deploy: 8867245

History.html

@@ -604,7 +604,7 @@ <h3><a class="reference external" href="https://en.wikipedia.org/wiki/Photoelect
 <p>Einstein decided to take Planck’s mathematical expression at face value, and interpret light as coming in chunks called photons, with energy given by Planck’s expression:</p>
 <div class="math notranslate nohighlight">
 \[ E_{ph} = h \nu_{ph} \]</div>
-<p>This allowed him to describe the photoelectric effect. Specifically, whenever <span class="math notranslate nohighlight">\(h \nu \)</span> was less than a threshold value for the substance called its work function, <span class="math notranslate nohighlight">\(W\)</span>, no light would be emitted. (The work function can be thought of as the “ionization potential” for a metal.) But when <span class="math notranslate nohighlight">\(h \nu &gt; W\)</span>, then photons had enough energy to remove an electron from the metal, and the surplus energy from the photon could be imparted to the electron as kinetic energy. Ergo</p>
+<p>This allowed him to describe the photoelectric effect. Specifically, whenever <span class="math notranslate nohighlight">\(h \nu \)</span> was less than a threshold value for the substance called its work function, <span class="math notranslate nohighlight">\(W\)</span>, no electrons would be emitted. (The work function can be thought of as the “ionization potential” for a metal.) But when <span class="math notranslate nohighlight">\(h \nu &gt; W\)</span>, then photons had enough energy to remove an electron from the metal, and the surplus energy from the photon could be imparted to the electron as kinetic energy. Ergo</p>
 <div class="math notranslate nohighlight">
 \[ \text{Kinetic Energy}_{\text{max}} = h \nu - W \]</div>
 <p>As the light becomes more intense, the number of photons increases and the number of electrons emitted increases but, unless the light is very, very, very intense, the maximum kinetic energy of the electrons is left unaltered.</p>

OneElectronAtoms.html

@@ -635,7 +635,7 @@ <h2>Eigenenergies and Wavefunctions for One-Electron Atoms<a class="headerlink"
 <p>The following animations shows one can take linear combinations of the (complex) spherical harmonics to form the <span class="math notranslate nohighlight">\(p_x\)</span>, <span class="math notranslate nohighlight">\(p_y\)</span>, etc. orbitals one generally uses in chemistry.</p>
 <p><img alt="animation of 2p orbital" src="https://github.com/PaulWAyers/IntroQChem/blob/main/linkedFiles/Orbital_p1-px_animation.gif?raw=true" /></p>
 <p><img alt="animation of 3p orbital" src="https://github.com/PaulWAyers/IntroQChem/blob/main/linkedFiles/Orbital_3p1-3px_animation.gif?raw=true" /></p>
-<p>Using the <a class="reference external" href="https://winter.group.shef.ac.uk/orbitron/atomic_orbitals/7i/index.html">orbitron</a>, you can visualize the (real, Cartesian) spherical harmonics and the <a class="reference external" href="https://winter.group.shef.ac.uk/orbitron/atomic_orbitals/7f/7f_wave_function.html">radial wavefunctions</a> for hydrogenic orbitals. Most orbitals have very complicated formulas, but a few have simple equations, including:</p>
+<p>Most orbitals have very complicated formulas, but a few have simple equations, including:</p>
 <div class="math notranslate nohighlight">
 \[\begin{split}
 \psi_{\text{1s}}(r) = \psi_{100}(r) = \sqrt{\frac{Z^3}{\pi}}e^{-Zr}  \\
@@ -697,7 +697,8 @@ <h2>📚 References<a class="headerlink" href="#references" title="Link to this
 <li><p><a class="reference external" href="https://github.com/PaulWAyers/IntroQChem/blob/main/documents/DumontBook.pdf?raw=true">Randy’s book</a></p></li>
 <li><p>D. A. MacQuarrie, Quantum Chemistry (University Science Books, Mill Valley California, 1983)</p></li>
 <li><p><a class="reference external" href="https://github.com/PaulWAyers/IntroQChem/blob/main/documents/Hatom.pdf?raw=true">One-electron atoms</a> (my notes).</p></li>
-<li><p><a class="reference external" href="https://dpotoyan.github.io/Chem324/Lec5-0.html">Davit Potoyan’s Jupyterbook course</a>.</p></li>
+<li><p><a class="reference external" href="https://dpotoyan.github.io/Chem324/ch05/note01.html">Davit Potoyan’s Jupyterbook course</a>.</p></li>
+<li><p>It isn’t active anymore, but the <a class="reference external" href="https://winter.group.shef.ac.uk/orbitron/atomic_orbitals/7i/index.html">orbitron</a> let one visualize the (real, Cartesian) spherical harmonics and the <a class="reference external" href="https://winter.group.shef.ac.uk/orbitron/atomic_orbitals/7f/7f_wave_function.html">radial wavefunctions</a> for hydrogenic orbitals. I’m leaving the link here hoping that it, or something similar, becomes available.</p></li>
 </ul>
 <p>There are also some excellent wikipedia articles:</p>
 <ul class="simple">

