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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Generalizing Born-Oppenheimer approximations and observable effects of an induced gauge field

Chang-Pu Sun, Mo‐Lin Ge

Published Feb 15, 1990• 94 citations
Publisher

Abstract

By extending the Born-Oppenheimer approximation improved by Wilczek et al. to deal with separation of the spin and space coordinates of a particle in an external field, we generally discuss the direct effects of an induced gauge field and the higher-order corrections to the problem. It is shown that Bitter and Dubbers's experiment about Berry's phase is directly explained as an effect of the induced gauge potential in terms of the first-order approximation in this paper. The higher-order effects appearing in this experiment are also pointed out when the adiabatic conditions are broken.

DOI
10.1103/physrevd.41.1349

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By extending the Born-Oppenheimer approximation improved by Wilczek et al. to deal with separation of the spin and space coordinates of a particle in an external field, we generally discuss the direct effects of an induced gauge field and the higher-order corrections to the problem. It is shown that Bitter and Dubbers's experiment about Berry's phase is directly explained as an effect of the induced gauge potential in terms of the first-order approximation in this paper. The higher-order effects appearing in this experiment are also pointed out when the adiabatic conditions are broken.
By making use of the improved Born-Oppenheimer approximate method, we deal with the separation of spin variable and space variable of neutral particle with arbitrary spin in inhomogeneous magnetic field and non-adiabatic corrections for the problems are thereby computed as perturbations. We also analysis the geometrical properties of loop phase of the induced gauge potential and point out its observable effects relating to Bitter and Dubbers' experiment. Finally, we discuss the transitions between the instantaneous spin states as non-;-adiabatic effects resulting from inhomogeneity of the magnetic field.
In non-adiabatic cases the topological action associated with Berry's phase and the corresponding effective Hamiltonian are obtained by path-integral method. We also give the non-adiabatic transition probability amplitude in the first-order approximation. It is thereby shown that the Berry's phase and the induced gauge structure have universality of existence. As an example, dynamics of induced monople relating to the Bitter-Dubber's experiment is analysed in terms of induced gauge field.
Under non-relativistic conditions, the properties of adiabatic solutions of the Schrodinger equation in moving frame of reference and the behaviours of the corresponding Berry's Phase are analysed. Two cases of translation and rotation are discussed in detail, which show that the existence of Berry's phase depends on the choice of frame of reference. While Bitter and Dubbers's experiment is explained by the first-order approximation in our discussion, the nonadiabatic effects in this experiment are predicted by the second-order approximation when the adiabatic condition is broken.
Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian: