In this video I demonstrate calculations of quantum harmonic oscillator in Excel.
For the best experience, I recommend viewing it full screen!
Lennard-Jones Potential Equation $$ V(r) = 4 \epsilon \biggl[ \biggl (\frac{\sigma}{r}\biggr)^{12} - \biggl (\frac{\sigma}{r}\biggr)^6 \biggr] $$
$$ \sigma = \frac{r_m}{2^{1/6}} $$
Here, epsilon is the energy minimum and r_m is the distance of the energy minimum.
Note the part of the equation inside the square brackets. Recall that negative energies represent a more favorable interaction. The attractive term (raised to the power of 6) is subtracted from the repulsive term (raised to the power of 12).
Introduction In my post “R: Deep Learning Organic Chemistry Again,” I trained a convolutional neural network based on VGG16 to recognize a benzene ring diagram, a crucial structure in many organic chemistry molecules. The classification problem I posed to the convnet was a binary classification to separate diagrams of molecules that contain a benzene ring from those that do not. However, near the end of that post, I found images I had mistakenly put in the wrong training and validation folders.
Introduction In my post “Python: Deep Learning Organic Chemistry," I trained a convolutional neural network to recognize a diagram of a benzene ring, which is a crucial structure in many organic chemistry molecules. The classification problem I posed to the convnet was a binary classification to separate diagrams of molecules that contain a benzene ring from those that do not. Using Python, TensorFlow, and Keras, my experiment proceeded in three steps:
In this post, I define a class to model the behavior of a hydrogen atom. In the process, I get to solve integrals like the following numerically to test my code:
\[ \int_0^{\pi} \int_0^{2\pi} \lvert Y_{l, m_l} \rvert ^2 \sin \theta d \theta d \phi = 1 \]
This post consists of a arge block of Python code up front, and then explanations and plots below. Let’s get started!
Hydrogen spectra using Rydberg Formula The Rydberg formula is used to predict emission spectrum lines from hydrogen. The significance of the Rydberg formula is that it was one of the first studies of quantum effects of energy transitions in atoms. Furthermore, it demonstrates that energy emitted is in specific wavelengths, corresponding to a particular energy level transition.
\[ \tilde\nu = R_H\Biggl(\frac{1}{n_{1}^2} + \frac{1}{n_{2}^2}\Biggr)\]
Where \(\) is the wavenumber in \(cm^{-1}\) and \(R_H = 109,677 cm^{-1}\).
Introduction In this post, I will define Python code that models the quantum harmonic oscillator. This page follows page 290 to 297 in Physical Chemistry, 8th Ed. by Peter Atkins and Julio de Paula for the math to create and examples to test the code in this post.
The following equations describe its energy levels:
\[ \omega = \sqrt{\frac{k}{m}} \]
\[ E_{\nu} = \Bigl(\nu + \frac{1}{2}\Bigr) \hbar \omega \]
