Roughly - Classical mechanics deals with quantities on a macroscopic scale and quantum mechanics deals with quantities on a sub atomic scale. Classical mechanics use the idea that a particle is at a particular place at a given time and thus uses the concept of a trajectory. Quantum mechanics deals only with probabilities of where a partilce will be found, the momentum and energy it will have etc. Pete
Quantum mechanics is the more detailed and more correct accounting of the "what and why" of matter. All classical problems can be solved using quantum rules, but when your dealing with large numbers of particles, the "probabilities" of quantum mechanics start to appear to be more absolute. For example, in quantum mechanics, you would be interested in what happens to a single neutron. Depending on the condition, there are various probabilities to the outcomes (80% likely to do this, 20% likely to do that). When you have billions of billions of billions of particles, the most probable things seem dominant and the least probable things go unnoticed {WARNING: QUANTUM OVERSIMPLIFICATION} Who cares if a few of your electrons suddenly "tunnel" into the next galaxy. THere goes one right now! But you're fine, because most of the rest did what we expected them to do.
Quantum mechanics deals with things very differently to classical mechanics. Quantum is all about waves, the dynamics of all things is determined by waves. Classical things like momentum and position dont have the same meaning, they are just borrowed terms. A typical example is spin, this is an intrinsic property of particles, but it doesnt mean the particle actualy spins. However, for waves with short wavelength, things get classical. Its like rays in optics. You can think about how a magnifying glass works just with a simple ray picture, you dont need to worry about interference of the light due to its wave nature. You cant explane diffraction in the same way.
Er... no. A key conclusion of QM is that waves and particles are just different sides of the same coin. Mathematically, Schrodinger wave equations are ultimately equivalent to quantum particle equations. The schrodinger equation gets most limelight because it is generally considered as easier, and making more sense.
Actualy there are sugestions on how to measure wave functions experimentaly. Mathematicaly you done need them its true. And I dont think there are any final conclusions about waves and particles.
These questions always risk a confusion between the quantum amplitude wave - the thing that is supposed to collapse - and the wave nature of the particle. The particle's wave nature is a real thing that exists in spacetime and has physical things - like diffraction - happen to it. The amplitude "wave" is a thing expressed in complex numbers, that cannot be a thing in spacetime, and exists primarily in an abstract Hilbert Space. Operators act on it and produce "eigenvalues" and these eigenvalues generate the probabilities that something will happen in spacetime. It is possible to imagine the amplitude is just a mathematical construct, part of a calculating algorithm. But the wave nature is experimentlly visible.