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Sabatier Agrawal Machado Advances in Fractional Calculus Episode 5

Tham khảo tài liệu 'sabatier agrawal machado advances in fractional calculus episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | SEMI-INTEGRALS AND SEMI-DERIVATIVES IN PARTICLE PHYSICS 149 L Ll Ly L -h2 A3 ọ -Ã 2 1 Ổ Y n ổ Y 1 Ổ2 I sin 9 I -I---- - sin 9Ổ 1 Ổ9j sin2 9 ỔỌ2 36 and commutes with all three components of L 0 37 These commutation rules tell us that the magnitude of the angular momentum can always be exactly determined but only one of its components can be measured at the same time with arbitrary accuracy. The other two remain indeterminable. Usually Lz is taken as the measurable component. For a stationary state the part of the wave function TL which describes the angular momentum has to be an eigenfunction of the angular momentum operators thus it has to satisfy the two equations L2 T. L2 T. 38 and Lz Tl Lz Tl 39 where L2 and Lz represents the associated eigenvalues. The spherical harmonics Ỹlm 9 ọ l l - m 2l 1 m z . Yl m 9 . T hl P l s 3 exp m 40 ỵ 4n l ml with the associated Legendre polynomials satisfy the relations LYm 9 ọ l l 1 Ã 2TZm 3 Ọ 41 LzYlm Ọ mhYlm Ọ 42 yielding the eigenvalues L2 l l 1 Ã2 and Lz mh 43 for L2 and Lz. The magnitude of the angular momentum L hj l l 1 44 as well as its z-component Lz are quantised with l and m as quantum numbers. The latter can be changed applying the ladder operators 150 Krempl L Lx iLy 45 to create or annihilate - a magnetic quantum according to Lz L Tl h m 1 L Tl 46 but do not change the magnitude L2 L Tl h2l l 1 L l 47 Since Lx and Ly are both Hermitian which implies and L Ỹlm 2 0 we have the well known restrictions -l m l between the so-called angular l and magnetic m quantum numbers. 4.2 Spin-1 2-particles in fractional description Since the discovery of the electron spin 1925 this quantum state keeps some mystery since it cannot be described like an angular momentum but shows experimental evidences to be something like an angular momentum due to the facts that spin acts experimentally like something rotating in space having an intrinsic angular momentum has the physical dimension of an angular momentum couples with the normal .