## Quantum Mechanics Courses in the Paradigms

We use an unusual content order for the early paradigms on quantum mechanics:

Nearly all of our students have had a descriptive course in Modern Physics, before they enter the paradigms, which covers a traditional historical approach to the Schroedinger equation, including the double-slit experiment and qualitative descriptions of the solutions of a particle-in-a-box.

In the paradigms, we start with a mathematical interlude (7 contact hours) covering basic linear algebra in both matrix and Dirac bra-ket language. Students learn that eigenvectors are the vectors that are unchanged by a transformation (or at most scaled) and how to find eigenvectors and eigenvalues for matrix operators.

Next comes an extended description of quantum measurement in the context of repeated Stern-Gerlach apparatuses. Many faculty believe that the topic of spin is too-abstract for students to grasp as their first substantive example of a quantum mechanical system. We have not found this to be the case. We devote an entire paradigm (21 contact hours) to spin 1/2 and spin 1 systems. This is the course where we introduce the postulates of quantum mechanics and students first do extensive calculations. Special features of this course are:

• Students use software that simulates a succession of Stern-Gerlach apparati.
• Some of the calculations that students do infer the quantum state for the spin 1/2 system from the results of simulated experiments, as happens in experiments; rather than predicting the results of experiments from a known quantum state, as only happens in textbook problems.

Immediately following the spins paradigm, is a paradigm on one-dimensional waves. Much of this content (10 contact hours) covers the classical mechanics of waves on ropes and waves in electrical circuits, introducing the basic conceptual and mathematical language used to describe waves: standing and traveling waves, wave packets and dispersion, energy, reflection and transmission, impedance, fourier analysis. This is followed by (11 contact hours) covering the quantum mechanics of the 1-dimensional Schroedinger equation, including reflection and transmission at a barrier, and the solution of a particle-in-a-finite box, including calculation of the matching conditions at the boundary. Special features of this course are:

• Students are immediately able to compare classical and quantum waves in the context of a single course.
• Students use both wave function and bra-ket notations to describe quantum states.

The third paradigm in this sequence is Central Forces. Students spend about 1/3 of their time (7 contact hours) exploring the classical mechanics of orbits and 2/3 of their time (14 contact hours) exploring the quantum mechanics of the unperturbed hydrogen atom.

• Students are immediately able to compare the classical and quantum aspects of central forces within a single course. Of particular interest are the similarities and differences in the ways that the symmetries of the problem lead to conservation of energy and conservation of angular momentum in the two contexts.
• The role of the effective potential in both the classical and quantum contexts is highlighted.
• The solutions of the Schroedinger equation for the hydrogen atom are built up, one dimension at a time, by first considering a particle confined to a ring, then a particle confined to a sphere, and finally, the complete hydrogen atom solutions.

In the spring term of the junior year, most students take a paradigm (21 contact hours) on Periodic Potentials. This course extends ideas from the waves paradigm to situations with a periodic potential and acts as our introduction to solid state physics.

By the end of the junior year, students have gained extensive experience with all of the basic one-particle quantum examples except the quantum harmonic oscillator. The Capstone in Quantum Mechanics (30 contact hours), is taken by most students in their senior year. This course starts by comparing and contrasting these simple quantum systems, with an emphasis on the role of eigenstates of the Hamiltonian, by examining the quantum harmonic oscillator and simultaneously reviewing the junior year examples. The course then combines these simple systems in various ways: two particles, the hydrogen atom coupled to spin, perturbation theory, etc.

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