ParticleIn1DBox.html

@@ -645,7 +645,7 @@ <h2>The Wavefunctions of the Particle in a Box (solution)<a class="headerlink" h
 \[
 0 = \psi(a) = A \sin(ca)
 \]</div>
-<p>requires us to recall that <span class="math notranslate nohighlight">\(\sin(x) = 0\)</span> whenever <span class="math notranslate nohighlight">\(x\)</span> is an integer multiple of <span class="math notranslate nohighlight">\(\pi\)</span>. So <span class="math notranslate nohighlight">\(c=n\pi\)</span> where <span class="math notranslate nohighlight">\(n=1,2,3,\ldots\)</span>. The wavefunction for the particle in a box is thus,</p>
+<p>requires us to recall that <span class="math notranslate nohighlight">\(\sin(x) = 0\)</span> whenever <span class="math notranslate nohighlight">\(x\)</span> is an integer multiple of <span class="math notranslate nohighlight">\(\pi\)</span>. So <span class="math notranslate nohighlight">\(c=\tfrac{n\pi}{a}\)</span> where <span class="math notranslate nohighlight">\(n=1,2,3,\ldots\)</span>. The wavefunction for the particle in a box is thus,</p>
 <div class="math notranslate nohighlight">
 \[
 \psi_n(x) = A_n \sin\left(\tfrac{n \pi x}{a}\right) \qquad \qquad n=1,2,3,\ldots
@@ -702,7 +702,7 @@ <h2>Normalization of Wavefunctions<a class="headerlink" href="#normalization-of-
 \[
 A_n = \left(\cos(\theta) \pm i \sin(\theta) \right) \sqrt{\tfrac{2}{a}}
 \]</div>
-<p>where <span class="math notranslate nohighlight">\(k\)</span> is any real number. The arbitrariness of the <em>phase</em> of the wavefunction is an important feature. Because the wavefunction can be imaginary (e.g., if you choose <span class="math notranslate nohighlight">\(A_n = i \sqrt{\tfrac{2}{a}}\)</span>), it is obvious that the wavefunction is not an observable property of a system. <strong>The wavefunction is only a mathematical tool for quantum mechanics; it is not a physical object.</strong></p>
+<p>where <span class="math notranslate nohighlight">\(\theta\)</span> is any real number. The arbitrariness of the <em>phase</em> of the wavefunction is an important feature. Because the wavefunction can be imaginary (e.g., if you choose <span class="math notranslate nohighlight">\(A_n = i \sqrt{\tfrac{2}{a}}\)</span>), it is obvious that the wavefunction is not an observable property of a system. <strong>The wavefunction is only a mathematical tool for quantum mechanics; it is not a physical object.</strong></p>
 <p>Summarizing, the (normalized) wavefunction for a particle with mass <span class="math notranslate nohighlight">\(m\)</span> confined to a one-dimensional box with length <span class="math notranslate nohighlight">\(a\)</span> can be written as:</p>
 <div class="math notranslate nohighlight">
 \[

