Theory of Angular Momentum and Spin


 Myron Francis
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1 Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO3 of the associated rotation matrices and the corresponding transformation matrices of spin states forming the group SU occupy a very important position in physics. The reason is that these transformations and groups are closely tied to the properties of elementary particles, the building blocks of matter, but also to the properties of composite systems. Examples of the latter with particularly simple transformation properties are closed shell atoms, e.g., helium, neon, argon, the magic number nuclei like carbon, or the proton and the neutron made up of three quarks, all composite systems which appear spherical as far as their charge distribution is concerned. In this section we want to investigate how elementary and composite systems are described. To develop a systematic description of rotational properties of composite quantum systems the consideration of rotational transformations is the best starting point. As an illustration we will consider first rotational transformations acting on vectors r in 3dimensional space, i.e., r R 3, we will then consider transformations of wavefunctions ψ r of single particles in R 3, and finally transformations of products of wavefunctions like N j= ψ j r j which represent a system of N spinzero particles in R 3. We will also review below the wellknown fact that spin states under rotations behave essentially identical to angular momentum states, i.e., we will find that the algebraic properties of operators governing spatial and spin rotation are identical and that the results derived for products of angular momentum states can be applied to products of spin states or a combination of angular momentum and spin states. 5. Matrix Representation of the group SO3 In the following we provide a brief introduction to the group of threedimensional rotation matrices. We will also introduce the generators of this group and their algebra as well as the representation of rotations through exponential operators. The mathematical techniques presented in this section will be used throughout the remainder of these notes. 97
2 98 Theory of Angular Momentum and Spin Properties of Rotations in R 3 Rotational transformations of vectors r R 3, in Cartesian coordinates r = x, x, x 3 T, are linear and, therefore, can be represented by 3 3 matrices R ϑ, where ϑ denotes the rotation, namely around an axis given by the direction of ϑ and by an angle ϑ. We assume the convention that rotations are righthanded, i.e., if the thumb in your right hand fist points in the direction of ϑ, then the fingers of your fist point in the direction of the rotation. ϑ parametrizes the rotations uniquely as long as ϑ < π. Let us define the rotated vector as r = R ϑ r. 5. In Cartesian coordinates this reads 3 x k = [R ϑ] kj x j k =,, j= Rotations conserve the scalar product between any pair of vectors a and b, i.e., they conserve a b = 3 j= a jb j. It holds then a b = 3 j,k,l= Since this holds for any a and b, it follows [R ϑ] jk [R ϑ] jl a k b l = 3 a j b j. 5.3 j= 3 [R ϑ] jk [R ϑ] jl = δ kl. 5.4 j= With the definition of the transposed matrix R T this property can be written [R T ] jk R kj 5.5 R ϑ R T ϑ = R T ϑ R ϑ =. 5.6 This equation states the key characteristic of rotation matrices. From 5.6 follows immediatety for the determinant of R ϑ using deta B = deta detb and deta T = deta detr ϑ = ±. 5.7 Let us consider briefly an example to illustrate rotational transformations and to interpret the sign of detr ϑ. A rotation around the x 3 axis by an angle ϕ is described by the matrix cosϕ sinϕ 0 R ϑ = 0, 0, ϕ T = ± sinϕ cosϕ In case of a prefactor +, the matrix represents a proper rotation, in case of a prefactor a rotation and an inversion at the origin. In the latter case the determinant of R ϑ = 0, 0, ϕ T is negative, i.e. a minus sign in Eq.5.7 implies that a rotation is associated with an inversion. In the following we want to exclude inversions and consider only rotation matrices with detr =.
