Heisenberg's & Schrodinger's Picture

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SUMMARY

The Heisenberg picture and Schrödinger picture in quantum mechanics are indeed equivalent, producing the same results, but they differ fundamentally in their treatment of time and operators. In the Heisenberg formulation, operators are time-dependent while the state vector remains fixed, making it more adaptable to relativity and congruous with quantum field theory. Conversely, the Schrödinger formulation features time-dependent states with a potentially time-independent Hamiltonian. This distinction is crucial when considering scenarios involving particle creation and annihilation, where the state vector cannot be treated in a Hilbert space.

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Is the Heisenberg's picture really equivalent to Schrödinger's ?
It may seem so at a first glance & both produce the same results.But space & time are treated equally in Heisenberg's formulation : the operators are time dependent.In Schrödinger's formulation, states are time-dependent while the hamiltonian may be time-independent. This makes the former congruous & adaptable to relativity.
Also, the initial state vector remains fixed & operators move in Heisenberg's formulation. The initial state is not of so great a consequence as in Schrödinger's picture. Hence, it is more congruous with quantum field theory .(When particle creation & annihilation is involved, we can't work with a state vector in a Hilbert space).
I wonder how the two become totally compatible in the non-relativistic formulation.
 
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It is easy to show mathematically that the two pictures give equal results.

A simple ways to visualize it is to consider that the Schrödinger picture corresponds to a lab-fixed reference frame, in which the state of the quantum system will evolve, while the Heisenberg picture corresponds to a system-fixed reference frame, where the state remains unchanged but the world around (including measuring apparatuses) move as a function of time.
 

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