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DPotoyan committed Nov 4, 2024
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31 changes: 23 additions & 8 deletions _sources/ch04/note04.ipynb
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Expand Up @@ -203,29 +203,44 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
":::{admonition} **Example**\n",
":class: note\n",
"\n",
"Find eigenfunctions and eigenvalues of $\\hat{L}_z$\n",
"Find the eigenfunctions and eigenvalues of the operator $\\hat{L}_z$:\n",
"\n",
"$${\\hat{L}_z = -i\\hbar\\frac{\\partial}{\\partial\\phi}}$$\n",
"$$\\hat{L}_z = -i\\hbar\\frac{\\partial}{\\partial\\phi}$$\n",
"\n",
":::\n",
"\n",
":::{admonition} **Solution**\n",
":class: note, dropdown\n",
"\n",
"Lets write the eigenfunction-eigenvalue problem\n",
"To solve the eigenvalue problem for $\\hat{L}_z$, we write:\n",
"\n",
"$$\\hat{L}_z f(\\phi) = a f(\\phi)$$\n",
"\n",
"$$-i\\hbar\\frac{\\partial f}{\\partial\\phi} = a f$$\n",
"Substituting $\\hat{L}_z = -i\\hbar\\frac{\\partial}{\\partial\\phi}$:\n",
"\n",
"$$-i\\hbar \\frac{\\partial f}{\\partial \\phi} = a f$$\n",
"\n",
"Rearranging, we find:\n",
"\n",
"$$\\frac{1}{f} \\frac{\\partial f}{\\partial \\phi} = \\frac{a}{-i\\hbar}$$\n",
"\n",
"This equation can be solved by recognizing that $\\frac{\\partial \\ln f}{\\partial \\phi} = \\frac{a}{-i\\hbar}$:\n",
"\n",
"$$f(\\phi) = e^{\\frac{a}{-i\\hbar} \\phi} = e^{im\\phi}$$\n",
"\n",
"where \\(m\\) is an integer for the function $f(\\phi)$ to be single-valued over $0 \\leq \\phi < 2\\pi$.\n",
"\n",
"Thus, we identify the eigenvalues $a = \\hbar m$, giving:\n",
"\n",
"$$\\frac{-i\\hbar}{f}\\frac{\\partial f}{\\partial\\phi} f = -i\\hbar\\frac{\\partial log f}{\\partial\\phi} = m $$\n",
"$$a = \\hbar m \\quad \\text{and} \\quad f(\\phi) = e^{im\\phi}$$\n",
"\n",
"$$f = e^{im \\phi}$$\n",
"where $m$ is the quantum number associated with the angular momentum eigenvalue.\n",
"\n",
"$$a = \\hbar m$$\n",
"$$\\hat{L}_z f(\\phi) = \\hbar m \\cdot f(\\phi)$$\n",
"\n",
":::"
]
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"\n",
"$${{\\hat{L}}^2 Y_l^m(\\theta,\\phi) = \\hbar^2 l(l+1)\\cdot Y_l^m(\\theta,\\phi)}$$\n",
"\n",
"$${\\hat{L}_zY^m_l(\\theta,\\phi) = m\\hbar \\cdot Y_l^m(\\theta,\\phi)}$$\n",
"$${\\hat{L}_zY^m_l(\\theta,\\phi) = \\hbar m \\cdot Y_l^m(\\theta,\\phi)}$$\n",
"\n",
"- **Eigenfunctions:** $Y_l^m(\\theta,\\phi) $\n",
"- **Angular quantum number:** $l = 0,1,2,3...$ \n",
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6 changes: 3 additions & 3 deletions ch01/BlackBody.html
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12 changes: 6 additions & 6 deletions ch01/demo_complex_nums.html
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Expand Up @@ -759,7 +759,7 @@ <h2>Polar plot of complex numbers<a class="headerlink" href="#polar-plot-of-comp
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[&lt;matplotlib.lines.Line2D at 0x7f0000feb7f0&gt;]
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[&lt;matplotlib.lines.Line2D at 0x7fb403f2f970&gt;]
</pre></div>
</div>
<img alt="../_images/ee45d03eabdfceef8711fa6413ebd066e53e9a1f98627fed3abd7fcdc5171724.png" src="../_images/ee45d03eabdfceef8711fa6413ebd066e53e9a1f98627fed3abd7fcdc5171724.png" />
Expand Down Expand Up @@ -797,7 +797,7 @@ <h2>Polar plot of complex numbers<a class="headerlink" href="#polar-plot-of-comp
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7f0000f5f4c0&gt;
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7fb403f822e0&gt;
</pre></div>
</div>
<img alt="../_images/9c1820ba33cdcd3b3a364a819675c4e3159216fc14810013c3d3dd60b70324e9.png" src="../_images/9c1820ba33cdcd3b3a364a819675c4e3159216fc14810013c3d3dd60b70324e9.png" />
Expand Down Expand Up @@ -937,12 +937,12 @@ <h1>Complex numbers, Julia sets and fractals.<a class="headerlink" href="#comple
<div class="cell_output docutils container">
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">KeyboardInterrupt</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">26</span><span class="p">],</span> <span class="n">line</span> <span class="mi">31</span>
<span class="g g-Whitespace"> </span><span class="mi">29</span> <span class="c1"># Do the iterations</span>
<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">26</span><span class="p">],</span> <span class="n">line</span> <span class="mi">32</span>
<span class="g g-Whitespace"> </span><span class="mi">30</span> <span class="k">while</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">zabs_max</span> <span class="ow">and</span> <span class="n">nit</span> <span class="o">&lt;</span> <span class="n">nit_max</span><span class="p">:</span>
<span class="ne">---&gt; </span><span class="mi">31</span> <span class="n">z</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span>
<span class="g g-Whitespace"> </span><span class="mi">32</span> <span class="n">nit</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="g g-Whitespace"> </span><span class="mi">31</span> <span class="n">z</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span>
<span class="ne">---&gt; </span><span class="mi">32</span> <span class="n">nit</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="g g-Whitespace"> </span><span class="mi">34</span> <span class="n">shade</span> <span class="o">=</span> <span class="mi">1</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">nit</span> <span class="o">/</span> <span class="n">nit_max</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">35</span> <span class="n">ratio</span> <span class="o">=</span> <span class="n">nit</span> <span class="o">/</span> <span class="n">nit_max</span>

<span class="ne">KeyboardInterrupt</span>:
</pre></div>
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