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Accepting the postulates of quantum mechanics, so promoting the classical dynamical variables to operators with appropriate commutation relations, is it possible to "derive" Schrödinger's equation

$$ih\frac{\partial}{\partial t} |\psi(t)\rangle = H |\psi (t)\rangle.\tag{1}$$

Directly from Hamilton's equations of motion? So I am asking for a derivation that avoids the use of Poisson's brackets and avoids the use of the Heisenberg picture. Is it feasible? Or are we forced to introduce the concept of Poisson's brackets to show the parallelism between the classical EOM and the QM EOM in the Heisenberg picture?

Qmechanic
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Noumeno
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1 Answers1

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  1. First of all, it should be stressed that the TDSE (1) cannot be derived from classical physics alone, cf. e.g. this Phys.SE post.

  2. However, assuming that the classical system at hand can be quantized, the operator form of Hamilton's equations $$ d_tA(t)~=~\{A(t), H(t)\}_{PB} + \partial_tA(t) \tag{CM}$$ is the Heisenberg EOM $$d_t\hat{A}_H(t)~=~\frac{1}{i\hbar}[\hat{A}_H(t),\hat{H}_H(t)] +(\partial_t\hat{A}_S)_H.\tag{QM}$$ in the Heisenberg picture, which in turn is equivalent to the TDSE (1) in the Schrödinger picture.

Qmechanic
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