3 5.: Matrix Representation of SO3 99 Definition of the Group SO3 We will consider then the following set SO3 = { real 3 3 matrices R; R R T = R T R =, detr = } 5.9 This set of matrices is closed under matrix multiplication and, in fact, forms a group G satisfying the axioms i For any pair a, b G the product c = a b is also in G. ii There exists an element e in G with the property a G e a = a e = a. iii a, a G a G with a a = a a = e. iv For products of three elements holds the associative law a b c = a b c. We want to prove now that SO3 as defined in 5.9 forms a group. For this purpose we must demonstrate that the group properties i iv hold. In the following we will assume R, R SO3 and do not write out explicitly since it represents the matrix product. i Obviously, R 3 = R R is a real, 3 3 matrix. It holds R T 3 R 3 = R R T R R = R T R T R R = R T R =. 5.0 Similarly, one can show R 3 R T 3 =. Furthermore, it holds It follows that R 3 is also an element of SO3. detr 3 = det R R = detr detr =. 5. ii The group element e is played by the 3 3 identity matrix. iii From detr 0 follows that R is nonsingular and, hence, there exists a real 3 3 matrix R which is the inverse of R. We need to demonstrate that this inverse belongs also to SO3. Since R T = R T it follows which implies R SO3. R T R = R T R = R R T = = 5. iv Since the associative law holds for multiplication of any square matrices this property holds for the elements of SO3. We have shown altogether that the elements of SO3 form a group. We would like to point out the obvious property that for all elements R of SO3 holds R = R T 5.3
4 00 Theory of Angular Momentum and Spin Exercise 5..: Test if the following sets together with the stated binary operation form groups; identify subgroups; establish if  homomorphic mappings exist, i.e. mappings between the sets which conserve the group properties: a the set of real and imaginary numbers {, i,, i} together with multiplication as the binary operation ; b the set of matrices {M, M, M 3, M 4 } = { 0 0 0, 0 0, 0 0, 0 } 5.4 together with matrix multiplication as the binary operation ; c the set of four rotations in the x, x plane {rotation by O o, rotation by 90 o, rotation by 80 o, rotation by 70 o } and consecutive execution of rotation as the binary operation. Generators, Lie Group, Lie Algebra We want to consider now a most convenient, compact representation of R ϑ which will be of general use and, in fact, will play a central role in the representation of many other symmetry transformations in Physics. This representation expresses rotation matrices in terms of three 3 3 matrices J, J, J 3 as follows R ϑ = exp i ϑ J 5.5 Below we will construct appropriate matrices J k, presently, we want to assume that they exist. We want to show that for the property R R T = to hold, the matrices L k = i J k 5.6 must be antisymmetric. To demonstrate this property consider rotations with ϑ = ϑ k ê k where ê k is a unit vector in the direction of the Cartesian k axis. Geometric intuition tells us Using the property which holds for matrix functions Rϑ k ê k = R ϑ k ê k = exp ϑ k L k 5.7 [fr] T = fr T 5.8 we can employ 5.3 Rϑ k ê k = Rϑ k ê k T = [exp ϑ k L k ] T = exp ϑ k L T k 5.9 and, due to the uniqueness of the inverse, we arrive at the property from which follows L T k = L k. exp ϑ k L T k = exp ϑk L k 5.0
5 5.: Matrix Representation of SO3 0 We can conclude then that if elements of SO3 can be expressed as R = e A, the real matrix A is antisymmetric. This property gives A then only three independent matrix elements, i.e., A must have the form 0 a b A = a 0 c. 5. b c 0 It is this the reason why one can expect that threedimensional rotation matrices can be expressed through 5.5 with three real parameters ϑ, ϑ, ϑ 3 and three matrices L, L, L 3 which are independent of the rotation angles. We assume now that we parameterize rotations through a threedimensional rotation vector ϑ = ϑ, ϑ, ϑ 3 T such that for ϑ = 0, 0, 0 T the rotation is the identity. One can determine then the matrices L k defined in 5.5 as the derivatives of R ϑ with respect to the Cartesian components ϑ k taken at ϑ = 0, 0, 0 T. Using the definition of partial derivatives we can state L = lim ϑ 0 ϑ R ϑ = ϑ, 0, 0 T and similar for L and L 3. Let us use this definition to evaluate L 3. Using 5.8 we can state lim R[0, 0, ϑ 3 T ] = ϑ 3 0 lim ϑ 3 0 cosϑ 3 sinϑ 3 0 sinϑ 3 cosϑ = ϑ 3 0 ϑ Insertion into 5. for k = 3 yields L 3 = ϑ 3 0 ϑ 3 0 ϑ = One obtains in this way i J = L = i J = L = i J 3 = L 3 = The matrices L k are called the generators of the group SO3. Exercise 5..: Determine the generator L according to Eq.5..
