5½ Examples in Quantum Mechanics

The new theories, if one looks apart from their mathematical setting, are built up from physical concepts which cannot be explained in terms of things previously known to the student, which cannot even be explained adequately in words at all. Like the fundamental concepts (e.g., proximity, identity) which every one must learn on his arrival into the world, the newer concepts of physics can be mastered only by long familiarity with their properties and uses.

P.A.M. Dirac (1930) Preface The Principles of Quantum Mechanics

We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous with it... You know how it is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem... I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem.

R. P. Feynman as quoted in Genius (1992)


As the above quotations suggest, quantum mechanics is difficult, or perhaps impossible to understand (Steven Weinberg: "no one fully understands" [quantum mechanics]). Nevertheless, quantum mechanics is used on a daily basis by thousands of physicists, chemists and engineers. (E.g., Nobel prize winning chemist Linus Pauling: "There is no part of chemistry that does not depend, in its fundamental theory, upon quantum principles.") The ability to use quantum mechanics depends in part on mechanical mathematical skills ("doing the algebra", now made much easier with programs such as Mathematica) but more importantly on "physical intuition". Unfortunately our bedrock intuition -- based on classical mechanics -- is often at odds with quantum mechanics. This does not mean you must discard your hard-won classical intuition, rather you should treasure those parts that survive into quantum mechanics. These pages try find common ground between Newton's explanations and Schrödinger's explanations. In addition because of the revolution in computer algebra (e.g., Mathematica), these pages try to formulate problems in ways that allow easy translation into Mathematica code. Often simple practices (like the use of dimensionless variables) make understanding easier for both the computer and the human. Finally I should note that these pages are aimed at folks who want to do quantum mechanics. Like Dirac, I believe that the apprentice quantum mechanic gains facility by practice. I hope that you will do the examples, not just read the examples. A set of problems at the end of each "chapter" provide extentions of the examples. Feel free to write me if you have questions or comments!

Introduction: Classical Mechanics First!

Usually books on quantum mechanics start with quantum mechanics. Instead I start with a brief review of examples in classical mechanics. Note that all of the below descriptions of classical motion report how the particle's position and velocity change in time. Quantum descriptions must be quite different because quantum mechanics asserts that a particle does not have a position and a velocity. Instead the particle has, in some sense, simultaneously a range of possible positions and velocities. The particle has some chance of being found here, another chance of being found there, etc. Position is not a property that a particle has any more than a die has the property of the number that comes up. The die actually has is a range of possible outcomes (1-6) with a probability for each outcome (1/6). So too with position, but rather than just having a finite list of possibilities, usually a particle's position will be found within a continuous range of possibilities. Thus we seek the probability density for a particle's position. The aim of quantum mechanics is to calculate this range of possible particle positions and the relative probability of those positions. This sounds nothing like classical mechanics!

In classical mechanics if we say that the particle has a position of 100±1, we mean that the particle has a position in the range: 99-101, we're just not sure where. In quantum mechanics if we say that the particle has a position of 100±1, we mean that the particle is simultaneously all over the range: 99-101. This ability to be "spread out" is not surprising for waves (in fact it would make little sense to say the wave is localized to a region smaller than a wavelength... a wave needs at least a wavelength to "wave"); here we apply this wave property to things like electrons, which are traditionally called "particles".

Almost all standard courses in calculus-based introductory physics describe three examples of motion:
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