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arXiv:quant-ph/0001111

Noether Theorem and the quantum mechanical operators

Wai Bong Yeung

Published Jan 30, 2000• 0 citations
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Abstract

We show that the quantum mechanical momentum and angular momentum operators are fixed by the Noether theorem for the classical Hamiltonian field theory we proposed.
DOI
10.48550/arxiv.quant-ph/0001111

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When a classical particle moves in a central field the energy and the three components of angular momentum are all constants of the motion. The motion is an orbit whose orientation in space is fixed. To discuss the corresponding situation in quantum mechanics we must examine the commutation properties of the associated operators: If they all commute, then the quantities which they represent may all simultaneously be constants of the motion; if not, we must find as many mutually commuting operators as we can and obtain a maximal set of constants of the motion.
Programm (1) Classical mechanics: Principle of the least action. Lagrangian mechanics. Symmetries and Noether’s theorem. The Kepler problem. The Legendre transform. Hamiltonian mechanics. The classical action. Poisson brackets. Canonical transformations and generating functions. Symplectic manifolds. Hamilton’s and Liouville’s representations. [1][Chapter 1] (2) Hilbert spaces: Basic denitions. Bounded operators. Compact operators. The spectral theorem for self-adjoint compact operators. Trace-class and HilbertSchmidt operators. Closed operators. Symmetric and self-adjoint operators. Resolvent and spectrum. The spectrum of self-adjoint operators. The spectral theorem for (bounded and unbounded) self-adjoint operators. Stone’s theorem. [2, 3] (3) Basic principles of quantum mechanics: Observables, states, and dy
The theory of angular momentum and spin in quantum mechanics seems to defy common-sense intuition.We render the theory intelligible again by pointing out that this apparent impenetrability merely stems from an {\em undue} parallel interpretation of the algebraic expressions for the angular-momentum and spin operators in the group representation theory of SO(3) and SU(2). E.g. the correct meaning of the quantumoperator L^z=ℏı (x∂∂y−y∂∂x){\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )L^z​=ℏ​(x∂y∂​−y∂x∂​) is not that it is the operatorfor the zzz-component LzL_{z}Lz​ of the angular momentum L{\mathbf{L}}L, but rather the expression of the operator for the angular momentum L{\mathbf{L}}L when it is aligned with the zzz-axis. Hence what we are used to note (erroneously) as L^z{\hat{L}}_{z}L^z​ is not a scalar but a vector operator.The same applies{\em mutatis mutandis} for the spin operators. In the correct interpretation, the wholealgebraic formalism is just the group representation theory for the rotationsof three-dimensional Euclidean geometry. It is thus mere,elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data and the algebra quantum mechanics remains correct. It is all only a matter of the correct geometrical meaning of the algebra.All these statements are proved within the framework of the group representation theory for SO(3) and SU(2) which is the basic tool used to describe rotational motion in quantum mechanics.
A rigorous application of the correspondence rules shows that the operator of the angular momentum of a quantum particle---corresponding to the classical magnitude l=mr∧v\mathbf{l}= m \mathbf{r} \wedge \mathbf{v}l=mr∧v---is given by l^=r∧(−iℏ∇−ecA)\mathbf{\hat{l}}=\mathbf{r}\wedge(-i\hbar\mathbf{\nabla} -\frac{e}{c}\mathbf{A})l^=r∧(−iℏ∇−ce​A) in the presence of an electromagnetic field. Thus, despite the general opinion on the corresponding rules of quantization, the eigenvalues of the angular momentum depend on the configuration of the electromagnetic field. The usual rules of commutation [l^i,l^j]=iℏεijkl^k[{\hat{l}}_i,{\hat{l}}_j]=i\hbarε_{ijk}{\hat{l}}_k[l^i​,l^j​]=iℏεijk​l^k​, that are at the foundation of the calculus of angular momentum and of the theory of \emph{spin}---and Bohm's example of the EPR argument---are not valid in the presence of an electromagnetic field. The expected value of the operator l^=−iℏr∧∇\mathbf{\hat{l}}=-i\hbar\mathbf{r}\wedge\mathbf{\nabla}l^=−iℏr∧∇ is not gauge invariant, it depends on the calibration of the electrodynamic potentials.