ParticleInMultiD.html

@@ -903,11 +903,11 @@ <h3>The Schrödinger Equation in Polar Coordinates<a class="headerlink" href="#t
 \[
 E r^2 +\frac{\hbar^2}{2m}\frac{r^2}{R(r)} \left(\frac{d^2 R(r)}{dr^2} + \frac{1}{r} \frac{d R(r)}{dr} \right) - r^2 V(r)=-\frac{\hbar^2}{2m} \frac{1}{\Theta(\theta)}\left(\frac{d^2 \Theta(\theta)}{d \theta^2} \right) 
 \]</div>
-<p>The right-hand-side depends only on <span class="math notranslate nohighlight">\(r\)</span> and the left-hand-side depends only on <span class="math notranslate nohighlight">\(\theta\)</span>; this can only be true for all <span class="math notranslate nohighlight">\(r\)</span> and all <span class="math notranslate nohighlight">\(\theta\)</span> if both sides are equal to the same constant. This problem can therefore be solved by separation of variables, though it is a slightly different form from the one we considered previously.</p>
+<p>The left-hand-side depends only on <span class="math notranslate nohighlight">\(r\)</span> and the right-hand-side depends only on <span class="math notranslate nohighlight">\(\theta\)</span>; this can only be true for all <span class="math notranslate nohighlight">\(r\)</span> and all <span class="math notranslate nohighlight">\(\theta\)</span> if both sides are equal to the same constant. This problem can therefore be solved by separation of variables, though it is a slightly different form from the one we considered previously.</p>
 </section>
 <section id="the-angular-schrodinger-equation-in-polar-coordinates">
 <h3>The Angular Schrödinger equation in Polar Coordinates<a class="headerlink" href="#the-angular-schrodinger-equation-in-polar-coordinates" title="Link to this heading">#</a></h3>
-<p>To find the solution, we first solve the set the left-hand-side equal to a constant, which gives a 1-dimensional Schrödinger equations for the angular motion of the particle around the circle,</p>
+<p>To find the solution, we first solve the set the right-hand-side equal to a constant, which gives a 1-dimensional Schrödinger equations for the angular motion of the particle around the circle,</p>
 <div class="math notranslate nohighlight">
 \[
 -\frac{\hbar^2}{2m} \frac{d^2 \Theta_l(\theta)}{d \theta^2} = E_{\theta;l} \Theta_l(\theta)
@@ -1024,7 +1024,7 @@ <h3>Eigenvalues and Eigenfunctions for a Particle Confined to a Circular Disk<a
 <li><p><a class="reference external" href="https://demonstrations.wolfram.com/ParticleInAnInfiniteCircularWell/">Interactive Demonstration of the States of a Particle-in-a-Circular-Disk</a></p></li>
 <li><p><a class="reference external" href="https://www.reddit.com/r/dataisbeautiful/comments/mfx5og/first_70_states_of_a_particle_trapped_in_a/?utm_source=share&amp;amp;utm_medium=web2x&amp;amp;context=3">Movie animation of the quantum states of a particle-in-a-circular-disk</a></p></li>
 </ul>
-<p>A subtle result, <a class="reference external" href="https://en.wikipedia.org/wiki/Bessel_function#Bourget's_hypothesis">originally proposed by Bourget</a>, is that no two Bessel functions ever have the same zeros, which means that the values of <span class="math notranslate nohighlight">\(\{x_{n,l} \}\)</span> are all distinct. A corollary of this is that eigenvalues of the particle confined to a circular disk are either nondegenerate (<span class="math notranslate nohighlight">\(l=0\)</span>) or doubly degenerate (<span class="math notranslate nohighlight">\(|l| /ge 1\)</span>). There are no accidental degeneracies for a particle in a circular disk.</p>
+<p>A subtle result, <a class="reference external" href="https://en.wikipedia.org/wiki/Bessel_function#Bourget's_hypothesis">originally proposed by Bourget</a>, is that no two Bessel functions ever have the same zeros, which means that the values of <span class="math notranslate nohighlight">\(\{x_{n,l} \}\)</span> are all distinct. A corollary of this is that eigenvalues of the particle confined to a circular disk are either nondegenerate (<span class="math notranslate nohighlight">\(l=0\)</span>) or doubly degenerate (<span class="math notranslate nohighlight">\(|l| \ge 1\)</span>). There are no accidental degeneracies for a particle in a circular disk.</p>
 <p>The energies of an electron confined to a circular disk with radius <span class="math notranslate nohighlight">\(a\)</span> Bohr are:</p>
 <div class="math notranslate nohighlight">
 \[

