Does quantum mechanics need complex numbers?

Question

the Schrödinger equation, has an i explicitly in it and y is inherently a complex-valued function.

Does quantum mechanics need complex numbers? Couldn't one express y as the linear combination of two real fields, f and g, where y = f + i g. By substituting this expression into the Schrödinger equation, you could develop two equations for the two unknown fields f and g. Do you think that quantum mechanics demands complex numbers, or complex numbers are simply a convenient way of expressing things?

Answer 1

That's a very interesting question. Most applications of complex numbers, like AC circuit theory, use them as just a mathematical trick to get a useful result easily. Phase-shifted alternating current doesn't really have an imaginary component. But the one exception in the whole of physics that I'm aware of, is Schroedinger's equation. The wave function psi really does have a real and imaginary component. There's no other way to express it. And when you multiply psi by its complex conjugate, you get a simple, easily-understood physical quantity; the probability of finding the particle at a particular location in space. Philosophically your question is very significant.

Answer 2

it really is conceivable to describe QM with little more beneficial than Algebra lower than your belt. in spite of the indisputable fact that, that understanding would not come til you draw close Diff Eq's. i'd advise starting up with a psychological workout, like explaining how regulation of Conservation and Uncertainty account for textile manifestations. no count number how stepped ahead that sounds, any clarification will bring about further understanding. Then, try employing that understand-the thanks to sorts of fields, consisting of the Quantum plane. if you're nevertheless with me, arising a mathematical formulation on your observations will boost your understanding 100 fold.