.PhV Tutorium Gruppe 3.md

Jupyter-Notebooks zu den Tutorien in Physik V.

(PS: manchmal ist GitHub etwas überfordert mit der Vorschau der ganzen Notebooks — am Besten runterladen, oder im Binder anschauen!)

Tutorium 1.ipynb

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   "source": [
    "# Übungsgruppe 3 — Tutorium #1\n",
    "\n",
    "> ralf@uni-bonn.de\n",
    "\n",
    "## Einheiten in Python\n",
    "\n",
    "Es gibt verschiedene Bibliotheken, die die Arbeit mit Einheiten in Python vereinfachen, zB `pint` oder `unyt`. Ich benutze dafür `astropy` ([Dokumentation](https://docs.astropy.org/en/stable/units/)):"
   ]
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   "source": [
    "import astropy.units as u\n",
    "import astropy.constants as const\n",
    "\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Viele Konstanten sind, mit den korrekten Einheiten, schon eingebaut:"
   ]
  },
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$2.9979246 \\times 10^{8} \\; \\mathrm{\\frac{m}{s}}$"
      ],
      "text/plain": [
       "<<class 'astropy.constants.codata2018.CODATA2018'> name='Speed of light in vacuum' value=299792458.0 uncertainty=0.0 unit='m / s' reference='CODATA 2018'>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "const.c"
   ]
  },
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1.0545718 \\times 10^{-34} \\; \\mathrm{J\\,s}$"
      ],
      "text/plain": [
       "<<class 'astropy.constants.codata2018.CODATA2018'> name='Reduced Planck constant' value=1.0545718176461565e-34 uncertainty=0.0 unit='J s' reference='CODATA 2018'>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "const.hbar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Man kann mit `.to(...)` in andere Einheiten wechseln:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$6.5821196 \\times 10^{-16} \\; \\mathrm{eV\\,s}$"
      ],
      "text/plain": [
       "<Quantity 6.58211957e-16 eV s>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "const.hbar.to(\"eV.s\")  # äquivalent: const.hbar.to(u.eV * u.s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A1. Natürliche Einheiten\n",
    "### c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Um eine *Quantität* zu erhalten, multipliziert man die Einheit einfach hinten dran:"
   ]
  },
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1000 \\; \\mathrm{MeV}$"
      ],
      "text/plain": [
       "<Quantity 1000. MeV>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E = 1000 * u.MeV\n",
    "E"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Photon**\n",
    "\n",
    "Das Photon hat $M_\\gamma = 0$, damit ist\n",
    "\n",
    "$$\n",
    "p_\\gamma = E = 1000\\,\\text{MeV}\n",
    "$$\n",
    "\n",
    "**Pion**\n",
    "\n",
    "Die *Quantitäten* kann man dann ganz einfach mit den arithmetischen Funktionen von Python (`+`, `**`, ...) oder numpy (`np.sqrt`, ...) verwenden — die Einheiten werden automatisch mitgeschleift."
   ]
  },
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$990.1515 \\; \\mathrm{MeV}$"
      ],
      "text/plain": [
       "<Quantity 990.15150356 MeV>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m_pion = 140 * u.MeV\n",
    "p_pion = np.sqrt(E**2 - m_pion**2)\n",
    "\n",
    "p_pion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proton**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
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     "shell.execute_reply.started": "2020-11-05T11:33:57.620257Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$346.63525 \\; \\mathrm{MeV}$"
      ],
      "text/plain": [
       "<Quantity 346.63525499 MeV>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m_proton = 938 * u.MeV\n",
    "p_proton = np.sqrt(E**2 - m_proton**2)\n",
    "\n",
    "p_proton"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Austauschteilchen\n",
    "\n",
    "> siehe Vorlesung 3, Slide #95\n",
    "\n",
    "Die maximale Reichweite ist definiert als\n",
    "\n",
    "$$\n",
    "R \\leq \\frac{\\hbar}{M_X c}\n",
    "$$\n",
    "\n",
    "hier ist $M_X = m_W$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Im PDG unter [https://pdglive.lbl.gov](https://pdglive.lbl.gov) findet man die Masse des W-Bosons, angegeben in natürlichen Einheiten als\n",
    "\n",
    "$$\n",
    "\\text{W MASS} \\qquad 80.379\\pm0.012\\,\\text{GeV} \n",
    "$$\n",
    "\n",
    "`astropy` kommt leider nicht so ganz mit natürlichen Einheiten klar, deswegen nehmen wir den Umweg und verwenden $\\text{GeV}/c^2$ als Einheit für die Masse:"
   ]
  },
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$8.9012004 \\times 10^{-16} \\; \\mathrm{\\frac{s^{2}\\,GeV}{m^{2}}}$"
      ],
      "text/plain": [
       "<Quantity 8.90120045e-16 GeV s2 / m2>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mW = 80 * u.GeV/const.c**2\n",