6 0 Theory of Angular Momentum and Spin Sofar it is by no means obvious that the generators L k allow one to represent the rotation matrices for any finite rotation, i.e., that the operators 5.5 obey the group property. The latter property implies that for any ϑ and ϑ there exists a ϑ 3 such that exp i ϑ J exp i ϑ J = exp i ϑ 3 J 5.8 The construction of the generators through the limit taken in Eq. 5. implies, however, that the generators represent infinitesimal rotations with >> ϑ around the x , x , and x 3 axes. Finite rotations can be obtained by applications of many infinitesimal rotations of the type R ϑ = ɛ, 0, 0 T = expɛ L, R ϑ = 0, ɛ, 0 T = expɛ L and R ϑ = 0, 0, ɛ 3 T = expɛ 3 L 3. The question is then, however, if the resulting products of exponential operators can be expressed in the form 5.5. An answer can be found considering the BakerCampbellHausdorf expansion exp λa exp λb = exp λ n Z n 5.9 where Z = A + B and where the remaining operators Z n are commutators of A and B, commutators of commutators of A and B, etc. The expression for Z n are actually rather complicated, e.g. Z = [A, B], Z 3 = [A, [A, B]] + [[A, B], B]. From this result one can conclude that expressions of the type 5.5 will be closed under matrix multiplication only in the case that commutators of L k can be expressed in terms of linear combinations of generators, i.e., it must hold 3 [L k, L l ] = f klm L m m= Groups with the property that their elements can be expressed like 5.5 and their generators obey the property 5.30 are called Lie groups, the property 5.30 is called the Lie algebra, and the constants f klm are called the structure constants. Of course, Lie groups can have any number of generators, three being a special case. In case of the group SO3 the structure constants are particularly simple. In fact, the Lie algebra of SO3 can be written [L k, L l ] = ɛ klm L m 5.3 where ɛ klm are the elements of the totally antisymmetric 3dimensional tensor, the elements of which are 0 if any two indices are identical for k =, l =, m = 3 ɛ klm =. 5.3 for any even permutation of k =, l =, m = 3 for any odd permutation of k =, l =, m = 3 For the matrices J k = i L k which, as we see later, are related to angular momentum operators, holds the algebra [J k, J l ] = i ɛ klm J m We want to demonstrate now that exp ϑ L yields the known rotational transformations, e.g., the matrix 5.8 in case of ϑ = 0, 0, ϕ T. We want to consider, in fact, only the latter example and n=
7 5.: Matrix Representation of SO3 03 determine expϕl 3. We note, that the matrix ϕl 3, according to 5.7, can be written 0 ϕl 3 = A 0 0, A = ϕ One obtains then for the rotational transformation [ ] 0 exp A 0 = n=0 = = ν=0 ν! Aν ν= ν= ν! ν! = 0 A A ν e A ν To determine expa = ν= Aν /ν! we split its Taylor expansion into even and odd powers e A = n! An + n +! An The property 0 0 allows one to write n 0 = n and, accordingly, e A = n=0 n 0 n! ϕn 0 n=0 =, n=0 0 0 Recognizing the Taylor expansions of cos ϕ and sin ϕ one obtains e A cos ϕ sin ϕ = sin ϕ cos ϕ n+ = n 0 0 n 0 n +! ϕn and, using 5.34, 5.35, which agrees with 5.8. e ϕl 3 = cos ϕ sin ϕ 0 sin ϕ cos ϕ Exercise 5..3: Show that the generators L k of SO3 obey the commutation relationship 5.3.
