* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Line integrals
Survey
Document related concepts
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Casimir effect wikipedia , lookup
Hidden variable theory wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Feynman diagram wikipedia , lookup
Double-slit experiment wikipedia , lookup
Atomic theory wikipedia , lookup
Bra–ket notation wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
History of quantum field theory wikipedia , lookup
Canonical quantization wikipedia , lookup
Renormalization wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Transcript
Lecture 3: Line Integrals • We start with two (atypical) examples where integrand is (i) a scalar field, integrated w.r.t. a scalar 2 1 and L goes from point 1 to point 2 In general, it’s the area under the following curve 1 Physical example might be mass of a “chain” where 2 is mass/unit length in plane z=d y • (ii) integrand is (ii) a vector field, integrated w.r.t. a scalar increasing t x Integral is a vector Physical example might be “average” vector force on a moving particle Common type of line integral 2 For example, work done by a force is Add up all the which is the work done by the force during a small displacement Note that no work is done if the everywhere orthogonal and are This underlies an interesting “problem” in quantum mechanics. Early theoretical ideas for quantum mechanics suggested that electrons moved in perfect circles around the Hydrogen nucleus. No work is done ( ) so how does the electron radiate energy? If is not always perpendicular to , how does the electron avoid spiralling into the nucleus? 1 Example Quic kTime™ and a TIFF (Unc ompres sed) dec ompres sor are needed to see this pic ture. (2,1) B2 Route A (1,0) B1 Route B = B1+B2. B1 B2 Since this line integral depends on the path of integration. (2,0) Conservative fields QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. • Important difference between line integrals that (can) depend on both start/end points and path, and those that depend only on start/end points 2 • Consider exact differential of some function 1 is independent of path, and we can construct a contour map of • and vector field is said to be Conservative • Around any closed loop, we begin and end at same point Examples QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Could write as So is independent of path • Commonly in physics we have scalar potential functions (normally, but not exclusively) potentials are single-valued functions of position Electric Field +ve near +ve Q Gravitational Field -ve near +ve M • In both cases defines zero point of potential r Inverse Square Law Due to +ve charge at origin. Take another +ve charge from point 1 to infinity and back again (all in z=0 plane). Path A, along x axis (a,,0) y (a,0,0) x (,0,0) So work done against force is Path B, around large loop, force and path perpendicular so no work done Path C, parallel to y axis So work done against force is Total work done is zero, as expected for a closed-loop line integral in a conservative vector field Summary In a Conservative Vector Field Which gives an easy way of evaluating line integrals: regardless of path, it is difference of potentials at points 1 and 2. Obvious provided potential is single-valued at the start and end point of the closed loop.