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Appendix A. Angular Momentum and Spherical Harmonics

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Appendix A. Angular Momentum and Spherical Harmonics Appendix A. Angular Momentum and Spherical Harmonics The total angular momentum of an isolated system is conserved; it is a constant of the motion. If we change to a new, perhaps more convenient, coordinate system related to the old by a rotation, the wave function for the system trans- forms in a defmite way depending upon the value of the angular momentum. If the system is made up of two or more parts, each of them may have angular momentum, and we need to know the quantum rules for combining these to a resultant. We cannot do more here than introduce and summarise a few salient points about these matters. More detailed discussions of the quantum theory of angular momentum are available in several treatises (for example, Brink and Satchler, 1968; see also Messiah, 1962). A general word of caution is in order. The literature on this subject presents a wide variety of conventions for notation, normalisation and phase of the various quantities which occur. Sometimes these conventions represent largely arbitrary choices, sometimes they emphasise different aspects of the quantities involved. If the reader is aware of this possibility, and if he adopts a consistent set of formulae, little difficulty should be encountered. AI ANGULAR MOMENTUM IN QUANTUM THEORY An isolated system will have a total angular momentum whose square has a discrete value given by j(j + 1)1i 2 , where1i is Planck's constant divided by 21T and where j is an integer or integer-plus-one-half according to whether the 283 284 INTRODUCTION TO NUCLEAR REACTIONS system as a whole is a boson or a fermion.* That is to say, the wave function of the system will be an eigenfunction of 12 , the operator for the square of the angular momentum J2 lj) = j(j + l)lj) (AI) (Henceforth, we shall use natural units so that1i = 1.) Note that we often speak of 'the angular momentum j' when strictly we mean that the square of the angu- lar momentum isj(j + 1). The state lj) cannot be an eigenfunction of the operator J because the various components of the vector J do not commute with each other. Instead their commutation relations can be summarised by the formula J X J = iJ (A2) Hence a system cannot simultaneously be an eigenfunction of the operators for more than one component of J. Each component of J commutes with J2 , how- ever, so we are free to choose J2 and one component to generate angular momen- tum eigenfunctions. It is conventional to choose the z-component Jz and we label its eigenvalues by m Jz ljm) =mljm) (A3) The lack of commutativity between the components J; can then be pictured by using the vector model, Figure Al. In this model, the vector J has a definite magnitude (j (j + 1)] 1/ 2 and a definite projection m on the z-axis, but its direction is uncertain because of the uncertainty in Jx and Jy. This uncertainty is represented by imagining the vector to be precessing around the z-axis so that its x andy components fluctuate. The prototypical angular momentum operators are those corresponding to the orbital angular momentum L of a particle with momentum p. Explicitly L = r X p, with Px = -i atax, etc. or in polar coordinates L± =Lx ±iLy = ± e±il/> {aao ± i cot 8 ~} at/> . a L =-r- (A4) z atJ> *A nucleus is made up of A nucleons, each of which is a fermion and has an intrinsic spin of 111. Their relative orbital motions can have angular momenta which are integral multiples of1r. The rules of angular momentum addition (gee below) then ensure that the total angular momentum or 'nuclear spin' is an integral multiple of11 if A is even or an integer-plus-one- half times11 if A is odd. Therefore nuclei with A even are bosons, while those with A odd are fermions. APPENDIX A 28S I ' ' z' ' ' ' m ' ' ' ' ' ' Figure Al Vector model of an angular momentum vector /with a magnitude j [iU + 1)] 1•/2 and a z-projection equal tom. It precesses about the z-axis so that its x- andy-components are variable. Its projection m' upon another z' -axis also fluctuates; classically only the range of m' values shown is allowed but quantum indeterminacy allows -j .;;; m' .;;; j and 2 22 {AS) L =- ti~ o aao (sin 8 a:)+ si~ 2 8 a: } The expression AS for L2 is one which occurs when the Schrodinger equation for a particle moving in a central field is separated into radial and angular equa- tions (Messiah, 1962). The angular part of the wave function is an eigenfunction of Lz and L2 ; these eigenfunctions are called spherical harmonics and denoted Yf{' (8, </>). They obey the eigenvalue