8 04 Theory of Angular Momentum and Spin 5. Function space representation of the group SO3 In this section we will investigate how rotational transformations act on single particle wavefunctions ψ r. In particular, we will learn that transformation patterns and invariances are connected with the angular momentum states of quantum mechanics. Definition We first define how a rotational transformation acts on a wavefunction ψ r. For this purpose we require stringent continuity properties of the wavefunction: the wavefunctions under consideration must be elements of the set C 3, i.e., complexvalued functions over the 3dimensional space R 3 which can be differentiated infinitely often. In analogy to R ϑ being a linear map R 3 R 3, we define rotations R ϑ as linear maps C 3 C 3, namely R ϑψ r = ψr ϑ r. 5.4 Obviously, the transformation R ϑ: C 3 C 3 is related to the transformation R ϑ: R 3 R 3, a relationship which we like to express as follows R ϑ = ρ R ϑ ρ conserves the group property of SO3, i.e. for A, B SO3 holds ρa B = ρa ρb This important property which makes ρ SO3 also into a group can be proven by considering ρ R ϑ R ϑ ψ r = ψ[ R ϑ R ϑ ] r 5.45 = ψ[r ϑ ] [R ϑ ] r = ρ R ϑ ψ[r ϑ ] r 5.46 = ρ R ϑ ρ R ϑ ψ r Since this holds for any ψ r one can conclude ρ R ϑ ρ R ϑ = ρ R ϑ R ϑ, i.e. the group property 5.44 holds. Exercise 5..: Assume the definition ρ R ϑ ψ r = ψr ϑ r. Show that in this case holds ρ A B = ρ B ρ A. Generators One can assume then that R ϑ should also form a Lie group, in fact, one isomorphic to SO3 and, hence, its elements can be represented through an exponential R ϑ = exp i ϑ J 5.48
9 5.: Function Space Representation of SO3 05 where the generators can be determined in analogy to Eq. 5. according to i J = lim ϑ 0 ϑ R ϑ = ϑ, 0, 0 T 5.49 note the minus sign of ϑ which originates from the inverse of the rotation in Eq.5.4 and similarly for J and J 3. The generators J k can be determined by applying 5.49 to a function f r, i.e., in case of J 3, i J 3 f r = lim ϑ 3 0 ϑ 3 R ϑ = 0, 0, ϑ 3 T f r f r = lim ϑ 3 0 ϑ 3 f R0, 0, ϑ 3 r f r Using 5.3, 5.4 one can expand R0, 0, ϑ 3 r ϑ 3 0 ϑ x x x 3 = x + ϑ 3 x ϑ 3 x + x. x Inserting this into 5.50 and Taylor expansion yields i J 3 f r = lim ϑ 3 0 ϑ 3 f r + ϑ 3 x f r ϑ 3 x f r f r = x x f r. 5.5 Carrying out similar calculations for J and J one can derive i J = x 3 x i J = x 3 x i J 3 = x x Not only is the group property of SO3 conserved by ρ, but also the Lie algebra of the generators. In fact, it holds [J k, J l ] = i ɛ klm J m We like to verify this property for [J, J ]. One obtains using and f r = f [J, J ] f r = [ i J, i J ] f r = [ x 3 x 3 x 3 f x 3 f x 3 x 3 x 3 f x 3 f ] = [ +x x 3 3 f x x 3 f x 3 f + x x 3 3 f + x f x f x x 3 3 f + x 3 f + x x 3 f x x 3 3 f ] = x x f r = i J 3 f r Exercise 5..: a Derive the generators J k, k =, by means of limits as suggested in Eq. 5.49; show that 5.56 holds for [J, J 3 ] and [J 3, J ].
10 06 Theory of Angular Momentum and Spin Exercise 5..3: Consider a wave function ψφ which depends only on the azimutal angle φ of the spherical coordinate system r, θ, φ, i.e. the system related to the Cartesian coordinates through x = r sinθ cosφ, x = r sinθ sinφ, and x 3 = r cosθ. Show that for J 3 as defined above holds exp iα J 3 ψφ = ψφ + α. 5.3 Angular Momentum Operators The generators J k can be readily recognized as the angular momentum operators of quantum mechanics. To show this we first note that can be written as a vector product of r and i J = r From this follows, using the momentum operator ˆp = i, J = r i = r ˆp 5.59 which, according to the correspondence principle between classical and quantum mechanics, identifies J as the quantum mechanical angular momentum operator. Equation 5.56 are the famous commutation relationships of the quantium mechanical angular momentum operators. The commutation relationships imply that no simultaneous eigenstates of all three J k exist. However, the