert H_1\vert\phi_n\rangle$	$E_n^{(2)}=\sum\limits_{k\neq n}{\vert\langle \phi_k\vert H_1\vert\phi_n\rangle \vert^2\over E_n^{(0)}-E_k^{(0)}}$	$C_{nk}^{(1)}={\langle \phi_k\vert H_1\vert\phi_n\rangle \over E_n^{(0)}-E_k^{(0)}}$
$c_m(t) = {1\over i\hbar} \int\limits_{0}^{t}\; dt'\; e^{-i(E_m-E_k)t'/\hbar}\langle \phi_m\vert V(t')\vert\phi_k\rangle$
Fermi's Golden Rule:	$W={2\pi\over\hbar}\vert\langle \psi_f\vert V\vert\psi_i\rangle \vert^2\times$ (density of final states)
$\Gamma_{i\rightarrow f} = {2\pi\over \hbar} \int\; \prod\li... ... ({Vd^3{p_k}\over \rm (2\pi \hbar)^3}) \;\vert M_{fi}\vert^2\; \delta^3($momentum conservation $) \; \delta$(Energy conservation)
$\Gamma^{rad}_{m\rightarrow k} = {\alpha\over 2\pi m^2c^2}\i... ...\vec{k}\cdot\vec{r}}\vec{\epsilon}\cdot\vec{p}\vert\phi_k\rangle \vert^2$
$\Gamma^{E1}_{m\rightarrow k} = {\alpha\over 2\pi c^2}\int d... ...t\langle \phi_m\vert\vec{\epsilon}\cdot\vec{r}\vert\phi_k\rangle \vert^2$		$\Delta l=\pm 1$, $\Delta s=0$
$\hat{\epsilon}\cdot \hat{r}=\sqrt{4\pi\over 3}(\epsilon_zY_... ...ilon_y\over\sqrt{2}}Y_{11}+{\epsilon_x+i\epsilon_y\over\sqrt{2}}Y_{1-1})$		$\vec{\epsilon}\cdot\vec{k}=0$
$I(\omega)\propto{\Gamma/2\over (\omega-\omega_0)^2+(\Gamma/2)^2}$	$\Gamma_{collision}=P\sigma\sqrt{3\over mkT}$	$({\Delta\omega\over\omega})_{Dopler}=\sqrt{kT\over mc^2}$
$({d\sigma\over d\Omega})_{BORN}={1\over 4\pi^2 \hbar^4}{p_f\over p_i}m_f m_i \vert{\tilde V}(\vec{\Delta}$ $)\vert^2$	${\tilde V}(\vec{\Delta}$ $) = \int \; d^3 \vec{r}\;e^{-i \vec{\Delta}\cdot\vec{r}}\;V(\vec{r}$ $)$	$\vec{\Delta}={\vec{p_f}-\vec{p_i}\over\hbar}$
ATOMS AND MOLECULES
Hund: 1) max $s$	2)max $\ell$ (allowed)	3) min $j$ $(\leq {1\over 2}$ shell) else max $j$
$E_{rot}={\ell (\ell +1)\hbar^2\over 2I}\approx {1\over 2000}$ eV	$E_{vib}=(n+{1\over 2})\hbar\omega\approx {1\over 50}$ eV