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My professor argues that one of the fundamentally unique properties of Quantum Mechanics is that the imaginary unit i is not removable (you can't avoid using it, unlike in other areas of physics like AC circuits where it is merely a convenience).

However, I'm not convinced. Isn't the reason i appears in the schrodinger equation that the equation needs to relate the wavefunction to its derivative? For convenience, we represent the wavefunction as a (possibly infinite) linear combination of complex exponentials, and this makes taking the derivative extremely easy which is where the factor of i comes from. However, can't complex exponentials always be replaced by sines and cosines? Wouldn't there be some way to state schrodinger's equation without using i as a result? The i merely represents the phase shift between the wavefunction and its derivative, so it makes the schrodinger equation much easier to write down (it would be more difficult to write a differential equation with an explicit phase shift). However, I don't see why a version of the schrodinger equation communicating the phase shift directly rather than by using i would be any less correct. Can you provide an argument for why the use of complex numbers is essential in quantum mechanics?

Marko Bakić
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    Possible duplicates: http://physics.stackexchange.com/q/8062/2451 , http://physics.stackexchange.com/q/32422/2451 , http://physics.stackexchange.com/q/11396/2451 , http://physics.stackexchange.com/q/105278/2451 and links therein. – Qmechanic Feb 15 '17 at 22:00
  • Try constructing the Pauli spin matrices without imaginary numbers. In what way are numbers real as opposed to imaginary? in that that they are all products of our imagination. –  Feb 15 '17 at 23:37
  • -1. Lack of research effort. See the numerous possible duplicates given by QM, and other related questions listed on the right. – sammy gerbil Feb 15 '17 at 23:47
  • Ok, fair enough – Marko Bakić Feb 15 '17 at 23:51

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The math of (formal systems describing) points in a plane and those of complex numbers are equivalent. This means that you are correct, you can always chose to describe your math either using complex numbers or pair of real numbers. The choice between these two descriptions is not only a matter of simplicity in the equations, but also a matter of taste (intuitions, etc). It is obvious that today's physicists do not have any problem with describing (real?) physical entities using complex numbers. I am not that sure that Newton would have liked it though.