David J. Raymond and Alan M. Blyth
Physics Department
New Mexico Tech
Socorro, New Mexico
raymond@kestrel.nmt.edu
Mechanics is generally considered to be the mother of all physics.
However, seeing the origins of quantum mechanics in classical
mechanics is no easy task -- on the conventional track one first
learns Newtonian mechanics, followed by Lagrangian mechanics, followed
by Hamiltonian mechanics, followed by Hamilton-Jacobi theory. At the
very end of Goldstein's graduate mechanics text we finally learn that
classical mechanics is isomorphic to geometrical optics, from which we
infer the existence of a ``physical optics'' of material particles,
i. e., quantum mechanics.
Louis de Broglie showed us an alternate path in his Nobel Prize Lecture, but few of us have been listening. De Broglie is known for the relationship between momentum and wavelength in elementary texts, but the reasoning which leads to this relationship is rarely discussed. This reasoning is based on an elegant and subtle application of the principle of relativity which is nevertheless accessible even to beginning physics students. Furthermore, de Broglie laid the groundwork for the Schrödinger equation via an analogy between the behavior of light in a medium with variable refractive index and the motion of a material particle in a spatially varying potential.
We have developed a new introductory physics course called ``A Radically Modern Approach to Physics'' which attempts to take advantage of de Broglie's insights. The course begins with optics and waves where key ideas about diffraction, interference, dispersion relations, phase and group velocity, and geometrical optics are introduced. Relativistic kinematics is covered next. De Broglie's argument is then invoked to obtain the dispersion relation for the free relativistic matter wave. Bragg scattering of X-rays and electrons is then used to tie these ideas to observation, resulting in the introduction of the idea of mass as the relativistic scalar proportional to the ``length'' of the four-momentum and wave four-vector. Elementary quantum mechanics follows.
Asking how one can introduce interactions between particles leads to the idea of potential energy. Extending this to the relativistic domain makes the existence of a ``potential momentum'' inevitable. From here it is a short jump to gauge theory, the elementary consequences of which are developed using analogs of Snell's law. Electromagnetism falls out of this discussion in a natural way, as does a qualitative but nontrivial discussion of the Standard Model.
Mechanics is treated as the geometrical optics limit of quantum mechanics, a line of inference which is much easier to develop than the reverse sequence. As little or as much mechanics is then done as time and the inclinations of the teacher allow. Since the idea of quantum mechanical states is familiar to the students, the development of some of the elementary ideas of statistical mechanics, e. g., the relationship between the number of accessible states and entropy, is straightforward. Frictional macroscopic systems and heat engines are also introduced at this point, rounding out the course.
Physics is not a deductive subject like mathematics. However, in our opinion a logical as opposed to a historical development appeals more to the students and allows a more compact presentation, which means that more of physics can be introduced in the usual two semesters available to the introductory course without overloading the student. Most importantly, non-physics majors get an overview of all of physics from a modern point of view, while hopefully acquiring the skills needed for engineering and other technical disciplines. Our experience with this course leads us to believe that optics and relativistic kinematics together constitute a much firmer basis for physics than does classical mechanics.
Student reactions to the course have been generally enthusiastic and we are in the process of evaluating the ability of the course to compete with traditional courses in classical areas using standard testing tools. If these results are satisfactory, we would like to make this course our standard introductory course.