# Eigenvalues and eigenfunctions

Computing the eigenvectors $\psi(x)$ and eigenvalues $E$ of some Hamiltonian $H$ belongs to the key problems of quantum mechanics. For the one-dimensional Schrödinger equation we have to solve $$E\psi(x) = H\psi(x) = -\frac{1}{2}\frac{\mathrm{d}^2\psi(x)}{\mathrm{d} x^2} + V(x)\psi(x)$$for some given potential $V(x)$. Analytical solutions, however, are scare goods. Thus, one has to rely on approximations… Continue reading Eigenvalues and eigenfunctions

Python

# Visualizing streamlines

In Visualizing vector fields I showed how to plot vector fields using Python and Matplotlib. Streamlines are a concept that is closely related to vector fields. Mathematically speaking streamlines are continuous lines whose tangent at each point is given by a vector field. Each line, and therefore also streamlines, can be parametrized by some parameter $t$.… Continue reading Visualizing streamlines

Python

# Visualizing vector fields

A vector field assigns to every point in space some vector. Vector fields are used in many branches of physics, e.g., to describe force fields, electromagnetic fields or velocity fields. Two-dimensional vector fields can be easily visualized using Python and the popular matplotlib package. As an example let us visualize the vector field \vec… Continue reading Visualizing vector fields

Python

# Point and interval estimation in Python

Let us consider a random sample $x_1,x_2,\dots,x_N$ drawn from a normal distribution with unknown mean $\mu$ and unknown variance $\sigma^2$. The best point estimates for the mean and the variance are given by the sample mean$$\hat\mu = \frac{1}{N}\sum_{i=1}^{N} x_i$$ and the sample variance $$\hat\sigma^2 = \frac{1}{N-1}\sum_{i=1}^{N} (x_i-\hat\mu)^2\,,$$ respectively. The sample mean… Continue reading Point and interval estimation in Python

# Scientific Python

From time to time students ask me what might be the best programming language for computational physics and numerical applications. For a long time my standard answer to this question was twofold: The language does not matter, learn C++! On the one hand, I think one should not bother to much with this issue because… Continue reading Scientific Python