operator commutes with all three J k, k =,, 3, i.e., J = J + J + J [J, J k ] = 0, k =,, We demonstrate this property for k = 3. Using [AB, C] = A[B, C] + [A, C]B and 5.56 one obtains [J, J 3 ] = [J + J, J 3 ] 5.6 = J [J, J 3 ] + [J, J 3 ] J + J [J, J 3 ] + [J, J 3 ] J = i J J i J J + i J J + i J J = 0 According to 5.6 simultaneous eigenstates of J and of one of the J k, usually chosen as J 3, can be found. These eigenstates are the wellknown spherical harmonics Y lm θ, φ. We will show now that the properties of spherical harmonics J Y lm θ, φ = ll + Y lm θ, φ 5.63 J 3 Y lm θ, φ = m Y lm θ, φ 5.64 as well as the effect of the socalled raising and lowering operators J ± = J ± i J 5.65
11 5.3: Angular Momentum Operators 07 on the spherical harmonics, namely, J + Y lm θ, φ = l + m + l m Y lm+ θ, φ 5.66 J + Y ll θ, φ = J Y lm+ θ, φ = l + m + l m Y lm θ, φ 5.68 J Y l l θ, φ = are a consequence of the Lie algebra For this purpose we prove the following theorem. An Important Theorem Theorem. Let L k, k =,, 3 be operators acting on a Hilbert space H which obey the algebra [L k, L l ] = ɛ klm L m and let L ± = L ± i L, L = L + L + L 3. Let there exist states l l in H with the property L + l l = L 3 l l = i l l l 5.7 then the states defined through L l m + = i β m l m 5.7 have the following properties i L + l m = i α m l m ii L 3 l m = i m l m 5.74 iii L l m = ll + l m 5.75 To prove this theorem we first show by induction that i holds. For this purpose we first demonstrate that the property holds for m = l by considering i for the state L ll ll. It holds L + L ll = L + L L L + + L L + l l = [L +, L ] + L L + l l. Noting [L +, L ] = [L + il, L il ] = il 3 and L + l l = 0 one obtains L + L ll = il 3 ll = l ll, i.e., in fact, L + raises the mvalue of ll from m = l to m = l. The definitions of the coefficients α l and β l yield L + L ll = L + iβ l ll = α l β l ll, i.e. α l β l = l We now show that property ii also holds for m = l proceeding in a similar way. We note L 3 L ll = [L 3, L ] + L L 3 ll. Using [L 3, L ] = [L 3, L il ] = L + il = i L il = il and L 3 ll = il ll we obtain L 3 L ll = il + L L 3 ll = i l + L ll. From this follows that L ll is an eigenstate of L 3, and since we have already shown ll = iβ l L ll we can state L 3 ll = il ll. Continuing our proof through induction we assume now that property i holds for m. We will show that this property holds then also for m. The arguments are very similar to the ones used above and we can be brief. We consider L + L lm = [L +, L ] + L L + lm = il 3 +
12 08 Theory of Angular Momentum and Spin L L + lm = m α m β m lm. This implies that i holds for m, in particular, from L + L lm = α m β m lm follows the recursion relationship α m β m = m + α m β m We will show now that if ii holds for m, it also holds for m. Again the argumemts are similar to the ones used above. We note L 3 L lm = [L 3, L ] + L L 3 lm = il i m L lm = i m L lm from which we can deduce L 3 lm = im lm. Before we can finally show that property iii holds as well for all l m l we need to determine the coefficients α m and β m. We can deduce from and α l β l = l α l β l = l + l α l 3 β l 3 = l + l + l l Obviously, it holds α m β m = k=m+ k. Using the familiar formula n k=0 k = n n + / one obtains α m β m = l + l m + m = l + l + m l l m m + m = l + m + l m 5.78 One can normalize the states lm such that α m = β m, i.e., finally α m = β m = l + m + l m We can now show that iii holds for all proper m values. We note that we can write L = L +L + L L + + L 3. It follows then L lm = α m β m + α mβ m + m lm = l + m + l m + l + ml m + + m = ll + lm. This completes the proof of the theorem. We note that Eqs. 5.70, 5.7, 5.73, 5.79 read L + lm = i l + m + l m lm L lm + = i l + m + l m lm, 5.8 i.e., yield properties 5.66, We will have various opportunities to employ the Theorem just derived. A first application will involve the construction of the eigenstates of the angular momentum operators as defined in 5.63, For this purpose we need to express the angular momentum operators in terms of spherical coordinates r, θ, φ. Angular Momentum Operators in Spherical Coordinates We want to express now the generators J k, given in , in terms of spherical coordinates defined through x = r sin θ cos φ 5.8 x = r sin θ sin φ 5.83 x 3 = r cos θ 5.84
