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As we know we all say that quantum mechanics is "wave mechanics", and particles are described as waves or associated with every particle a wave nature; the behavior of such waves are described by Schrodinger's equation of motion. However, we already have the classical wave equation for classical waves.

So what makes us to use a different equation for quantum mechanical waves compared to the classical wave? What is peculiar about quantum waves that can not be described by a classical wave equation?!

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There are two major differences.

The first difference is that quantum wavefunction is not observable by itself. It only allows you to compute probability distributions for observables.

The other important difference is entanglement. If you need to describe two particles with help of wavefunctions classically you may expect one of two possibilities. On the one hand, you may think that they correspond to some two wavepackets of the single classical field $\Psi(x)$. On the other hand, you may think that they are described by two separate classical fields $\Psi_1(x_1)$ and $\Psi_2(x_2)$. The quantum reality is more interesting - you have to describe them by single wavefunction depending on both coordinates generally not factorizable into to separate pieces $\Psi(x_1,x_2)\neq \Psi_1(x_1)\Psi_2(x_2)$

It may not be very obvious but combined those two traits present huge obstacle (in the form of Bell theorem and its relatives) for trying to dismiss quantum theory as just approximation to some more fundamental classical behaviour.

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    $\begingroup$ It might be worth mentioning that a classical wave also need not factorize into functions of individual components. The difference is that the quantum "wave" is not the analogon to a classical wave (which would be a field theory), but a classical point particle described by a unique phase space point $(x,p)$. $\endgroup$
    – ACuriousMind ♦
    Nov 28, 2016 at 17:23
  • $\begingroup$ @ACuriousMind: I know it in this way that:" particles at quantum level treat as if they are waves!". is it the same thing as what you say?!! $\endgroup$
    – P.A.M
    Nov 28, 2016 at 21:00
  • $\begingroup$ @P.A.M It is an often repeated way of putting things which I personally think is rather confusing to the point of being wrong. It's not the wave part of quantum mechanics that is different, it's the compatible marginal distributions cannot exist for all composite state part, which is what I think was trying to get at, but it has (almost) nothing to do with the use of wavefunctions. $\endgroup$
    – G. Bergeron
    Dec 29, 2016 at 11:28
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    $\begingroup$ @Ooker The classical wave can transmit energy, as per the Poynting vector, and likewise, the QM wavefunction can be interpreted as being able to transmit probabilities: see probability current. However, I suggest to take those interpretations with a grain of salt as they can easily lead to erroneous thinking. The really different part of QM is in the use of probabilities the rest is really just a concretization of this. The fact that the equations are not for observables but for the amplitude has fundamental implications, such as the U(1) symmetry of QM which leads to quantum statistics. $\endgroup$
    – G. Bergeron
    Jul 25, 2017 at 16:30
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    $\begingroup$ @Ooker The fact that QM is not about wavefunctions but rather about state vectors. The whole ''wavefunction'' thing adds a lot of, sometimes useful, properties that are not what QM is about. For instance, consider the OPs question... $\endgroup$
    – G. Bergeron
    Apr 28, 2018 at 2:31
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Classically, every quantity (irrespective whether it is real- or complex-valued) which solves the wave-equation is called a wave. This can e.g. be the electric field, the air-density or the quantum mechanical so-called wave-function.

In quantum mechanics, a particle is described by the wave-function $\psi$, which generally is a complex-valued quantity. The wave-function needn't be a wave (so, the name is a little bit misleading) but it can be a wave.

So, the wave-function in quantum mechanics hasn't much in common with a classical wave. Possibly the name wave-function stems from the fact that the wave-function oscillates

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