Ch. 1 notes, part1 1 of 20 8/17/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Quantum Mechanics Quantum Mechanics Introductory Remarks: Q.M. is a new (and absolutely necessary) way of predicting the behavior of microscopic objects. It is based on several radical, and generally also counter-intuitive, ideas: 1) Many aspects of the world are essentially probabilistic, not deterministic. 2) Some aspects of the world are essentially discontinuous Bohr: "Those who are not shocked when they first come across quantum theory cannot possibly have understood it." Humans have divided physics into a few artificial categories, called theories, such as • classical mechanics (non-relativistic and relativistic) • electricity & magnetism (classical version) • quantum mechanics (non-relativistic) • general relativity (theory of gravity) • thermodynamics and statistical mechanics • quantum electrodynamics and quantum chromodynamics (relativistic versions of quantum mechanics) Each of these theories can be taught without much reference to the others. (Whether any theory can be learned that way is another question.) This is a bad way to teach and view physics, of course, since we live in a single universe that must obey one set of rules. Really smart students look for the connections between apparently different topics. We can only really learn a concept by seeing it in context, that is, by answering the question: how does this new concept fit in with other, previously learned, concepts? Each of these theories, non-relativistic classical mechanics for instance, must rest on a set of statements called axioms or postulates or laws. Laws or Postulates are statements that are presented without proof; they cannot be proven; we believe them to be true because they have been experimentally verified. (E.g. Newton's 2nd Law, € Fnet= ma, is a postulate; it cannot be proven from more fundamental relations. We believe it is true because it has been abundantly verified by experiment. ) Actually, Newton's 2nd Law has a limited regime of validity. If you consider objects going very fast (approaching the speed of light) or very small (microscopic, atomic), then this "law" begins to make predictions that conflict with experiment. However, within its regime of validity, classical mechanics is quite correct; it works so well that we can use it to predict the time of a solar eclipse to the nearest second, hundreds of year in advance. It works so well, that we can send a probe to Pluto and have it arrive right on target, right on schedule, 8 years after launch. Classical mechanics is not wrong; it is just incomplete. If you stay within its well-prescribed limits, it is correct.Ch. 1 notes, part1 2 of 20 8/17/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Each of our theories, except relativistic Quantum Mechanics, has a limited regime of validity. As far as we can tell (to date), QM (relativistic version) is perfectly correct. It works for all situations, no matter how small or how fast. Well... this is not quite true: no one knows how to properly describe gravity using QM, but everyone believes that the basic framework of QM is so robust and correct, that eventually gravity will be successfully folded into QM without requiring a fundamental overhaul of our present understanding of QM. String theory is our current best attempt to combine General Relativity and QM, but "String Theory" is perhaps not yet really a theory, since it cannot yet make (many) predictions that can be checked experimentally. Roughly speaking, our knowledge can be divided into regimes like so: In this course, we will mainly be restricting ourselves to the upper left quadrant of this figure. However, we will show how non-relativistic QM is completely compatible with non-relativistic classical mechanics. (We will show how QM agrees with classical mechanics, in the limit of macroscopic objects.) In order to get some perspective, let's step back, and ask What is classical mechanics (C.M.)? It is, most simply put, the study of how things move! Given a force, what is the motion? So, C.M. studies ballistics, pendula, simple harmonic motion, electrons in E and B fields, etc. Then, one might use the concept of energy (and conservation laws) to make life easier. This leads to new tools beyond just Newton's laws: e.g. the Lagrangian, L, and the Hamiltonian,H, describe systems in terms of different (but still conventional) variables. With these, C.M. becomes more economical, and solving problems is often simpler. (At the possible cost of being more formal) Of course, what one is doing is fundamentally the same as Newton's F=ma! Mechanics (non-relativistic) speed c 0 big small 1/size Relativistic Mechanics QM (non-relativistic) Relativistic QMCh. 1 notes, part1 3 of 20 8/17/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) The equations of motion are given in these various formalisms by equations like: € ddt∂L∂˙ x −∂L∂x= 0 or ∂H∂x= −px∂H∂px= x       , or F = ma (If you've forgotten the Lagrangian or Hamiltonian approaches, it's ok for now…) Just realize that the general goal of C.M. is to find the equation of motion of pointlike objects: Given initial conditions, find x(t) and p(t), position and momentum, as a function of time. Then, you can add complications: E.g. allow for more complicated bodies which are not pointlike, ask questions about rotation (introduce the moment of inertia, and angular moment L=rxp), move to many-body systems (normal modes), etc… Q.M. is about the same basic thing: Given a potential, what is the motion? It's just that Q.M. tends to focus on small systems. (Technically, systems with small action, € Ldt∫<≈ ) And the idea of "motion" will have to be generalized a bit (as we shall see soon!) Having just completed C.M., your initial reaction may be "but, size doesn't matter"! After all, neither L nor H cares about size, and C.M. often deals with so called "point objects". (Isn't a point plenty small?!) Unfortunately, it turns out that in a certain sense, everything you learned in 2210 and 3210 is WRONG! To be a little more fair, those techniques are fine, but only if applied to real-world sized objects. (As I said above, there's a regime of validity) Size doesn't matter up to a point, but ultimately, C.M. breaks down: if