Author: Haroon Khalil
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Electron Clouds
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,…
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The Hydrogen Atom
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…
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The Particle on a Spring
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…
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Barriers and Tunneling
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…
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The Particle in a Box
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…
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The Free Particle
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…
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Introduction
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…
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Heisenberg’s Matrix Mechanics
It is interesting to note that Werner Heisenberg was working on his own quantum mechanics at the same time Schroedinger was coming up with his equation. In fact, together with colleagues Max Born and Pascual Jordan, Heisenberg actually developed and published his theory a short time before Schroedinger. This alternative theory was based on the…
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Probability and Expectation Values
What the Schroedinger theory doesn’t do is provide us with a physical interpretation for the value of the wave function Ψ at any particular location and time. Because it involves imaginary numbers, we can’t expect to assign any measurable property to Ψ anyway, so maybe it is just as well. Wave functions can be deduced…
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Eigenfunctions, Eigenvalues, and Quantization
So far, this could be just a bunch of abstract math. These rules apply to any differential equation of the kind we are discussing. But we are interested in what is physically represented by the quantities in the equation. Well, as we’ve already alluded, Schroedinger was able to show that the right side of his…