Why is it interesting?

Classical theories are useful approximations of quantum theories.

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Examples

Example1

Example2:

FAQ

How can we understand the transition from quantum to classical physics?

For a nice discussion, see chapter 2 in Sleeping Beauties in Theoretical Physics by Thanu Padmanabhan:

"Particles do not follow trajectories. They are described by wavefunctions but under appropriate circumstances the constructive interference of the phases of the wavefunction will single out a path which we call a classical trajectory. The Hamilton-Jacobi Equation is just the lowest-order Schrodinger equation if we use the ansatz in Eq. (2.1). The ¨ mysterious procedure in Hamilton-Jacobi theory — of differentiating the solution to Hamilton-Jacobi equation and equating it to a constant — is just the condition for constructive interference of the phases of waves differing slightly in the parameter E. The procedure based on Hamilton-Jacobi theory works in classical mechanics because it is supported by the Schrodinger equation"

See also: Decoherence and the Transition from Quantum to Classical—Revisited by Zurek and The quantum-to-classical transition and decoherence by Maximilian Schlosshauer

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