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arXiv:1405.4614

Oscillator Model of Spin

Yitian Ding, Miaomiao Xu

Published May 19, 2014• 0 citations
arXivHTMLPDF

Abstract

The Schwinger's representation of angular momentum(AM) relates two important fundamental models, that of AM and that of harmonic oscillator(HO). However, the representation offers only the relations of operators but not states. Here, by developing a graphic method to calculate the states, we show that the Schwinger's model is valid only with certain spin. With the relation of states, we also demonstrate how the superposition states in AM map to entangled states in HO. This study has promising applications in quantum computation, and may cast light on the relation between superposition and entanglement.

DOI
10.48550/arxiv.1405.4614

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The Schwinger's representation of angular momentum(AM) relates two important fundamental models, that of AM and that of harmonic oscillator(HO). However, the representation offers only the relations of operators but not states. Here, by developing a graphic method to calculate the states, we show that the Schwinger's model is valid only with certain spin. With the relation of states, we also demonstrate how the superposition states in AM map to entangled states in HO. This study has promising applications in quantum computation, and may cast light on the relation between superposition and entanglement.
In this chapter, we develop the background needed to move our quantum mechanical formalism fully into three dimensions. We discuss angular momentum, which is the motional complement to spin angular momentum, that we developed earlier. There are some critical differences between orbital and spin angular momentum, and we need to carefully discuss how they relate and how they differ. Why is angular momentum so important? Because in problems where we have central forces (forces that depend only on the distance between two particles and act along the line connecting the two particles), the angular momentum is conserved. So, it becomes an integral part of how we solve the problem. The most important problem we have to solve, the hydrogen atom, is a central force problem; hence, we need this work before we can move forward to some really interesting physics that will come in Chap. 11 . We cover the angular momentum operators, eigenvalues, and eigenstates, their matrix representations, how to combine angular momentum, and spherical harmonics. We revisit the EPR experiment using our new formalism. We also discuss some unique representations of angular momentum by mapping to two-dimensional harmonic oscillators, as developed by Schwinger.
The Jordan–Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan–Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.
The various representations of rotations in physical space and Hilbert space are developed in detail. This leads to an in-depth treatment of the representation of states of well-defined spin and angular momentum in quantum systems. The peculiarities of the physics of spin— systems (spinors) is outlined. The tensorial character of representations is implicit in the treatment. The Schwinger and Bargmann representations are introduced in some detail; and this leads to SU(2) coherent states (which are important for more advanced group representation theory).