What is the difference between a quantum mechanical wave and a classical wave? 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?!
 A: 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.
A: 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 
