Quantum theory of angular momentum

Transcription

1 Quantum theoy of angula momentum Igo Mazets (Atominstitut TU Wien, Stadionallee 2, 1020 Wien Time: Fiday, 13:00 14:30 Place: Feihaus, Sem.R. DA gün 06B (exception date 18 Nov.: Sem.R. DA gün 06A Lectue notes will appea at:

2 Recommended liteatue 1. Vashalovich, Moskalev, Khesonsky Quantum theoy of angula momentum othe specialized liteatue: 2. Edmomds Angula momentum in quantum mechanis 3. Fano & Racah Ieducible tensoial sets 4. Wigne Goup theoy some mateial in standad couses: 5. Landau & Lifshitz Quantum Mechanics 6. Akhieze & Beestetsky Quantum Electodynamics...and in any othe q.m.-textbook whee you can find it...

3 0. Notation Vecto Vecto component Unit vecto Lowe index covaiant e e Uppe index contavaiant

4 Co-odinate system Catesian spheical Levi-Civita tenso

5 Cyclic coodinates Covaiant Contavaiant

6 Cyclic unit vectos Covaiant Contavaiant

7 I. Rotation opeation An isolated quantum system in a 3D space ( = all vaiables Enegy Paity Squae of the total angula momentum Ang.momentum pojection to z axis How is the wave function tansfomed unde a otation of the co-odinate system?

8 We conside passive otations: the physical system emains at est, but the coodinate system is otated. The otation is given by 1 the otation angle ω; 2 the diection of the otation axis: unit vecto n = e x sin Θ cos Φ + + e y sin Θ sin Φ + + e z cos Θ Pove: n α = n α Conside ω as a paamete of a continuous tansfomation fom S to S : de α /dω = [n x e α ] dx α /dω = d(e α /dω = de α /dω = [n x e α ]

9 What we do (stating with a function of co-odinates of one paticles: 1. Make co-odinate system otation, calculate the new = (x, y, z as a function of = (x, y, z, ω, and n. 2. Choose an abitay (diffeentiable function Ψ. 3. Find, which opeato elates this function in the new coodinates to the same function in the old co-odinates: (, ˆ ( ' ( n D Again, conside as a value paametized by the vaiable ω and finally eaching. Then (, ˆ ( ( n D d d ( ] [ ( ] [ ( ] [ ( ( (,,,, n n e n x x d dx d d d d z y x z y x

10 Recall the obital momentum opeato Lˆ i[ ]. Then one can see that Dˆ (, n inlˆ, Dˆ (, n exp( inlˆ Dˆ (0, n 1 The latte equation holds in a case of a function of co-odinates of N paticles. In that case Lˆ N Lˆ j1 ( j is the total obital momentum. [ Lˆ, Lˆ 2 ] Theefoe an eigenfunction of the opeato with the eigenvalue = L(L+1 is tansfomed afte a otation into a linea combination of the eigenfunctions with the same total obital momentum L. Thee ae in total 2L+1 diffeent functions fo a given L, chaacteized by L z = L, L+1,..., L 1, L. How to intepete and extend this obsevation? 0 ˆL 2

11 Rotations as a goup Definition of a goup. Goup is a set G. An opeation (the goup law is defined so that 1. If a G and b G then a b G. 2. Associativity: { a, b, c} G we have (a b c = a (b c. 3. Identity element: e G such that a G we have e a = a e = a. In fact, the identity element is always unique Invese element: a G an element a G exists, such that a 1 a = a a 1 = e. The otations satisfy all these equiements, the opeation being a subsequent pefomance of two otations.

12 Evey goup can be chaacteized by its ieducible epesentations. An ieducible epesentation is a set of objects whee a linea combination is defined and (i that is mapped to itself unde action of any element of the goup (epesentation, but (ii it is impossible to make (by constucting linea combinations its subset, which also would be a epesentation (ieducibility. The ieducible epesentations (IRs of the otation goup have dimensions 1, 2, 3, 4,..., each dimension appeaing only once. Odd-dimensional IRs can be associated with a system chaacteized by an intege angula momentum (ealizable by the obital momentum. The even-dimensional IRs can be associated with a system with a half-intege angula momentum (ealizable by the spin. Dˆ (, n exp( injˆ Then in a geneal case, Ĵ whee is the angula momentum opeato (without concetization of its obital, spin o composite natue.

13 Eule angles Altenatively, a otation may be defined with thee Eule angles. Scheme A: (i Rotation aound z by α (0 < α < 2π. (ii Rotation aound new y 1 by β (0 < β < π. (iii Rotation aound final z 2 = z by γ (0 < γ < 2π.

14 Scheme B (equivalent to A, angles ae the same. Thee otations aound the old axes. (i Rotation aound z by γ (0 < γ < 2π. (ii Rotation aound y by β (0 < β < π. (iii Rotation aound z by α (0 < α < 2π. The same otation is achieved also with The pola angles of an abitay diections in and ae elated as

15 Fom now on, we put Eule angles as the aguments of the otation opeato A function in the new coodinate and an opeato ae expessed as (scheme A (3 Rotation aound final z 2 = z by γ (0 < γ < 2π. α (0 < α < 2π. (2 Rotation aound new y 1 by β (0 < β < π. (1 Rotation aound z by Ty to pove thei equivalence! o, equivalently, (scheme B (3 Rotation aound z by α (0 < α < 2π. (2 Rotation aound y by β (0 < β < π. (1 Rotation aound z by γ (0 < γ < 2π.

16 Unitaity: Wigne D-function (definition We denote by X the angula (obital & spin vaiables of a system. X ' JM ' X M Dˆ (,, JM ' X X JM JM J '' M J '' M Dˆ (,, JM ' M X J '' M JM Dˆ (,, JM ' D J MM ' (,,