Category: Wave Functions Illustrated

Principal Quantum Number n Angular Momentum Magnitude l Angular Momentum Projection m 1 0 0 2 0 0 2 1 -1, 0, 1 3 0 0 3 1 -1, 0, 1 3 2 -2, -1, 0, 1, 2 4 0 0 4 1 -1, 0, 1 4 2 -2, -1, 0, 1, 2 4 3 -3, -2, -1,… Continue reading The Quantum Numbers of the Hydrogen Atom

When we presented our examples in one dimension, the Schroedinger equation gave us bound states with quantized energy. In case you didn’t notice, only one quantity ended up being quantized: energy. Why do we care? Well, it turns out that when you solve the Schroedinger equation in higher dimensions, you end up with more than… Continue reading Quantum Numbers and Degeneracy

Once again, the energy of the electron can only take certain values; energy is quantized. This happens because only certain wave functions satisfy the Schroedinger equation and all boundary conditions, and these Eigenfunctions each have their own energy Eigenvalues. The great thing for fans of quantum physics is that the spacing of these energies that… Continue reading The Bohr Atom Revisited

Because we know exactly what the electric field looks like surrounding a positively charged proton, it is easy to specify the potential energy we need to put into the three-dimensional version of the Schroedinger equation. But now it turns out to be more useful to use a potential energy scale where zero is the maximum,… Continue reading Electron Clouds

At this point, we’ve more or less exhausted the number of interesting examples for one-dimensional quantum systems. Since we live in a three-dimensional world, let’s go there to see if quantum physics has anything still hidden up its sleeve. Fortunately, nature is full of three-dimensional systems to explore. The most instructive problem is that of… Continue reading The Hydrogen Atom

High school and college physics teachers love to torture their students with something known as the “simple harmonic oscillator.” A good, classical example is a simple mass attached to an idealized spring that oscillates back and forth when given a little tug. The idealized spring is one which doesn’t have any mass itself and does… Continue reading The Particle on a Spring

We’ve just demonstrated that quantum particles can actually seep into classically forbidden regions. We’ll delve into the philosophical aspects of that soon enough, but for now we simply ask, could this strange behavior actually be detected in the laboratory? To answer that, let’s consider a slightly different case–that of a potential “wall” instead of a… Continue reading Barriers and Tunneling

In the case of a free particle, Schroedinger had little to offer that hadn’t already been understood since de Broglie’s time. To get the real bang for Schroedinger’s buck, we need a more interesting potential energy function. The simplest case would be a region of zero potential energy surrounded by a region of constant but… Continue reading The Particle in a Box

The simplest case we can look at is that of a free particle—that is, a particle traveling through space under no influence of external forces. A real-world example would be the electrons that emerged from J. J. Thomson’s cathode ray tube before they were acted upon by his electromagnetic fields. To find the wave function in… Continue reading The Free Particle

We will take a brief detour from the quantum expressway to enjoy a scenic route past a number of beautiful wave functions. We will look at several specific solutions of the Schroedinger equation, and see how different wave functions emerge under different conditions. We’ll ditch the math on this detour and paint a few portraits… Continue reading Introduction