Author Archives: wdengquantum.me

The purpose of this post is to explain the Born-Oppenheimer approximation. Ignoring relativistic fine/hyperfine structure effects, the gross structure molecular Hamiltonian \(H\) (i.e. the “theory of everything”) is: \[H=T_{\text n}+H_{\text e}\] where the nuclear kinetic energy is: \[T_{\text n}=\sum_{i}\frac{\textbf P_i^2}{2M_i}\] … Continue reading →

In this interview, well-known theoretical physicist Frank Wilczek commented that “still to this day I think the quantum theory of angular momentum is one of the absolute pinnacles of human achievement. Just beautiful”. The purpose of this post is to … Continue reading →

The purpose of this post will be to review a standard method for passing from the classical to the quantum world (or more poetically, turning \(\hbar=0\) on to \(\hbar=1\) in natural units). This procedure is called canonical quantization. It is … Continue reading →

The Shapes and Structures of Molecules Also, spin angular momentum coupling between \(^{13}\text C\) nuclei and \(^1\text H\) nuclei are typically suppressed via broadband proton decoupling. Also, due to low isotopic abundance, spin angular momentum coupling among \(^{13}\text C\) nuclei … Continue reading →

If one has a collection \(Q=\int\rho d^3x>0\) of positive charge \(\rho>0\) in some region of space, then the electric dipole moment \(\textbf p\) of \(Q\) may be viewed as \(\textbf p=Q\textbf X\), where \(\textbf X=\frac{1}{Q}\int\textbf x\rho d^3x\) is the center … Continue reading →

Problem: Describe the wrong way to define a (Bravais) lattice. Solution: A wrong way to define a Bravais lattice proceeds by \(3\) steps. First, one constructs the concept of a Delone set as any subset of a metric space with … Continue reading →

Problem: What does it mean for a collection of feature vectors \(\mathbf x_1,…,\mathbf x_{T}\) to represent a form of sequence data. Give some examples of sequence data. Solution: It means that the feature vectors are not i.i.d.; indeed, they are … Continue reading →

Problem: Where and why should one create an __init__.py file? Solution: Inside a folder/directory that’s meant to be a Python package containing a bunch of Python modules with useful functions, etc. that other Python scripts would be importing from (not … Continue reading →

The purpose of this post is to compile solutions to select exercises from Steven Strogatz’s textbook Chaos and Nonlinear Dynamics. Chapter #\(1\): Chapter #\(2\): Chapter #\(3\): Chapter #\(4\): Chapter #\(5\): Chapter #\(6\): Chapter #\(7\): Chapter #\(8\):