Physics 515 Problem Set #5 April 29, 2004
Due Thursday, May 13, 2004

Midterm: Thursday May 6

1. (40 pts) Astronomer B is moving relative to astronomer A. Choose coordinate systems so that B's reference frame is related to A's by a boost with velocity $v$ in the $\hat z$ direction.
   1. Astronomer A observes photons with frequency $\omega$ coming from a source which, in his spherical coordinates, lies at angular position $(\theta,\phi)$. Astronomer B, passing close by Astronomer A, observes photons from the same source and describes them as having frequency $\omega'$ and source localtion $(\theta',\phi')$. How are $\omega'$, $\theta'$, and $\phi'$ related to $\omega$, $\theta$, and $\phi$?
   2. Astronomer A observes a frequency-dependent flux of photons $I(\omega,\theta,\phi)$. I.e., $I(\omega,\theta,\phi) \> d\omega \> d\Omega$ is the number of photons per second with frequency and direction lying within a frequency interval $d\omega$ and solid angle $d\Omega$ centered on the given frequency and direction. Astronomer B observes a corresponding flux $I'(\omega',\theta',\phi')$. How is $I'$ related to $I$?
   3. Suppose the flux $I$ observed by astronomer A is that of isotropic blackbody radiation at temperature $T$. What flux does astronomer B observe? Is the flux which B sees coming from a particular direction thermal? If so, what is $T'(\theta',\phi')$?
2. (30 pts) Prove that the following expressions are all valid representations for the retarded Green's function for the wave equation, defined by $-\partial^2 D_R(x,x') = \delta^4(x{-}x')$ and $D_R(x,x') = 0$ for $x^0 \lt x'^0$.
   1. $D_R(x,x') = \int {d^4 k \over (2\pi)^4} \> {e^{ik\cdot (x-x')} \over k^2 - i \epsilon \mathop {\rm sgn}(k^0) }$, with $\epsilon \to 0^+$, and $\mathop {\rm sgn}x \equiv +1$ if $x \gt 0$, $-1$ otherwise,
   2. $D_R(x,x') = \Theta(x^0{-}x'^0) \int {d^3\k \over (2\pi)^3} \> e^{i \k \cdot (\x{-}\x')} {\sin (\vert\k\vert(x^0 {-}x'^0)) \over \vert\k\vert}$, with $\Theta(x) \equiv 1$ for $x \gt 0$, 0 otherwise.
   3. $D_R(x,x') = \int {d\omega \over 2\pi c} \> {e^{-i\omega (t - t' - \vert\x{-}\x'\vert/c)} \over 4\pi \vert\x{-}\x'\vert}$,
   4. $D_R(x,x') = {\delta (x^0 {-} x'^0 - \vert\x {-} \x'\vert) \over 4\pi \vert\x{-}\x'\vert}$,
   5. $D_R(x,x') = {1 \over 2\pi} \Theta(x^0{-}x'^0) \delta((x{-}x')^2)$.
   6. $D_R(x,x') = \int {dk \over 2\pi} \> e^{-i\omega (t - t')} \> \sum_{l,m} i k j_l(k r_\lt) h^{(1)}_l(k r_\gt) Y_{lm}(\hat \x) Y_{lm}(\hat \x')^*$, with $r_\lt \equiv \min(\vert\x\vert,\vert\x'\vert)$, $r_\gt \equiv \max(\vert\x\vert,\vert\x'\vert)$, and $\omega \equiv c k$.
3. (60 pts) Multipole radiation. Recall the representation for electromagnetic fields outside a bounded source which we derived last quarter:
   $$\begin{eqnarray*} \E &=& -\M \alpha -i \L \dot\beta/c -i \N L^{-2} \rho/\... ...artial^2 \beta = \mu_0 L^{-2} [- i c \L\cdot\j] , \end{eqnarray*}$$
   
   
   
   
   
   with $\p \equiv -i \grad$, $\L\equiv \r \times \p$, $\M = \p \times \L$, and $\N = -\r \times \L$. (Compared to our earlier treatment, $\beta$ has been rescaled by a factor of $c$ to make it have the same dimensions as $\alpha$.) Assume that the sources ($\rho$ and $\j$) are harmonically varying in time at frequency $\Omega$.
   1. At long distance, the function $\alpha$ has the form $\alpha(t,\x) \sim \frac 1r e^{i k r - i \Omega t} \sum_{l,m} \alpha_{lm} Y_{lm}(\hat x)$, for some set of coefficients $\{\alpha_{lm}\}$, and similarly $\beta(t,\x) \sim \frac 1r e^{i k r - i \Omega t} \sum_{l,m} \beta_{lm} Y_{lm}(\hat x)$ for some $\{\beta_{lm}\}$. Here $k \equiv \Omega/c$ and $r \equiv \vert\x\vert$. Justify this, and describe the domain of validity of these forms.
   2. Compute the time-averaged power radiated (by integrating the Poynting flux over an arbitrarily large sphere). Show that the radiated power is $P_{\rm rad} = {\epsilon_0 \Omega^2 \over 2c} \sum_{lm} l(l{+}1) \left[ \vert\alpha_{lm}\vert^2 + \vert\beta_{lm}\vert^2 \right]$.
   3. For a compact source (whose size $R$ is small compared to the wavelength $\lambda$ of emitted radiation), we found last quarter that in the near zone ( $\lambda \gg r \gg R$),
      $$\begin{eqnarray*} \alpha(t,\x) &=& -{1\over\epsilon_0} e^{-i\Omega t} \sum... ...{1 \over l (2l{+}1)} \tilde {\cal Y}_{lm}(\x) m_{lm} , \end{eqnarray*}$$
      
      
      
      
      
      up to corrections suppressed by $r/\lambda$ or $R/r$. The multipole moments appearing here are $q_{lm} = \int d^3\r \; {\cal Y}_{lm}^*(\r) \rho(\r)$, $q'_{lm} = {i k \over c (l{+}1)} \int d^3\r \> {\cal Y}_{lm}^*(\r) \r\cdot \j(\r)$, and $m_{lm} = -{1 \over l{+}1} \int d^3\r \> {\cal Y}_{lm}^*(\r) \grad \cdot (\r \times \j(\r))$. Determine how $\alpha_{lm}$ and $\beta_{lm}$ are related to these moments.
   4. Specialize your results to the case of pure dipole radiation, and verify that you recover the correct result.
   5. If the first non-zero multipole moment for some system of oscillating charges is order $\ell$, give a dimensionally consistent estimate for the power radiated which shows how the power scales with the frequency $\Omega$, source size $R$, and amount of oscillating charge $Q$.
   6. Describe simple arrangements of oscillating charges whose radiation is (predominately) dipole, quadrupole, or octopole.
4. (30 pts) An electron in a highly excited Rydberg state of a hydrogen-like atom with principal quantum number $n = \ell + 1 \gg 1$ may be regarded, to a good approximation, as orbiting the nucleus in a classical circular orbit. (A classical description becomes arbitrarily accurate for sufficiently large $n$.) If the nucleus has charge $Ze$, recall that the (mean) radius of a hydrogenic state is $r_n = n^2 \hbar / (Z \alpha m c)$, and the binding energy is $E_n = -\half (Z \alpha)^2 mc^2 / n^2$, with $\alpha = e^2 / (4\pi \epsilon_0 \hbar c)$ the fine structure constant and $m$ the electron mass. Determine the lifetime $\tau_n$ of the state by calculating the rate at which the electron radiates energy, and determining how long it takes for the electron to lose energy $\Delta E = E_n - E_{n-1}$.