On the (Non)Equivalence of the Schr\"odinger and Heisenberg Pictures of Quantum Mechanics

TL;DR

This paper demonstrates that the Schrödinger and Heisenberg pictures of quantum mechanics are not equivalent unless a specific quantization rule by Born and Jordan is applied, highlighting the importance of this rule for energy conservation.

Contribution

It clarifies the conditions under which the Schrödinger and Heisenberg pictures are equivalent, emphasizing the role of the Born-Jordan quantization rule for energy conservation.

Findings

Schrödinger and Heisenberg pictures are not generally equivalent.

The Born-Jordan quantization rule ensures their equivalence.

Weyl quantization does not guarantee equivalence or energy conservation.

Abstract

The aim of this short Note is to show that the Schr\"odinger and Heisenberg pictures of quantum mechanics are not equivalent unless one uses a quantization rule clearly stated by Born and Jordan in their famous 1925 paper. This rule is sufficient and necessary to ensure energy conservation in Heisenberg's matrix mechanics. It follows, in particular, that Schr\"odinger and Heisenberg mechanics are not equivalent if one quantizes observables using the Weyl prescription.