Exactness of semiclassical quantization rule for broken supersymmetry
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Semiclassical methods provide important tools for approximating solutions in quantum mechanics. In several cases these methods are intriguingly exact rather than approximate, as has been shown by direct calculations on particular systems. In this paper we prove that the long-conjectured exactness of the supersymmetry-based semiclassical quantization condition for broken supersymmetry is a consequence of the additive shape invariance for the corresponding potentials.Keywords:
Semiclassical physics
Quantum superposition
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Abstract The soliton of the effective chiral action of QCD is quantized in a semiclassical fashion using the “cranking approach” developed for the description of rotating nuclei. It is shown that for the Skyrmion model this approach is equivalent to the semiclassical quantization proposed by Adkins et al. Contrasting the assumptions of the cranking approach to the experimental excitation spectrum of the nucleon leads to the conclusion that the semiclassical quantization is inapplicable to the chiral soliton if the latter is to describe the physical nucleon.
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Skyrmion
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The "Bohr-Sommerfeld quantization rule" is used in the treatment of the massive Thirring model. A semiclassical bound-state spectrum is achieved. This spectrum is similar to that of the sine-Gordon theory.
Semiclassical physics
Thirring model
Bohr model
Sine
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Semiclassical physics
Adiabatic Quantum Computation
Adiabatic invariant
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One reason for studying supersymmetric quantum mechanics is that there are a class of superpotentials W(x) which behave at large x as x/sup ..cap alpha../ for which we know from general arguments whether SUSY is broken or unbroken. Thus one can use these superpotentials to test various ideas about how to see if supersymmetry is broken in an arbitrary model. Recently, Witten proposed a topological invariant, the Witten index ..delta.. which counts the number of bosons minus the number of fermions having ground state energy zero. Since if supersymmetry is broken, the ground state energy cannot be zero, one expects if ..delta.. is not zero, SUSY is preserved and the theory is not a good candidate for a realistic model. In this study we evaluate ..delta.. for several examples, and show some unexpected peculiarities of the Witten index for certain choice of superpotentials W(x). We also discuss two other nonperturbative methods of studying supersymmetry breakdown. One involves relating supersymmetric quantum mechanics to a stochastic classical problem and the other involves considering a discrete (but not supersymmetric) version of the theory and studying its behavior as one removes the lattice cuttoff. In this survey we review the Hamiltonian and path integralmore » approaches to supersymmetric quantum mechanics. We then discuss the related path integrals for the Witten Index and for stochastic processes and show how they are indications for supersymmetry breakdown. We then discuss a system where the superpotential W(x) has assymetrical values at +-infinity. We finally discuss nonperturbative strategies for studying supersymmetry breakdown based on introducing a lattice and studying the behavior of the ground state energy as the lattice cutoff is removed. 17 references.« less
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Hamiltonian (control theory)
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Semiclassical physics
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Semiclassical methods provide important tools for approximating solutions in quantum mechanics. In several cases these methods are intriguingly exact rather than approximate, as has been shown by direct calculations on particular systems. In this paper we prove that the long-conjectured exactness of the supersymmetry-based semiclassical quantization condition for broken supersymmetry is a consequence of the additive shape invariance for the corresponding potentials.
Semiclassical physics
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In a recent paper [Takatsuka, Takahashi, Koh and Yamashita, J. Chem. Phys. 126 (2007), 021104], it was shown that the semiclassical quantization of chaos can be accurately achieved only with the phases (the action and Maslov phases) without use of the amplitude (preexponential) factor, which is known to diverge exponentially in chaos. The aim of this paper is to analyze the role of the semiclassical amplitude in energy quantization from various points of view, and we actually show that it is indeed quite limited. Therefore even chaos is quantized mainly by the phase constructive and destructive interferences.
Semiclassical physics
Constructive
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A pedagogical review on supersymmetry in quantum mechanis is presented which provides a comprehensive coverage of the subject. First, the key ingredients on the quantization of the systems with anticommuting variables are discussed. The supersymmetric Hamiltotian in quantum mechanics is then constructed by emphasizing the role of partner potentials and the superpotentials. We also make explicit the mathematical formulation of the Hamiltonian by considering in detail the N=1 and N=2 supersymmetric (quantum) mechanics. Supersymmetry is then discussed in the context of one-dimensional problems and the importance of the factorization method is highlighted. We treat in detail the technique of constructing a hierarchy of Hamiltonians employing the so-called ‘shape-invariance’ of potentials. To make transparent the relationship between supersymmetry and solvable potentials, we also solve several examples. We then go over to the formulation of supersymmetry in radial problems, paying a special attention to the Coulomb and isotropic oscillator potentials. We show that the ladder operator technique may be suitably modified in higher dimensions for generating isospectral Hamiltonians. Next, the criteria for the breaking of supersymmetry is considered and their range of applicability is examined by suitably modifying the definition of Witten’s index. Finally, we perform some numerical calculations for a class of potentials to show how a modified WKB approximation works in supersymmetric cases.
Hamiltonian (control theory)
Isospectral
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Although the semiclassical quantization condition of a general (asymmetric) double minimum potential has been known for some time it has only been applied to the simpler symmetric case. In this work, new versions of a method for evaluating higher-order semiclassical phase integrals are applied to semiclassical quantization of an asymmetric double minimum potential. The quantization condition is discussed and energy eigenvalues for a model potential are determined in the first- , third- , and fifth-order phase integral approximations. Agreement of three, five, and seven significant digits, respectively, where the exact quantum mechanical eigenenergies are obtained.
Semiclassical physics
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