In accordance with this probability interpretation, it is not uncommon for the wavefunction to be called a ‘probability wave’. However, I think that this is a very unsatisfactory description. In the first place, ψ(x) itself is complex, and so it certainly cannot be a probability. Moreover, the phase of ψ (up to an overall constant multiplying factor) is an essential ingredient for the Schrödinger evolution.
Even regarding |ψ|² (or |ψ|² / ||ψ||) as a ‘probability wave’ does not seem very sensible to me. Recall that for a momentum state, the modulus |ψ| of ψ is actually constant throughout the whole of spacetime. There is no information in |ψ| telling us even the direction of motion of the wave! It is the phase, alone, that gives this wave its ‘wavelike’ character.
Moreover, probabilities are never negative, let alone complex. If the wavefunction were just a wave of probabilities, then there would never be any of the cancellations of destructive interference. This cancellation is a characteristic feature of quantum mechanics, so vividly portrayed (Fig. 21.4d) in the two-slit experiment!
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To be clear, it was the physicist Max Born who first interpreted the wavefunction as a probability wave, and this as giving the potential locations of a particle, not the direction of motion of an actual wave.
From the perspective of Systemic Functional Linguistic Theory, the notion of a probability wave is entirely consistent with the bands of constructive and destructive interference on the detector screen in the two-slit experiment. This is because the experiment sets up two waves of probability, one for each possibility — passing through one slit or the other — so that it is the probability waves that overlap. The frequency of particle locations observed on the detector screen is in line with the probable locations of particles as modelled by the overlapping wavefunctions.
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