Weyl quantization

This was the first indication of what [19] became the proposal to use inverse Fourier transforms for quantization, the now so-called Weyl quantization. We do not rely on this suggestion as Bom and Jordan had stated that Weyl s approach was too heavy for the introduction of quantum mechanics to physics. Having so stated, Bom and Jordan made their own approach. Thus, it became more incomprehensive to the chemists. 

In the quantum mechanical case we start with a Hamilton operator H which we assume to be obtained from the Weyl quantization of a classical Hamilton function H(q, p). Like in the previous section, q = pi, p2, , pf) and p = pi, p2,. .., pd) denote the canonical coordinates and momenta, respectively, of a Hamiltonian system with d DoEs. Eor convenience, we again choose atomic units, so that q and p are dimensionless. We denote the... 

The map H H(q, p fieff) leading to Eq. (A.25) is also called the Wigner map. It is the inverse of the transformation which yields a Hamilton operator H from the Weyl quantization, Op[H], of a phase space function H (the Weyl map) which, using Dirac notation, is given by... 

The reason for the equality of the classical transformations and the quantum transformations to second order is the commutativity of the Weyl quantization and affine linear symplectic transformations. For the nonlinear transformations corresponding to steps n > 3 the commutativity ceases to exist and the symbol calculus develops its full power. 

We thus see that the computation of the symbol of the QNF operator is very similar to the classical case. The major difference is the use of the Moyal bracket in the quantum case which is more complicated than the Poisson bracket in the classical case. What remains to be done to get the Nth order QNF operator is to compute the Weyl quantization of the symbol H p(z fteff) ... 

Quantum and Semiclassical Approaches A. The Wigner Function and Weyl s Rule Quantum Scars in Phase-Space Quantizing the ARRKM Theory... 

The canonical quantization of the field has introduced by Dirac [1] (see also Refs. 2-4,10,11,14,15,26,27) is provided by the substitution of the photon operators, forming a representation of the Weyl-Heisenberg algebra, into the... 

The multipole electromagnetic held can be quantized in much the same way as plane waves [2]. We have to subject the complex held amplitudes in the expansion (17) to the Weyl-Heisenberg commutation relations of the form... 

Now consider the set of Stokes operators that can be obtained by canonical quantization of (132). On the other hand, the Stokes operators should by definition represent the complete set of independent Hermitian bilinear forms in the photon operators of creation and annihilation. It is clear that such a set is represented by the generators of the SU(3) subalgebra in the Weyl-Heisenberg algebra of electric dipole photons. The nine generators have the form [46]... 

The quantization of the classical observables, p for P and q for Q, and the non-commutative P and Q led to a fundamental difficulty for an observable given as a function ft p, q) of the basic dynamical variables p and q. Weyl s unitary representation approach avoided this difficulty. The inverse operator of the Fourier transform (3.20) gave a unique well-determined assignment,/for F, of the Hermitian operators to the real-valued quantities. The same proposition can be advanced for electric variables such as the amount of electric charge, a as the classical observable, which after the quantization leads to Moreover, dynamical variables that are expressly related to the current intensity i as the classical observable yield after quantization to I. The same has to occur to the magnitude of the electric potential from v to V, after quantization of the classical observable. 

In the time concept of the pre-relativistic mechanics, the observable quantities, time t and energy E, have to be considered as another canonically conjugate pair, as in classical mechanics. The dynamic law (time-dependent energy term) of the Schrodinger equation will then completely disappear [19]. A good occasion for Weyl to introduce the relativistic view would have been his contributions to Dirac s electron theory. His other colleagues developed the method of the so-called second quantization that seemed easier for the entire community of physicists and chemists to accept. 

Weyl and Heisenberg s indeterminacy relation can be applied for the amount of charge, a, and the mass of the species in consideration, that is, m. After the corresponding quantizations, m to M and a to 2, we will have... 

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