Abstract
The version of classical mechanics based on forces and Newton’s laws resists any meaningful reformation into a quantum theory because it depends critically on such concepts (trajectory, acceleration, etc.) that do not correspond to any observable reality in the quantum world. More productive for finding links between classical and quantum realms is an alternative formulation, where energy rather than force takes the central role. There are two essential elements in this formulation of classical mechanics. One is the idea of canonical coordinates in the so-called phase space (as opposed to regular three-dimensional configuration space), and the other is the concept of Hamiltonian.
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- 1.
p2 is defined as usual as the square of the magnitude of the vector in Cartesian coordinates \(p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\).
- 2.
We all are used to deal with commutative multiplication of numbers: the result does not depend on the order, in which multiplication is performed. The lack of commutativity of multiplication was one of the features of the Heisenberg theory, which especially freaked out Schrödinger.
- 3.
I borrowed this fact without proof from the branch of mathematics called functional analysis that studies the properties of linear operators.
- 4.
This transformation is not as trivial as it might seem since taking absolute value of a vector involves operation of square root, which is not well defined for operators. Practically it is not a problem, however, because usually one works in the basis of the eigenvectors of the position operator, in which case \(\hat {r}^{-1}\) becomes simply 1/r. If you are not concerned with any of this, this note is not for you. I mention it here simply in order to avoid accusations in sweeping something under the rug.
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Deych, L.I. (2018). Observables and Operators. In: Advanced Undergraduate Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-71550-6_3
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DOI: https://doi.org/10.1007/978-3-319-71550-6_3
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