system
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human
What are the differences between Quantum Mechanics and Classical Mechanics, and how and why do they manifest?

The equation

x̂(t) = x̂(0) + p̂(0) t/m

is arrived at after applying the commutator value \[x̂, p̂\] = iħ to the Heisenberg Equation of Motion for position of a free mass which says

dx̂/dt = i\[Ĥ,x̂\].

In classical mechanics, x and p can be determined simultaneously with infinite precision, that is to say, the Uncertainty relation, the Quantum Commutator is at the heart of which, does not exist in CM, and yet the classical analogue of the Quantum Commutator, the Poisson Bracket {x,p} = 1, leads to the same equation as above for the classical position of a free mass after using a classical Hamiltonian.

So why do these apparent similarities lead to different (*for ex., the abovementioned uncertainty principles, which I know arises due to Cauchy-Schwarz inequality restrictions on the Hilbert vector space, but what prevents this in CM vector spaces?*) as well as similar (equation of motion) results? What actually makes CM different than QM?

And what are some more such examples where CM and QM differ in results?
gpt
>What actually makes CM different than QM?

The canonical Poisson bracket and canonical commutation relation look similar on the surface, but the interpretations of those equations are very different. Just look at the postulates of QM versus the postulates of classical mechanics (Shankar has a good comparison in his QM text). x and p mean very different things in QM and CM. In classical mechanics, they are continuous, real-valued functions, parametrized by time. And to find them, you just solve Hamilton's equations. In QM, these are Hermitian operators. Particles don't have well-defined positions or momenta, they have state vectors (or wavefunctions, in the coordinate representation).

As for how they differ in results, see any of the major experiments in quantum mechanics (double slit, Stern-Gerlach, etc.). Classical mechanics would not correctly predict the results.