Classical to Quantum | Complex numbers in Fourier Series and Quantum Mechanics | Wild Egg Maths

This is a video from the Playlist "Classical to Quantum" which is at our sister channel Wild Egg Maths. In this series we are planning on looking at a variety of topics in modern physics, particularly Relativity, Quantum Mechanics and the Standard Model, from a pure maths viewpoint. Currently we are looking at Harmonic Analysis on Circles and Spheres, and showing how to rethink this theory using only a rational algebraic approach. This means no "real numbers", no "transcendental functions" and no "infinite processes"! This will present a major new opportunity for both pure mathematicss and theoretical physics research, and will extend in a variety of directions. In this video we introduce complex numbers as simplifying agents to understand just the elementary harmonic analysis on the unit circle in the plane. The Laplacian operator plays an important role, as does the notion of a harmonic polynomial (or polynumber as we prefer to call them). It will turn out that this approach allows us to completely rethink also integration theory on the circle, and in fact spheres and hyperbolas more generally. Check out the entire Playlist for "Classical to Quantum", (Members of Wild Egg Maths channel): https://www.youtube.com/watch?v=BL9GzEBtHrY&list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy Video Contents: 00:00 Fourier Series and spheres 4:13 The 0 Dimensional Sphere 6:23 A containment hierarchy 10:22 Hyperboloids 13:43 Bi-polynumbers 18:05 Why complex numbers in QM? 20:23 Complex bi-polys 23:20 The complex basis 24:24 Proof of fundamental differential relation 30:34 Harmonic bi-polynumbers *********************** My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated. Here are all the Insights into Mathematics Playlists: Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316 Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47 MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf Box Arithmetic: https://www.youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist? list=PL8403C2F0C89B1333 Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua Sociology and Pure Physics: https://www.youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist? list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8 Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF Playing GO: https://www.youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW Diffusion Symmetry: A bridge between mathematics and physics: https://www.youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7 ************************