{"title":"驱动自旋的平滑和振荡几何相位修正","authors":"Michael V Berry","doi":"10.1088/1361-6404/acf81e","DOIUrl":null,"url":null,"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.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Smooth and oscillatory geometric phase corrections for driven spins\",\"authors\":\"Michael V Berry\",\"doi\":\"10.1088/1361-6404/acf81e\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":50480,\"journal\":{\"name\":\"European Journal of Physics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-09-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"European Journal of Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1088/1361-6404/acf81e\",\"RegionNum\":4,\"RegionCategory\":\"教育学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"EDUCATION, SCIENTIFIC DISCIPLINES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1361-6404/acf81e","RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"EDUCATION, SCIENTIFIC DISCIPLINES","Score":null,"Total":0}
Smooth and oscillatory geometric phase corrections for driven spins
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.
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.
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