week1 - 1 Introduction/Series 1.1 Why this course exists...

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Unformatted text preview: 1 Introduction/Series 1.1 Why this course exists PHZ3113 was created in the early 1990s in the department of physics as part of a general revamping of the undergraduate physics curriculum. The perception of many faculty at that time was that more and more students were arriving at so-called ”majors” courses like statistical mechanics and quantum mechanics math- ematically unprepared to understand lectures and perform exercises. One generally recognized problem was what a physicists would refer to as an “impedance mis- match” (i.e., difficulty of transporting information, energy, current etc. through an interface between two media because of very different properties of these media) between the physics and mathematics curricula at UF. As at many universities, mathematics is generally taught from the point of view of a mathematician, i.e. with a view to the formal structure of the subject and emphasizing aspects of in- terest to majors who might be considering doing research in the field. Physicists teaching 15 years ago found that students had an appreciation for the beauty of mathematics, and could, under some circumstances, prove some elementary the- orems, but were hard pressed to integrate the function sin x , much less solve the differential equation for a damped harmonic oscillator. Thus PHZ3113 was cre- ated to teach students, not just from physics but other scientific or engineering disciplines as well, the nuts and bolts of performing mathematical operations of use to our field, so that when they encounter a Hamiltonian matrix in PHY4604 (Quantum mechanics), they will know in their gut what a matrix is, how to find its eigenvalues and eigenvectors, how to invert it, and take its Hermitian conjugate, not just be able to prove that the dual of a monomorphism is an epimorphism (aside: your instructor remembers this statement from his undergrad linear algebra course, but not what it means). At the same time, teaching this material gives us the opportunity to explore aspects of the relationship between physics and mathematics which go well beyond nuts and bolts. Mathematics is sometimes called the language of physics, but can also be thought of as an infrastructure, a sort of highway system which connects different branches of physics, enabling insights from one to be exported from one to another. Think of an example you already know: the remarkable set of conclusions which follow from Gauss’s law in electrostatics, which are rigorously analogous to those derived by Newton for the gravitational force. At the level of laboratory physics, gravity and electromagnetism are completely distinct physical phenomena, yet they are bound together by a common mathematical framework. The fields 1 associated with a mass and a charge both fall off with the inverse square of the distance: E grav = Gm r 2 E Coul = 1 4 π q r 2 , (1) where q is the charge, m is the mass, and 1 / (4 π and G are the universal constants of electrostatics and gravity, respectively. But forof electrostatics and gravity, respectively....
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This note was uploaded on 04/17/2008 for the course PHZ 3113 taught by Professor Hirshfeld during the Fall '07 term at University of Florida.

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week1 - 1 Introduction/Series 1.1 Why this course exists...

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