A brief introduction to the use of complex numbers in quantum mechanics. This video is intended mostly for people who are learning quantum mechanics and have some familiarity with things like the quantum harmonic oscillator, or the hydrogen atom, but might have some confusion around what all the complex numbers are all about. I hope this video provides you with an improved sense of familiarity with the complex numbers. These things are cool. They take a bit of getting used to, but they're cool.
My main goal in this video is to make the complex numbers feel as natural and accessible as possible, so I emphasize the perspective that the complex phase can be thought of as a generalization of positivity and negativity, and in particular that the phase oscillates between two poles (which I half-jokingly refer to as yin and yang). This approach, though real-part-biased, is motivated by the observation that the interference of two waves of the same frequency (constructive, destructive, and everything in between) provides a natural picture for one of the things the phase of a complex number might mean. I hope that helps to demystify how complex numbers are not an entirely absurd concept, because a stumbling block for many people, myself included for a while, is that the complex numbers seem too unrealistic for human intuition to sincerely glom on to. But, as I hope this video shows, the complex numbers can be made intuitive.
It should be noted, however, that the story does not end here. Once you are familiar with the complex numbers, you should stretch your mind out again by regarding the complex numbers as equipping a model with a circular degree of freedom. In particular, you can imagine a wavefunction as a section of a fiber bundle whose base is spacetime, whose fibers are circles of mysterious origin, and whose total space is some fragment of this thing we call reality. That should keep you up at night!
I should also add that the "U(1) Symmetry implies Electromagnetism" argument may well be completely backwards. It is true that, if one takes the Dirac field with minimal coupling to the photon field, and imposes local U(1) symmetry by fiat, then all the beauty of classical electromagnetism follows. But one can easily argue that such an imposition is contrived, and more indicative of a redundancy of our model than a genuine symmetry of physics. That argument is strengthened in light of Wigner's classification, pun proudly intended, since if we take the masslessness of the photon as our starting point, then the photon can only have helicity eigenvalues of +-1, not 0 (the photon has no rest frame), and therefore one must remove any physical contributions coming from longitudinal photon modes, since they cannot exist. This fictionalization of the longitudinal modes yields precisely the usual gauge symmetry of the four-potential (or so I am told... still need to work out for myself why this is true), and once you have the gauge symmetry of the four-potential, then your Dirac field better have local U(1) symmetry if you want to preserve minimal coupling!
Anyway, whichever direction of the argument is more true, it is still a beautiful idea that local U(1) symmetry of the Dirac field, and the usual gauge symmetry of A, and the masslessness of the photon are, for all intents and purposes, the same thing. It is still an open philosophical question as to whether all this symmetry and gauge freedom is a genuine reflection of natural symmetry, or of mere theoretical redundancy; that question boils down to whether the transformations involved are active or passive, respectively, and that quickly gets into some murky existential territory when you really think about it. Fiery debates are ongoing around these questions. But that's a topic for another time, and not one which is answerable within a RU-vid video description.
Thanks for watching & reading :)
Chapters:
0:00 Introduction
1:00 Real vs. Complex Numbers
2:48 A Wavy Wave, Waving
4:33 Complex Representation of the Wave
7:48 Complex Addition, Multiplication, and Interference
12:10 Fourier Analysis & Superpositions
12:47 Examples: Harmonic Oscillator and Hydrogen
14:30 Plane Waves
16:49 Probability Density
18:07 U(1) Symmetry Implies Electromagnetism
#physics #quantum #math
3 май 2024
