Quantum Mechanics #

This section describes topics related to quantum mechanics.

perturbation: perturbation method # Time-Independent Perturbation Theory (Non-Degenerate) Time-Independent Perturbation Theory (Non-Degenerate) # This content is primarily based on Section 8, “Approximation Method,” in Chapter 2 of Physics of Atoms and Molecules (2nd Edition) by B. H. Bransden and C. J. Joachain1. Physics of Atoms and Molecules B. H. Bransden and C. J. Joachain, Benjamin Cummings Amazon 1. Summary # We aim to solve the time-independent Schrödinger equation (TISE): $$ \begin{align} (\hat{H}_0+\hat{H}’)\varphi = E \varphi \end{align} $$ Two-Level System with Periodic Perturbation Two-Level System with Periodic Perturbation # This content is primarily based on Section 8, “Approximation Method,” of Chapter 2 in Physics of Atoms and Molecules (2nd Edition) by B.
Bra-Ket Notation: Bra-Ket Notation # This section describes the properties and handling of the bra-ket notation proposed by Paul Dirac in 1939. Operations in Bra-Ket Notation Operations in Bra-Ket Notation # This article describes how to handle bra-ket notation that appears in quantum mechanics. It focuses on the possible operations rather than providing a detailed explanation of the notation itself. 1. Representation of Functions # $$ \begin{align} \psi(\mathbf{r}) &= \langle\mathbf{r}|\psi\rangle \label{e1}\\ \psi(\mathbf{p}) &= \langle\mathbf{p}|\psi\rangle \label{e2} \end{align} $$ $|\psi\rangle$: Description of a state without using a specific basis1 $\psi(\mathbf{r})$: State description in position representation $\psi(\mathbf{p})$: State description in momentum representation Position and Momentum Representation of States The goal of this article is to derive $\langle x|p\rangle = e^{ipx}/\sqrt{2\pi\hbar}$ and understand the transformation between position and momentum representations.
Numerical method in quantum mechanics: Potential Scattering # In this section, we will address scattering problems. Although primarily concerned with quantum scattering, we will also deal with classical scattering problems in some cases. 1D-TISE Numerical Method for One-dimensional Time-independent Schrödinger Equation # This section describes the numerical solution methods for the one-dimensional time-independent Schrödinger equation. Finite Difference Finite Difference Method # The finite difference method approximates differential equations with difference equations for numerical solutions. A web tool implementing this method is available at the link below: Solving the 1D Time-Independent Schrödinger Equation Using the Finite Difference Method 1.
Potential Scattering: Potential Scattering # In this section, we will address scattering problems. Although primarily concerned with quantum scattering, we will also deal with classical scattering problems in some cases. Scattering by a Coulomb potential The pure Coulomb potential is a long-range potential and, unlike short-range potentials, it exerts an influence over infinite distances. Therefore, it is not possible to use the boundary condition that the potential is zero at a large distance and a different approach is required.
TISE Examples: Specific Solutions of the Time-Independent Schrödinger Equation # 1. 1D TISE # $$ \begin{align} \left[-\frac{1}{2}\frac{d^2}{dx^2}+V(x)\right]\psi(x)=E\psi(x) \end{align} $$ You can experiment with these solutions using the following web tool: 1D Time-Independent Schrödinger Equation Solver via Finite Difference Method. 1.1. Infinite Square Well Potential # Potential $$ \begin{eqnarray} V(x)=\left\{ \begin{aligned} &0 &&( 0\lt x \lt L)\\ &\infty &&( x\le 0\hspace{0.5em} \text{or}\hspace{0.5em} L\le x) \end{aligned} \right. \end{eqnarray} $$ Eigenvalues $$ \begin{align} E_n=\frac{(n+1)^2\pi^2}{2 L^2},\hspace{2em}(n=0,1,2,\cdots) \end{align} $$