Postulates.html

@@ -785,9 +785,7 @@ <h4>📝 Exercise: Show that the equality in the last equation is true<a class="
 <h4>📝 Exercise: Show that the Laplacian is a linear, Hermitian, negative-definite operator.<a class="headerlink" href="#exercise-show-that-the-laplacian-is-a-linear-hermitian-negative-definite-operator" title="Link to this heading">#</a></h4>
 <blockquote>
 <div><p>It is OK (but not essential) to assume that you are interested in the Laplacian operator in 1 dimension, <span class="math notranslate nohighlight">\(\nabla^2 = \frac{d^2}{dx^2}\)</span>. When we say that <span class="math notranslate nohighlight">\(\nabla^2\)</span> is negative definite, we mean that for <em>any</em> <span class="math notranslate nohighlight">\(\Psi\)</span>, it is <em>always</em> true that
-$<span class="math notranslate nohighlight">\(
-\langle \Psi | \nabla^2 | \Psi \rangle &lt; 0
-\)</span>$
+<span class="math notranslate nohighlight">\(\langle \Psi | \nabla^2 | \Psi \rangle &lt; 0\)</span>
 Note: You may assume that the wavefunction, and its first first and second derivatives, vanish at the ends of the region of integration. However, there are also ways to answer these questions without explicitly invoking those assumptions.
 <a class="reference external" href="https://github.com/QC-Edu/IntroQM/blob/master/book/documents/laplacianproperties.pdf">Solution.</a></p>
 </div></blockquote>
@@ -800,7 +798,7 @@ <h3>Wavefunction “Collapse”<a class="headerlink" href="#wavefunction-collaps
 \[
 \Psi_{cat} = \tfrac{1}{\sqrt{2}}|\text{alive} \rangle + \tfrac{1}{\sqrt{2}}|\text{dead} \rangle
 \]</div>
-<p>and you opened the box and observed that the cat was dead (so after you open the box, <span class="math notranslate nohighlight">\(\Psi_{cat} =  |\text{dead} \rangle\)</span>), then <em>you</em> killed Schrödinger’s cat. To mildly exaggerate, some physicists would have you believe that every dead animal was slaughtered by the person who first observes its corpse. (To diminish culpability, it must be said that it the cat in this example was only technically half-dead, so the observer was merely a halfway-cat-assassin.) Most modern <a class="reference external" href="https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics">interpretations of quantum mechanics</a> tend to <a class="reference external" href="https://plato.stanford.edu/entries/qt-issues/">deny such culpability</a>. The physicists alibi is to assert that assert that while the system was described, mathematically, by <span class="math notranslate nohighlight">\(\Psi(x)\)</span> prior to the measurement, this does not that the system existed in the state <span class="math notranslate nohighlight">\(\Psi(x)\)</span>. Similarly, after the measurement the system is in a state mathematically described by <span class="math notranslate nohighlight">\(\psi_k(x)\)</span>. While it would be weird for observing a system to be able to change its state, it is not weird for an observation to change our mathematical description of a system. For example, before you observe <a class="reference external" href="https://en.wikipedia.org/wiki/Lake_Wobegon">Lake Wobegon</a>, it is reasonable to assume that all the women are strong, all the men are good-looking, and all the children are above average. But were you to visit Lake Wobegon, then based on your observation you might have to change your model.</p>
