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I used to believe firmly that the essence of quantum mechanics is complex number. Complex number makes the calculation of probability in quantum theory very different from that in classical statistical mechanics. In QM, it is the probability amplitudes that are added, NOT the probability themselves.

However, now I am in doubt about the necessity to use complex numbers, after reading a very easy book about QM, which explains quantum ideas without using complex numbers. Here are what exactly I am thinking about:

  1. QM reveals the wavelike nature of matter. Complex numbers are effective tools when we are dealing with waves, but it is not impossible to get a deep understanding of wave without complex numbers. Simply use $cos(\omega x)$ instead of $e^{i\omega x}$, or something.
  2. It looks like a conspiracy to me that after lots of computation using complex number, many final results in QM appear clean and contain only real numbers. For example, the ground state of Harmonic Oscillators is just a simple Normal distribution. The wavefunction of a particle in a potential well can be a sine wave, or an exponential curve, which is a real valued function defined on $\mathbb{R}$.

So, Here are my questions:

  1. Are there any quantum results that are very difficult or impossible to be obtained and explained without using complex numbers?
  2. Are there any beautiful examples of complex valued wavefunctions, which cannot be replaced by real valued ones?

And finally, if complex number is not the essence of QM, what is the essence of QM?


marked as duplicate by StephenG, Qmechanic quantum-mechanics Aug 8 '18 at 4:23

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