exercises added

pyqm-week5/exercises/PyQM_Ex.5.ipynb

@@ -0,0 +1,274 @@
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+   },
+   "source": [
+    "<center>\n",
+    "\n",
+    "# Python for Quantum Mechanics: \n",
+    "# Week 5: Excercises\n",
+    "\n",
+    "</center>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exercise 1: Settlers of Histogram"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Imagine a game that involves rolling two dice (six sided) and adding the results. Knowing the most likely outcomes is important for doing well in this game so you want to run a simulation of the dice rolls. Make a historgam of the results for 1000 rolls.\n",
+    "\n",
+    "<img src=\"images/two_dice.png\" alt=\"two dice\" width=\"200\"/>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "cell_id": "00008-772c311d-21f4-410e-beaf-efffd80e4261",
+    "deepnote_cell_type": "markdown",
+    "tags": []
+   },
+   "source": [
+    "## Exercise 2: Normal Distribution"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "cell_id": "00009-cda71b4c-8576-4de0-b957-104af25d30c7",
+    "deepnote_cell_type": "markdown",
+    "tags": []
+   },
+   "source": [
+    "A gaussian curve with mean $\\mu$ and standard deviation $\\sigma$ is given by\n",
+    "$$g(x)=\\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp{\\left( - \\frac{1}{2} \\frac{(x-\\mu)^2}{\\sigma^2} \\right)}.$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "cell_id": "00009-df9016c5-a72f-49e2-9983-ce181e0c522f",
+    "deepnote_cell_type": "code",
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+    "execution_millis": 2,
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+   "outputs": [],
+   "source": [
+    "def gaus(x,sig=1,mu=0):\n",
+    "    return (1/(sig*np.sqrt(2*np.pi)))*np.exp(-.5*((x-mu)/sig)**2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Make a plot of a normal distribution histogram with a gaussian curve overlayed on it."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Plot the following function using `plot_surface()`\n",
+    "$$G(x,y)=g\\left( \\sqrt{x^2+y^2} \\right)$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def gaus3d(x,y,sig=1,mu=0):\n",
+    "    return gaus(np.sqrt(x**2 + y**2))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "cell_id": "00004-cde4d2de-e564-4d94-b44f-92a484a4db84",
+    "deepnote_cell_type": "markdown",
+    "tags": []
+   },
+   "source": [
+    "## Exercise 3: Particle in a Box"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Use the following functions to make an animation of the real and imaginary parts of a particle in a box for any $n$.\n",
+    "<img src=\"images/box_potential.png\" alt=\"two dice\" width=\"200\"/>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
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+    "execution_millis": 599,
+    "execution_start": 1625584445278,
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+    "tags": []
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+   "outputs": [],
+   "source": [
+    "def psi(n,x,l):\n",
+    "    return np.sqrt(2/l) * np.sin(n*np.pi*x/l)\n",
+    "\n",
+    "def energy(n):\n",
+    "    return -13.6/n**2\n",
+    "\n",
+    "def psi_im(n,x,t,l):\n",
+    "    return np.sin(-t*energy(n)) * psi(n,x,l)\n",
+    "\n",
+    "def psi_re(n,x,t,l):\n",
+    "    return np.cos(-t*energy(n)) * psi(n,x,l)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now put the real and imaginary parts on the $y$ and $z$ axes of a 3D plot."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "cell_id": "00010-fcb17d54-1761-4e8c-9c73-907da40219c1",
+    "deepnote_cell_type": "markdown",
+    "tags": []
+   },
+   "source": [
+    "## Exercise 4: Plotting a Sphere"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "You can parameterize a sphere of radius $r$ with parameters $u \\in [0,2\\pi]$ and $v \\in [0,\\pi]$ \n",
+    "\n",
+    "$x = r \\cos(u)\\sin(v)$\n",
+    "\n",
+    "$y = r \\cos(u)\\sin(v)$\n",
+    "\n",
+    "$z = r \\cos(u)\\sin(v)$\n",
+    "\n",
+    "Plot a 3D sphere of radius $r=3$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exercise 5: Snake Animation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Reproduce the following animation of a line going around a spiral.\n",
+    "![SegmentLocal](images/my_animation.gif \"snake\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
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