+<p>and you opened the box and observed that the cat was dead (so after you open the box, <span class="math notranslate nohighlight">\(\Psi_{cat} =  |\text{dead} \rangle\)</span>), then <em>you</em> killed Schrödinger’s cat. To mildly exaggerate, some physicists would have you believe that every dead animal was slaughtered by the person who first observes its corpse. (To diminish culpability, it must be said that it the cat in this example was only technically half-dead, so the observer was merely a halfway-cat-assassin.) Most modern <a class="reference external" href="https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics">interpretations of quantum mechanics</a> tend to <a class="reference external" href="https://plato.stanford.edu/entries/qt-issues/">deny such culpability</a>. The physicists alibi is to assert that while the system was described, mathematically, by <span class="math notranslate nohighlight">\(\Psi(x)\)</span> prior to the measurement, this does not that the system existed in the state <span class="math notranslate nohighlight">\(\Psi(x)\)</span>. Similarly, after the measurement the system is in a state mathematically described by <span class="math notranslate nohighlight">\(\psi_k(x)\)</span>. While it would be weird for observing a system to be able to change its state, it is not weird for an observation to change our mathematical description of a system. For example, before you observe <a class="reference external" href="https://en.wikipedia.org/wiki/Lake_Wobegon">Lake Wobegon</a>, it is reasonable to assume that all the women are strong, all the men are good-looking, and all the children are above average. But were you to visit Lake Wobegon, then based on your observation you might have to change your model.</p>
 <p>That said, you may find the aforementioned “Copenhagen interpretation” convenient. Before my mother visits my home, I always clean it thoroughly. Nonetheless, my thorough cleaning is not up to my mother’s standards, and she’s always scandalized to find dust-bunnies under the sofa. (Who moves the sofa to vacuum under it, just to put the sofa back the same place and obscure the now-clean carpet?) I always tell my mom that the dust-bunnies were not there until she observed them. Unfortunately, my mom taught quantum mechanics herself, and she tells me that the wavefunction was:</p>
 <div class="math notranslate nohighlight">
 \[
@@ -818,7 +816,7 @@ <h3>The Born Postulate, revisited<a class="headerlink" href="#the-born-postulate
 <blockquote>
 <div><p><span class="math notranslate nohighlight">\(\int_{-\infty}^{+\infty} f(x) \delta(x-x_0) dx = f(x_0)\)</span></p>
 </div></blockquote>
-<p>Now according to the Hermitian postulate, the probability of observing the particle at position <span class="math notranslate nohighlight">\(x_0\)</span> is given by</p>
+<p>Now according to the Born postulate, the probability of observing the particle at position <span class="math notranslate nohighlight">\(x_0\)</span> is given by</p>
 <div class="math notranslate nohighlight">
 \[
 \int_{-\infty}^{+\infty} \left(\Psi(x)\right)^* \delta(x-x_0) \Psi(x) dx = \left|\Psi(x_0)\right|^2
@@ -1076,7 +1074,7 @@ <h2>📚 References<a class="headerlink" href="#references" title="Link to this
 <li><p><a class="reference external" href="https://github.com/PaulWAyers/IntroQChem/blob/main/documents/DumontBook.pdf?raw=true">Randy’s book</a></p></li>
 <li><p>D. A. MacQuarrie, Quantum Chemistry (University Science Books, Mill Valley California, 1983)</p></li>
 <li><p><a class="reference external" href="https://github.com/PaulWAyers/IntroQChem/blob/main/documents/PinBox.pdf?raw=true">Mathematical Features of Quantum Mechanics</a> (my notes, starting page 6).</p></li>
-<li><p><a class="reference external" href="https://www.chem.iastate.edu/people/davit-potoyan">Davit Potoyan’s</a> Jupyter-book course discusses the posultates in <a class="reference external" href="https://dpotoyan.github.io/Chem324/Lec3-2.html">Chapter 3.2</a>.</p></li>
+<li><p><a class="reference external" href="https://www.chem.iastate.edu/people/davit-potoyan">Davit Potoyan’s</a> Jupyter-book course discusses the posultates in <a class="reference external" href="https://dpotoyan.github.io/Chem324/ch03/note03.html">Chapter 3.2</a>.</p></li>
 <li><p><a class="reference external" href="https://youtu.be/TQKELOE9eY4">TedEd video on the Heisenberg Uncertainty Principle</a></p></li>
 </ul>
 <p>There are also some excellent wikipedia articles:</p>