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Classical mechanics and quantum mechanics differ fundamentally in the way they predict the behavior of systems. Classical mechanics is deterministic. This means if you know the initial conditions and the forces acting on an object, you can predict exactly how it will move in the future. This predictability is rooted in Newton's laws of motion. For example, if you throw a ball, you can calculate where it will land, given its speed and angle of throw.

Quantum mechanics, however, is not deterministic but probabilistic. Even if you know the exact initial conditions, you can only predict the probability of where a particle, like an electron, might be. This uncertainty arises because at small scales, particles exhibit both wave-like and particle-like properties, complicating precise predictions.

Wave-particle duality is a key concept in quantum mechanics. It means that particles like electrons and photons exhibit both wave-like and particle-like properties. For example, electrons can create interference patterns in an experiment, similar to waves. But they also impact a screen one at a time, showing particle-like behavior.

This duality is absent in classical mechanics, where objects like balls or planets are treated only as particles with well-defined positions and velocities. Understanding wave-particle duality helps explain phenomena like the double-slit experiment, where particles create interference patterns, suggesting wave-like behavior even though they are individual particles.

The uncertainty principle is a hallmark of quantum mechanics, introduced by Werner Heisenberg. It states that you cannot simultaneously know both the exact position and exact momentum of a particle. The more precisely you know one, the less precisely you know the other. This is contrary to classical mechanics, where both position and momentum can be known exactly at the same time.

The uncertainty principle can be mathematically expressed as \[ \triangle x \triangle p \geq \frac{h}{4\pi} \] where \( \triangle x \) is the uncertainty in position, \( \triangle p \) is the uncertainty in momentum, and \( h \) is Planck's constant.

This principle highlights the intrinsic limitations of measurements at atomic scales and reinforces the idea that quantum mechanics deals with probabilities rather than certainties.