updated Ch 8

Chapter_8/statistical-physics.ipynb

@@ -1000,9 +1000,10 @@
     "\n",
     "Let's retain the \"free electron\" assumption of the Drude model and use the results of the 3-D infinite square-well potential because it corresponds physically to a cubic lattice of ions.  The allowed energies are\n",
     "\n",
-    "\\begin{align*}\n",
+    "```{math}\n",
+    ":label: box_energy\n",
     "E = \\frac{h^2}{8mL^2}\\left(n_1^2 + n_2^2 + n_3^2 \\right),\n",
-    "\\end{align*}\n",
+    "```\n",
     "\n",
     "where $L$ is the side-length of a cube and $n_i$ are the integer quantum numbers.  \n",
     "\n",
@@ -1268,6 +1269,87 @@
     "## Bose-Einstein Statistics\n",
     "\n",
     "### Blackbody Radiation\n",
+    "Recall the definition of an *ideal* blackboddy as a nearly perfect absorbing cavity that emits a spectrum of electromagnetic radiation (see Section [3.5](https://saturnaxis.github.io/ModernPhysics/Chapter_3/experimental-quantum-physics.html#blackbody-radiation)).  The problem is to find the intensity of the emitted radiation as a function of temperature and wavelength:\n",
+    "\n",
+    "```{math}\n",
+    ":label: spectral_density\n",
+    "\n",
+    "\\mathcal{I}(\\lambda,T) = \\frac{2\\pi c^2 h}{\\lambda^5} \\frac{1}{e^{hc/(\\lambda kT)}-1}. \n",
+    "```\n",
+    "\n",
+    "In quantum theory, we begin with the assumption that the electromagnetic radiation is a collection of photons with energy $hc/\\lambda$.  Photons are *bosons* with spin $1$.  We use the Bose-Einstein distribution to find how the photons are distributed by energy, and then convert the energy distribution into a function of wavlength via $E= hc/\\lambda.$\n",
+    "\n",
+    "The key to the problem is the density of states $g(E)$.  We model the photon gas just as we did for the electron gas: a collection of free particles within a 3-D infinite potential well.  We cannot use Eq. {eq}`box_energy` for the energy states because the photons are masseless.  \n",
+    "\n",
+    "We recast the solution to the particle-in-a-box problem in terms of mementum states rather than energy states.  For a free particle of mass $m$, the energy $E=p^2/(2m)$. We rewrite Eq. {eq}`box_energy` as\n",
+    "\n",
+    "\\begin{align}\n",
+    "p &= \\sqrt{p_x^2 + p_y^2 + p_z^2}, \\\\\n",
+    "&= \\frac{h}{2L}\\sqrt{n_1^2 + n_2^2 + n_3^2}.\n",
+    "\\end{align}\n",
+    "\n",
+    "The energy of a photon is $E=pc$ so that\n",
+    "\n",
+    "\\begin{align}\n",
+    "E = \\frac{hc}{2L}\\sqrt{n_1^2 + n_2^2 + n_3^2}.\n",
+    "\\end{align}\n",
+    "\n",
+    "Again by thinking of $n_i$ as the coordinates of a number space and defining $r^2 = n_1^2 + n_2^2 + n_3^2$, we find the number of allowed energy states within \"radius\" $r$ is\n",
+    "\n",
+    "$$ N_r = 2\\left(\\frac{1}{8}\\right) \\left( \\frac{4}{3} \\pi r^3\\right). $$\n",
+    "\n",
+    "This time the factor of $2$ comes from the two possible *photon polarizations*.  Also, the energy is proportional to $r$ by\n",
+    "\n",
+    "\\begin{align}\n",
+    "E &= \\frac{hc}{2L} r.\n",
+    "\\end{align}\n",
+    "\n",
+    "We can then rewrite $N_r$ in terms of $E$ as\n",
+    "\n",
+    "\\begin{align}\n",
+    "N_r = \\frac{1}{3} \\pi r^3 = \\frac{8\\pi L^3}{3h^3 c^3} E^3.\n",
+    "\\end{align}\n",
+    "\n",
+    "The density of states $g(E)$ is\n",
+    "\n",
+    "\\begin{align}\n",
+    "g(E) = \\frac{dN_r}{dE} = \\frac{8\\pi L^3}{h^3 c^3}E^2.\n",
+    "\\end{align}\n",
+    "\n",
+    "The energy distribution is the product of the density of states and the a statistical factor.  In this case, the Bose-Einstein factor:\n",
+    "\n",
+    "\\begin{align}\n",
+    "n(E) &= g(E)F_{\\rm BE} = \\frac{8\\pi L^3}{h^3 c^3} E^2 \\frac{1}{e^{E/(kT)}-1}.\n",
+    "\\end{align}\n",
+    "\n",
+    "The normalization factor $B_{\\rm BE} = 1$ because we have a non-normalized collection of photons.  As photons are absorbed and emitted by the walls of the cavity, the number of photons is not constant.\n",
+    "\n",
+    "The next step is to convert from a number distribution to an energy density distribution $u(E)$.  Instead of number density $N/V$, we use a factor $E/L^3$ (energy per unit volume) as a multiplicative factor:\n",
+    "\n",
+    "\\begin{align*}\n",
+    "u(E) = n(E)\\frac{E}{L^3} = \\frac{8\\pi}{h^3 c^3}E^3 \\frac{1}{e^{E/(kT)}-1}.\n",
+    "\\end{align*}\n",
+    "\n",
+    "For all photons between $E$ and $E+dE$,\n",
+    "\n",
+    "\\begin{align}\n",
+    "u(E)\\ dE =  \\frac{8\\pi}{h^3 c^3} \\frac{E^3\\ dE}{e^{E/(kT)}-1}.\n",
+    "\\end{align}\n",
+    "\n",
+    "Using $E=hc/\\lambda$ and $|dE| = \\left(hc/\\lambda^2\\right) d\\lambda$., we find\n",
+    "\n",
+    "\\begin{align}\n",
+    "u(\\lambda, T)d\\lambda = \\frac{8\\pi hc}{\\lambda^5} \\frac{ dE}{e^{E/(kT)}-1}.\n",
+    "\\end{align}\n",
+    "\n",
+    "In the SI system, multiplying by a factor $c/4$ is required to change energy density $u(\\lambda, T)$ to a spectral density $\\mathcal{I}(\\lambda, T)$, or Eq. {eq}`spectral_density`.\n",
+    "\n",
+    "A few notes:\n",
+    "\n",
+    "- Planck **did not** use the Bose-Einstein distribution to derive his radiation law.  However, it shows the power of the statistical approach.\n",
+    "- This problem was first solved by [Satyendra Nath Bose](https://en.wikipedia.org/wiki/Satyendra_Nath_Bose) in 1924 before the concept of spin in quantum theory.\n",
+    "- Einstein's name was added because he helped Bose publish his work in the West and later applied the distribution to other problems. \n",
+    "\n",
     "\n",
     "### Liquid Helium\n",
     "\n",

_config.yml

@@ -7,7 +7,7 @@
 # Book settings
 title                       : PHYS2700 Modern Physics  # The title of the book. Will be placed in the left navbar.
 author                      : Billy Quarles  # The author of the book
-copyright                   : "2022"  # Copyright year to be placed in the footer
+copyright                   : "2024"  # Copyright year to be placed in the footer
 logo                        : "bohr_gif.gif"  # A path to the book logo
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 exclude_patterns            : [_build, Thumbs.db, .DS_Store, "**.ipynb_checkpoints","*.png"]
@@ -35,6 +35,7 @@ parse:
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+    - colon_fence
 
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