13 5.3: Angular Momentum Operators 09 We first like to demonstrate that the generators J k, actually, involve only the angular variables θ and φ and not the radius r. In fact, we will prove i J = L = sin φ + cot θ cos φ θ φ i J = L = cos φ + cot θ sin φ θ φ i J 3 = L 3 = φ To derive these properties we consider, for example, L fθ, φ = x 3 x 3 fθ, φ one obtains, applying repeatedly the chain rule, = x3 x x x /x x 3 }{{} = 0 Using L fθ, φ = x 3 x 3 fθ, φ 5.88 x /x = x 3 x tan φ + x x 3 /r x cos θ tan φ x x 3/r x 3 cos θ fθ, φ tan φ x 3 x r 3 cos θ x x + x r 3 fθ, φ 5.90 cos θ This yields, using / tanφ = cos φ / φ and / cosθ = /sinθ / θ, cos θ L fθ, φ = sin θ cos φ cos φ φ cos θ sin θ sin φ sin θ sin φ + sin θ fθ, φ sin θ θ sin θ θ which agrees with expression We like to express now the operators J 3 and J, the latter defined in 5.60, in terms of spherical coordinates. According to 5.87 holds To determine J we note, using 5.85, L = sin φ + cot θ cos φ θ φ and similarly, using 5.86, L = cos φ θ J 3 = i φ. 5.9 = sin φ θ sin sinφ cosφ θ φ + cotθ sinφ cosφ θ φ + cotθ cos φ θ + cot θ cos φ φ cot θ sin φ φ
14 0 Theory of Angular Momentum and Spin = cos φ θ + sin sinφ cosφ θ φ cotθ sinφ cosφ θ φ + cotθ sin φ θ + cot θ sin φ φ It follows L + L + L 3 = θ + cotθ θ + cot θ + φ With cot θ + = /sin θ, sin θ sinθ sinθ θ θ = θ + cotθ θ, 5.96 and the relationship between L k and J k as given in 5.85, 5.86, 5.87, one can conclude J = sin θ [ sinθ ] + θ φ Kinetic Energy Operator The kinetic energy of a classical particle can be expressed p m = p r m + J class m r 5.98 where J class = r m v is the angular momentum and p r = mṙ is the radial momentum. The corresponding expression for the quantum mechanical kinetic energy operator is ˆT = m = m r r r + J m r, 5.99 which follows from the identity r = r + r r r = r + r r + r sin θ r [ sinθ ] + θ φ and comparision with Angular Momentum Eigenstates According to the theorem on page 07 above the eigenfunctions 5.63, 5.64 can be constructed as follows. One first identifies the Hilbert space H on which the generators L k operate. The operators in the present case are L k = i J k where the J k are differential operators in θ and φ given in 5.85, 5.86, These generators act on a subspace of C 3, namely on the space of complexvalued functions on the unit 3dim. sphere S, i.e. C S. The functions in C S have real variables
15 5.4: Angular Momentum Eigenstates θ, φ, 0 θ < π, 0 φ < π, and to be admissable for description of quantum states, must be cyclic in φ with period π. The norm in H is defined through f = dω f θ, φfθ, φ 5.0 S where dω = sinθ dθ dφ is the volume element of S. The function space endowed with this norm and including only functions for which the integral 5.0 exists, is indeed a Hilbert space. The eigenfunctions 5.63, 5.64 can be constructed then by seeking first functions Y ll θ, φ in H which satisfy L + Y ll θ, φ = 0 as well as L 3 Y ll θ, φ = il Y ll θ, φ, normalizing these functions, and then determining the family {Y lm θ, φ, m = l, l +,..., l} applying L Y lm+ θ, φ = i l + m + l my lm θ, φ 5.03 iteratively for m = l, l,..., l. One obtains in this way l m Y lm θ, φ = l, m J Y llθ, φ 5.04 [ ] l + m! l, m = 5.05 l! l m! Constructing Y ll Y ll Y l l We like to characterize the resulting eigenfunctions Y lm θ, φ, the socalled spherical harmonics, by carrying out the construction according to 5.04, 5.05 explicitly. For this purpose we split off a suitable normalization factor as well as introduce the assumption that the dependence on φ is described by a factor expimφ It must hold Y lm θ, φ = l + l m! 4π l + m! P lmcosθ e imφ L + Y ll θ, φ = L 3 Y ll θ, φ = i l Y ll θ, φ The latter property is obviously satisfied for the chosen φdependence. The raising and lowering operators L ± = L ± i L, 5.09 using , can be expressed L ± = i e ±iφ θ ± i cotθ φ. 5.0 Accordingly, 5.07 reads θ + i cotθ P ll cosθ e ilφ = 0 5. φ
