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Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a
by midjji 5y ago
Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.
- kevin_thibedeau 5y ago'i' is a 90-degree rotation. It isn't hidden at all. Similarly, everywhere a complex exponential shows up, there's also some spinny/rotational thing happening.
- gfody 5y agoyou can always imagine another dimension, with all the symmetry of every dimension so far plus some novel one
- dreamcompiler 5y agoThe other side of that same coin is that we're describing wave functions of probability (with interference between probabilities) and complex numbers are exceedingly handy for wave functions, as any EE can attest.
- phkahler 5y ago>> complex numbers are exceedingly handy for wave functions, as any EE can attest. Because they encode phase information. Also because they come about in the solution of differential equations. Physicists often talk about amplitudes, but I never hear them talk about phase. There was one paper that I can't find, complete with a diagram that suggested (to me) that phase was determining quite a bit.
- kgwgk 5y agoA wave function describes - usually at least - the quantum state of an isolated quantum system. The phase has no physical meaning. The relative phase between wave functions could mean something... but not if the systems are isolated.
- dreamcompiler 5y agoPhase doesn't really matter much in EE for an isolated sinusoid either. But when you compare the phase of a transmitted signal to a local reference, or you compare the phases of the sinusoids in an FFT to one another, or you're trying to synchronize the Texas electrical grid to the rest of the country, phase means a lot. Phases are almost always only useful in a relative sense.
- GoblinSlayer 5y agoRotational symmetry a straightforward consequence of the wave equation.
- azalemeth 5y agoThere are lots of equivalent representations of the same thing, and for non relativistic QM i is by far the simplest way of proceeding. Part of learning QM is learning that changing how you view the world without actually changing the world is a very powerful thing and sometimes more complex ways of thinking are actually easier "later on". A simple example is the equivalence of two complex parameters, alpha and beta, arranged in a 2-matrix and a real 3-matrix for the representation of rotation. Another example would be ladder operators for the simple harmonic oscillator -- arguably overcomplicated for the problem at hand, they form the basis of much of what follow (i.e. vacuum creation and annihilation operators). The whole point of the notational soup that one finds e.g. in an extended field theory is that it correctly generates a lot of these details "automatically". It's obtuse and makes doing simple things hard, but makes showing non-trivial relationships that are true in general very much easier than the alternative ;-).