Abstract
Invertible maps of operators of quantum observables onto functions of c-number arguments and their associative products are reviewed. In particular, the symplectic tomography map is discussed and an expression connecting an arbitrary operator and its tomographic symbol is written down. This formula is applied to obtain explicit expressions for tomographic symbols, which are symplectic tomograms of Green functions of stationary and nonstationary Schrödinger equations written for the case of harmonic oscillator. The connection between the so-called classical propagator Π(X,μ,ν,t,X′,μ′,ν′,0) and the tomographic symbol of the evolution operator of nonstationary Schrödinger equation is found. The spin tomography is presented as a map of operators acting in spinor space onto functions of c-arguments. As an example, the spin located in a magnetic field is considered and the tomographic symbol of resolventa is obtained. Tomographic symbols of hermitian conjugate operators are shown to be complex conjugate functions.
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Man'ko, V.I., Sharapov, V.A. Green Functions of the Stationary Schrödinger Equation and Tomographic Representation of Quantum Mechanics. Journal of Russian Laser Research 25, 370–382 (2004). https://doi.org/10.1023/B:JORR.0000035719.85571.0d
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DOI: https://doi.org/10.1023/B:JORR.0000035719.85571.0d