16 Theory of Angular Momentum and Spin or θ l cotθ P ll cosθ = 0 5. where we employed the definition of P ll cosθ in A proper solution of this equation, i.e., one for which is finite, is π 0 dθ sinθ P ll cosθ 5.3 P ll cos θ = N l sin l θ 5.4 as one can readily verify. The normalization factor N l is chosen such that holds. 5.06, 5.4 yield π 0 N l l + 4π dθ sinθ Y ll cosθ = 5.5 π l! π dθ sin l+ θ = The integral appearing in the last expression can be evaluated by repeated integration by parts. One obtains, using the new integration variable x = cos θ, π 0 = dθ sin l+ θ = + [x x l] + dx x l + l [ x 3 = l 3 x l. = = Accordingly, 5.6 implies from which we conclude l l 3 5 l l l 3 5 l ] dx x x l l l 3 dx x l l + = + dx x 4 x l l! [ 3 5 l ] l N l [ 3 5 l ] = 5.8 N l = l 3 5 l 5.9 where the factor l has been included to agree with convention. We note that 5.06, 5.4, 5.9 provide the following expression for Y ll Y ll θ, φ = l l + 4π l! l l! sinl θ e ilφ. 5.0 See, for example, Classical Electrodynamics, nd Ed. by J.D. Jackson John Wiley, New York, 975
17 5.4: Angular Momentum Eigenstates 3 We have constructed a normalized solution of 5.06, namely Y ll θ, φ, and can obtain now, through repeated application of 5.03, 5.0, the eigenfunctions Y lm θ, φ, m = l, l,... l defined in 5.63, Actually, we seek to determine the functions P lm cos θ and, therefore, use 5.03, 5.0 together with 5.06 i e iφ θ i cotθ φ l + = i l + m + l m 4π l m! l + m +! P lm+cos θ e im+φ l + l m! 4π l + m! P lmcos θ e imφ. 5. This expressions shows that the factor e imφ, indeed, describes the φdependence of Y lm θ, φ; every application of L reduces the power of e iφ by one. 5. states then for the functions P lm cos θ + m + cotθ P lm+ cos θ = l + m + l m P lm cos θ. 5. θ The latter identity can be written, employing again the variable x = cos θ, P lm x = l + m + l m 5.3 [ ] x x + m + P x x lm+ x. We want to demonstrate now that the recursion equation 5.3 leads to the expression P lm x = l + m! l l! l m! x m l m x l m x l 5.4 called the associated Legendre polynomials. The reader should note that the associated Legendre polynomials, as specified in 5.4, are real. To prove 5.4 we proceed by induction. For m = l 5.4 reads P ll x = l! l l! x l x l = l 3 5 l sin l θ 5.5 which agrees with 5.4, 5.9. Let us then assume that 5.4 holds for m +, i.e., Then holds, according to 5.3, P lm+ x = l + m +! l l! l m! x m+ P lm x = l + m! l l! l m! x m+ [ x x l m x l m x l. 5.6 x + m + x l m x l m x l. 5.7 ]
18 4 Theory of Angular Momentum and Spin By means of x = x m x x m+ l m x l m x l l m x l m x l x m + x x m+ l m x l m 5.8 one can show that 5.7 reproduces 5.4 and, therefore, that expression 5.4 holds for m if it holds for m +. Since 5.4 holds for m = l, it holds then for all m. We want to test if the recursion 5.3 terminates for m = l, i.e., if L Y ll θ, φ = holds. In fact, expression 5.4, which is equivalent to recursive application of L, yields a vanishing expression for m = l as long as l is a nonnegative integer. In that case x l is a polynomial of degree l and, hence, the derivative l+ / x l+ of this expression vanishes. Constructing Y l l Y l l+ Y ll An alternative route to construct the eigenfunctions Y lm θ, φ determines first a normalized solution of L fθ, φ = L 3 fθ, φ = i l fθ, φ, 5.3 identifies fθ, φ = Y l l θ, φ choosing the proper sign, and constructs then the eigenfunctions Y l l+, Y l l+, etc. by repeated application of the operator L +. Such construction reproduces the eigenfunctions Y lm θ, φ as given in 5.06, 5.4 and, therefore, appears not very interesting. However, from such construction emerges an important symmetry property of Y lm θ, φ, namely, Y lm θ, φ = m Y l m θ, φ 5.3 which reduces the number of spherical harmonics which need to be evaluated independently roughly by half ; therefore, we embark on this construction in order to prove 5.3. We first determine Y l l θ, φ using 5.06, 5.4. It holds, using x = cos θ, P l l x = l l! l! x l Since the term with the highest power of x l is x l, it holds l x l x l and, therefore, l x l x l = l! 5.34 P l l x = l l! x l. 5.35
