Smooth and oscillatory geometric phase corrections for driven spins

IF 0.6 4区教育学 Q4 EDUCATION, SCIENTIFIC DISCIPLINES

European Journal of Physics Pub Date : 2023-09-08 DOI:10.1088/1361-6404/acf81e

Michael V Berry

引用次数: 0

Abstract

For a quantum spin driven cyclically by a slowly-rotated magnetic field, geometric phases are well understood. If the cycle takes a long time T, the leading-order (dynamical) phase is proportional to T and the geometric phase is the contribution independent of T. The dynamical and geometric phases are the first two terms of a series in slowness 1/T. Here it is shown with an exactly solvable example that the corrections are of two types: smooth, proportional to powers of slowness, and oscillatory: essential singularities in 1/T, in the form of trigonometric functions of T divided by powers of T. The calculations are elementary and therefore suitable for presentation in graduate quantum theory courses.

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驱动自旋的平滑和振荡几何相位修正

对于由缓慢旋转的磁场循环驱动的量子自旋，几何相位是众所周知的。如果周期需要很长的时间T，则前导阶（动力学）相位与T成比例，几何相位是独立于T的贡献。动力学相位和几何相位是慢度1/T序列的前两项。这里用一个完全可解的例子表明，校正有两种类型：平滑的，与慢度成比例的，和振荡的：1/T中的本质奇点，以T的三角函数除以T的幂的形式。计算是初等的，因此适合在研究生量子理论课程中介绍。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

European Journal of Physics 物理-物理：综合

CiteScore

1.70

自引率

28.60%

发文量

128

审稿时长

3-8 weeks

期刊介绍： European Journal of Physics is a journal of the European Physical Society and its primary mission is to assist in maintaining and improving the standard of taught physics in universities and other institutes of higher education. Authors submitting articles must indicate the usefulness of their material to physics education and make clear the level of readership (undergraduate or graduate) for which the article is intended. Submissions that omit this information or which, in the publisher''s opinion, do not contribute to the above mission will not be considered for publication. To this end, we welcome articles that provide original insights and aim to enhance learning in one or more areas of physics. They should normally include at least one of the following: Explanations of how contemporary research can inform the understanding of physics at university level: for example, a survey of a research field at a level accessible to students, explaining how it illustrates some general principles. Original insights into the derivation of results. These should be of some general interest, consisting of more than corrections to textbooks. Descriptions of novel laboratory exercises illustrating new techniques of general interest. Those based on relatively inexpensive equipment are especially welcome. Articles of a scholarly or reflective nature that are aimed to be of interest to, and at a level appropriate for, physics students or recent graduates. Descriptions of successful and original student projects, experimental, theoretical or computational. Discussions of the history, philosophy and epistemology of physics, at a level accessible to physics students and teachers. Reports of new developments in physics curricula and the techniques for teaching physics. Physics Education Research reports: articles that provide original experimental and/or theoretical research contributions that directly relate to the teaching and learning of university-level physics.