Abstract
We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born–Jordan rule is the correct quantization scheme for quantum mechanics. For this purpose we use correct short-time approximations to the action functional, initially due to Makri and Miller, and show that these lead to the desired quantization of the classical Hamiltonian.
Highlights
Weyl versus Born and JordanThere have been several attempts in the literature to find the “right” quantization rule for observables using either algebraic or analytical techniques [1,2,3,4,5,6,7]
In a recent paper [8] we have analyzed the Heisenberg and Schrödinger pictures of quantum mechanics, and shown that if one postulates that both theories are equivalent, one must use the Born–Jordan quantization rule (BJ)
As we have proven in [8,12], Heisenberg’s matrix mechanics [13], as rigorously constructed by Born and Jordan in [14] and Born, Jordan, and Heisenberg in [15], explicitly requires the use of the quantization rule (1) to be mathematically consistent, a fact which apparently has escaped the attention of physicists, and philosophers or historians of Science
Summary
There have been several attempts in the literature to find the “right” quantization rule for observables using either algebraic or analytical techniques [1,2,3,4,5,6,7]. In a recent paper [8] we have analyzed the Heisenberg and Schrödinger pictures of quantum mechanics, and shown that if one postulates that both theories are equivalent, one must use the Born–Jordan quantization rule (BJ). The Born–Jordan and Weyl rules yield the same result only if m < 2 or < 2; for instance in both cases the quantization of the product xp is 12 ( xbpb + pbxb). It turns out that this question is just a little bit more than academic: There are simple physical observables which yield different quantizations in the Weyl and Born–Jordan schemes. The Weyl quantization of2x is (`b2x )W = xb pb2y + xby pb2x − 21 ( xbpbx + pbx xb)(ybpby + pby yb).
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