    "mW"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "Das sieht leider etwas unschön aus, da `astropy` gleich den Wert der Konstanten $3\\cdot 10^8$ mit dazumultipliziert — das ist zwar korrekt, aber etwas schöner ist es, wenn man `c` als *Einheit* definiert — das mache ich *immer* wenn ich Einheiten verwende, normalerweise ganz am Anfang meines Notebooks:"
   ]
  },
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   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
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     "shell.execute_reply.started": "2020-11-05T11:33:57.632122Z"
    }
   },
   "outputs": [],
   "source": [
    "u.c = u.def_unit('c', const.c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Damit kann man `mW` schreiben als"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
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    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$80 \\; \\mathrm{\\frac{GeV}{c^{2}}}$"
      ],
      "text/plain": [
       "<Quantity 80. GeV / c2>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mW = 80 * u.GeV/u.c**2  # <- das neue u.c, und nicht const.c!\n",
    "mW"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "…was etwas angenehmer ist ;)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Um den Wert in SI-Einheiten anzuzeigen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-05T11:41:36.123101Z",
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1.4261295 \\times 10^{-25} \\; \\mathrm{kg}$"
      ],
      "text/plain": [
       "<Quantity 1.42612954e-25 kg>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mW.to(\"kg\")  # oder:   mW.to(u.kg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir berechnen die Reichweite:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "execution": {
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     "shell.execute_reply.started": "2020-11-05T11:33:57.651195Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$4.3970912 \\times 10^{-45} \\; \\mathrm{\\frac{c^{2}\\,s^{2}\\,J}{GeV\\,m}}$"
      ],
      "text/plain": [
       "<Quantity 4.39709118e-45 c2 J s2 / (GeV m)>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = const.hbar / (mW * const.c)\n",
    "R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mit `.decompose()` kann man eine Quantität in Basis-Einheiten umwandeln, um die unnötigen Einheiten wegkürzen zu lassen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "execution": {
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     "shell.execute_reply.started": "2020-11-05T11:33:57.657941Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$2.4665873 \\times 10^{-18} \\; \\mathrm{m}$"
      ],
      "text/plain": [
       "<Quantity 2.46658726e-18 m>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.decompose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "…oder direkt mit der Wunscheinheit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "execution": {
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     "shell.execute_reply.started": "2020-11-05T11:33:57.664391Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$0.0024665873 \\; \\mathrm{fm}$"
      ],
      "text/plain": [
       "<Quantity 0.00246659 fm>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.to(\"fm\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "PS: wir hätten auch einfach alles in natürlichen Einheiten rechnen können:\n",
    "\n",
    "\n",
    "$$\n",
    "R = \\frac{1}{M_X} \\qquad\\text{(in natürlichen Einheiten)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-05T11:33:57.670236Z",
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     "shell.execute_reply.started": "2020-11-05T11:33:57.670213Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$0.0125 \\; \\mathrm{\\frac{1}{GeV}}$"
      ],
      "text/plain": [
       "<Quantity 0.0125 1 / GeV>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mW_nat = 80 * u.GeV  # wie im PDG angegeben\n",
    "\n",
    "R_nat = 1 / mW_nat    # da hbar=1\n",
    "R_nat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Und das dann anschließend in SI-Einheiten umrechnen, mit dem Faktor $\\hbar c$ (für die Länge):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "execution": {
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   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$3.9519085 \\times 10^{-28} \\; \\mathrm{\\frac{J\\,m}{GeV}}$"
      ],
      "text/plain": [
       "<Quantity 3.95190847e-28 J m / GeV>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_si = R_nat * (const.hbar * const.c)\n",
    "R_si"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "execution": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$2.4665873 \\times 10^{-18} \\; \\mathrm{m}$"
      ],
      "text/plain": [
       "<Quantity 2.46658726e-18 m>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_si.decompose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
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     "iopub.status.busy": "2020-11-05T11:33:57.687282Z",
     "iopub.status.idle": "2020-11-05T11:33:57.691945Z",
