One way to understand that the wave function of quantum mechanics has no classical limit is taken equal to 0 ± so naive in the Schrödinger equation. Immediately there was a contradiction since the kinetic term disappears before potential. This implies that the wave function does not have an expansion in powers of h around a classical function.
To be a little more concrete, consider the Schrödinger equation for stationary states of a particle potential V (x). The Schrödinger equation is reduced to H ψ = E ψ, where the Hamiltonian H = H_0 + H_i has two terms: one describes the kinetic energy of the particle H_0 =- h ^ 2 / (2m) ^ 2 d ^ 2/dx and other interactions with the potential H_i = V (x). So naive it seems to take h = 0 means to suppress the kinetic term. It follows that V (x) ψ = E ψ, ie V (x) = E which is impossible since any potential depends on x is not equal to a constant E.
The correct way to take a semi-classical limit of the wave function is what is called the WKB approximation. The central idea is to understand that the wave function has an essential singularity as h tends to 0. That is, ψ (x) ~ A (x) e ^ (i B (x) / h), where A (x) and B (x) do have expansions around classical values \u200b\u200b±. But the singularity in the exponential limit makes it impossible to naive. This approach allows a simple way to estimate such subtle phenomena such as tunneling.
obstruction to have a classical limit for the wave function implies that there is a classic description of the information in a system equivalent to the quantum paradigm. If that were true, would have a quantum-classical mechanics, profoundly different from the Newtonian concepts. The ideas of superposition and quantum entanglement are as consistently guarded by these singular limits.
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