equations L2 Y:' (8, </>) = Q(Q + 1) Y:' (8, </>) {A6) Lz Y{' (8, </>)=mY{{' (8, </>) {A7) Spherical harmonics and their properties are discussed in more detail in section A3. Another important set of angular momentum operators are those for spin-i particles. For these we have j =i and J =t d, with the vector o representing the three 2 x 2 matrices of Pauli. With lz diagonal these are 286 INTRODUCTION TO NUCLEAR REACTIONS {AS) ax= (01 01 )' ay = (0i -i)0 ' az = (10 -10) with {A9) Then the eigenfunctions for an intrinsic spin of 1/2 obey the eigenvalue equations {AlO) Besides the commutation properties embodied in equation A2, the a matrices obey the anti-commutation relations {All) in particular, a~ =a~ =a; =1 , where 1 is the 2 x 2 unit matrix. Together with 1 , the a1 are sufficient for a complete description of a spin-t system, i.e. a system with two possible states (m = ± t). For this reason they can also be used to represent the isospin of an isospin doublet such as the neutron- proton pair. When used to represent isospin, the a, matrices are usually denoted Tt (see Wilkinson, 1969). It is often convenient to use two different coordinate systems which are oriented in space in different directions. [One example is in the description of a nucleus with a permanently deformed (non-spherical) shape. It is helpful to use a set of axes fiXed along the principal axes of the nucleus {the body-fixed axes) as well as a set independently ftxed in space {the space-fzxed axes). If the nucleus is rotating, the body-ftxed axes will be rotating with respect to the space-fiXed ones.] Then a state with angular momentum j which has a definite projection m on one z-axis will have a distribvtion of projections m' with respect to the other z' -axis. This distribution comes from the quantum uncertainty in the x andy components and can also be visualised using the vector model as being due to the precession of J about the z-axis (Figure Al). The rotational transformation which determines this distribution in m' is described in standard texts {Brink and Satchler, 1968; Messiah, 1962). A2 ANGULAR MOMENTUM COUPLING AND SYSTEMS COMPOSED OF TWO OR MORE PARTS Often we have systems made up of two parts, each with angular momentum. These parts may be two different particles, or perhaps the spin and orbital properties of a single particle. Each part has associated with it an angular momentum operator and its z-component. Let these be J 1 , J1z, J2 , and J2z. Then we have two choices for a set of four commuting operators for the com· bined system. One choice is {Al2) APPENDIX A 287 The corresponding eigenfunctions are products of the eigenfunctions for each part and obey the eigenvalue equations Jlli1m1} lj2m2> = it(it + l)lhmt> lhm2> (A13) (Al4) where i = 1 or 2. The other choice of a set of commuting operators is in terms of the total angular momentum of the combined system, J = J 1 + J2 • It is (Al5) The eigenfunctions of the coupled system we will write as Ij d 2JM}. They obey the eigenvalue equations Jllitj2JM> = j;(j; + l)lid2JM>, i = 1, 2 (A16) 121iti2JM> =J(J + l)ljihJM> (A17) lz lithlM> =Miid2JM> (A18) For given i1 and j 2 , the values of J are restricted by the triangular condition of vector addition (Al9) and the allowed J ranges between these limits in integer steps. The choice Al2 would be especially useful if the system were isolated and if there was no interaction between the two parts. Then the angular momentum and its orientation for each part would remain constants of the motion, as is expressed by equations Al3, Al4. However, if there is an interaction between the two parts, it is likely that the individual components m; will not remain constant even if the magnitudes it do: the two vectors ji will tend to precess around their resultant J instead of each precessing independently about the z-axis. If the system as a whole is isolated, J2 and lz will remain constants. These two extremes are pictured using the vector model in Figure A2. These two descriptions are not independent but their eigenfunctions are related by a unitary transformation. This may be written explicitly as lid2JM} = ~ ljlml}lhm2} <iihm1m2lid2JM) (A20) m 1 m 2 or the inverse ljlml}lj2m2} = ~ liti2JM} <iti2JMij.j2mlm2} (A21) J These equations defme the (real and symmetric) Wigner or Qebsch-Gordan coefficient (A22) 288 INTRODUCTION TO NUCLEAR REACTIONS z --- --- (D) (b) Figure A2 Vector picture of two angular momentaj1 andj2 • (a) In the coupled (J, M) representation, i 1 and i 2 have a resultant J with projection M. J precesses about the z-axis, while j 1 and j 2 precess about J; m 1 and m 2 are indeterminate.
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