     "shell.execute_reply": "2020-11-05T11:33:57.691387Z",
     "shell.execute_reply.started": "2020-11-05T11:33:57.687431Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$0.0024665873 \\; \\mathrm{fm}$"
      ],
      "text/plain": [
       "<Quantity 0.00246659 fm>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_si.to(\"fm\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PS: Unter [https://www.seas.upenn.edu/~amyers/NaturalUnits.pdf](https://www.seas.upenn.edu/~amyers/NaturalUnits.pdf) gibt es eine Tabelle mit den Faktoren für die verschiedenen Einheiten... also $1/c^2$ für die Masse, $\\hbar$ für eine Länge, etc…"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Wirkungsquerschnitt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-05T11:33:57.692902Z",
     "iopub.status.busy": "2020-11-05T11:33:57.692734Z",
     "iopub.status.idle": "2020-11-05T11:33:57.696732Z",
     "shell.execute_reply": "2020-11-05T11:33:57.696105Z",
     "shell.execute_reply.started": "2020-11-05T11:33:57.692880Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "99.99999999999999"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.barn.to(\"fm^2\")  # oder:   u.barn.to(u.fm**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(Rundungsfehler………)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### c)\n",
    "\n",
    "\n",
    "$$\n",
    "\\sigma_{\\text{tot}} = \\pi R^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-05T11:33:57.697690Z",
     "iopub.status.busy": "2020-11-05T11:33:57.697515Z",
     "iopub.status.idle": "2020-11-05T11:33:57.700881Z",
     "shell.execute_reply": "2020-11-05T11:33:57.699989Z",
     "shell.execute_reply.started": "2020-11-05T11:33:57.697669Z"
    }
   },
   "outputs": [],
   "source": [
    "def sigma(radius):\n",
    "    display((np.pi * radius**2).to(\"barn\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-05T11:33:57.701878Z",
     "iopub.status.busy": "2020-11-05T11:33:57.701661Z",
     "iopub.status.idle": "2020-11-05T11:33:57.707467Z",
     "shell.execute_reply": "2020-11-05T11:33:57.706779Z",
     "shell.execute_reply.started": "2020-11-05T11:33:57.701847Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$3.1415927 \\times 10^{8} \\; \\mathrm{barn}$"
      ],
      "text/plain": [
       "<Quantity 3.14159265e+08 barn>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r_atom = 1*u.angstrom\n",
    "sigma(r_atom)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-05T11:33:57.708598Z",
     "iopub.status.busy": "2020-11-05T11:33:57.708399Z",
     "iopub.status.idle": "2020-11-05T11:33:57.714779Z",
     "shell.execute_reply": "2020-11-05T11:33:57.713911Z",
     "shell.execute_reply.started": "2020-11-05T11:33:57.708575Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$0.97464329 \\; \\mathrm{barn}$"
      ],
      "text/plain": [
       "<Quantity 0.97464329 barn>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r_nucleon = r0 = 1.2*u.fm\n",
    "A = 100  # Anzahl Nukleonen\n",
    "r_kern = r0 * A**(1/3)\n",
    "\n",
    "sigma(r_kern)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-05T11:33:57.718293Z",
     "iopub.status.busy": "2020-11-05T11:33:57.717883Z",
     "iopub.status.idle": "2020-11-05T11:33:57.724045Z",
     "shell.execute_reply": "2020-11-05T11:33:57.723391Z",
     "shell.execute_reply.started": "2020-11-05T11:33:57.718261Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$0.045238934 \\; \\mathrm{barn}$"
      ],
      "text/plain": [
       "<Quantity 0.04523893 barn>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r_proton = 1.2*u.fm\n",
    "sigma(r_proton)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><br><br><br><br><br><br><br><br>\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-11-05T11:38:41.651193Z",
     "iopub.status.busy": "2020-11-05T11:38:41.650949Z",
     "iopub.status.idle": "2020-11-05T11:38:41.656203Z",
     "shell.execute_reply": "2020-11-05T11:38:41.655386Z",
     "shell.execute_reply.started": "2020-11-05T11:38:41.651170Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "astropy==4.1\n",
      "numpy==1.19.2\n",
      "matplotlib==3.3.2\n",
      "sympy==1.6.2\n",
      "scipy==1.5.2\n",
      "pint==0.16.1\n"
     ]
    }
   ],
   "source": [
    "import astropy, numpy, matplotlib, sympy, scipy, pint\n",
    "print(f\"astropy=={astropy.__version__}\")\n",
    "print(f\"numpy=={numpy.__version__}\")\n",
    "print(f\"matplotlib=={matplotlib.__version__}\")\n",
    "print(f\"sympy=={sympy.__version__}\")\n",
    "print(f\"scipy=={scipy.__version__}\")\n",
    "print(f\"pint=={pint.__version__}\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "3.8.5 (Framework)",
   "language": "python",
   "name": "3.8.5-framework"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}

Tutorium 10.ipynb

Tutorium 11.ipynb

Tutorium 12.ipynb

Tutorium 13.ipynb

Tutorium 3.ipynb

Tutorium 4.ipynb

Tutorium 5.ipynb

environment.yml

Tutorium 6.ipynb

Tutorium 8.ipynb