19 5.4: Angular Momentum Eigenstates 5 Due to 5.06, 5.4 one arrives at Y l l θ, φ = l + 4π l! l l! sinl θ e ilφ We note in passing that this expression and the expression 5.0 for Y ll obey the postulated relationship 5.3. Obviously, the expression 5.36 is normalized and has the proper sign, i.e., a sign consistent with the family of functions Y lm constructed above. One can readily verify that this expression provides a solution of This follows from the identity θ i cotθ sin l θ e ilφ = φ One can use 5.36 to construct all other Y lm θ, φ. According to 5.80 and 5.0 applies the recursion e iφ θ + i cotθ Y lm θ, φ = l + m + l m Y lm+ θ, φ φ Employing 5.06, one can conclude for the associated Legendre polynomials or l m! l + m! θ m cotθ P lm cos θ = l m! l + m + l m l + m +! P lm+cos θ 5.39 P lm+ cos θ = θ m cotθ P lm cos θ Introducing again the variable x = cos θ leads to the recursion equation [c.f. 5., 5.3] P lm+ x = x x m x x We want to demonstrate now that this recursion equation leads to the expression P lm x. 5.4 P lm x = m l l! x m l+m x l+m x l 5.4 For this purpose we proceed by induction, following closely the proof of eq We first note that for m = l 5.4 yields P l l x = l l l! x l x l = l l! x l 5.43
20 6 Theory of Angular Momentum and Spin wich agrees with We then assume that 5.4 holds for m. 5.4 reads then P lm+ x = 5.44 x x m x m x l x m l+m l! x l+m x l. Replacing in 5.8 m + m or, equivalently, m m yields Hence, 5.44 reads x x x m = x m+ P lm+ x = m+ l l! l+m l+m+ x l+m x l x l+m+ x l x + m x m x x m+ l+m xl+m l+m+ x l+m+ x l, 5.46 i.e., 5.4 holds for m + if it holds for m. Since 5.4 holds for m = l we verified that it holds for all m. The construction beginning with Y l l and continuing with Y l l+, Y l l+, etc. yields the same eigenfunctions as the previous construction beginning with Y ll and stepping down the series of functions Y ll, Y ll, etc. Accordingly, also the associated Legendre polynomials determined this way, i.e., given by 5.4 and by 5.4, are identical. However, one notes that application of 5.4 yields m l m! l + m! P l mx = m l x m l! the r.h.s. of which agrees with the r.h.s. of 5.4. Hence, one can conclude l+m x l+m x l, 5.47 m l m! P lm x = l + m! P l mx According to 5.06 this implies for Y lm θ, φ the identity 5.3. The Legendre Polynomials The functions P l x = P l0 x 5.49 are called Legendre polynomials. According to both 5.4 and 5.4 one can state P l x = l l! l x l x l. 5.50
21 5.4: Angular Momentum Eigenstates 7 The first few polynomials are P 0 x =, P x = x, P x = 3x P 3 x = 5x3 3x. 5.5 Comparision of 5.50 and 5.4 allows one to express the associated Legendre polynomials in terms of Legendre polynomials P lm x = m x m m x m P lx, m and, accordingly, the spherical harmonics for m 0 l + l m! Y lm θ, φ = 4π l + m! m sin m θ 5.53 m P l cos θ e imφ. cos θ The spherical harmonics for m < 0 can be obtained using 5.3. The Legendre polynomials arise in classical electrodynamics as the expansion coefficients of the electrostatic potential around a point charge q/ r r where q is the charge, r is the point where the potential is measured, and r denotes the location of the charge. In case q = and r < r holds the identity r r where γ is the angle between r and r. Using x = cos γ, t = r /r, and 5.54 can be written wx, t = = l=0 r l r l+ P l cos γ 5.54 r r = r x t + t, 5.55 = xt + t l=0 P l x t l wx, t is called a generating function of the Legendre polynomials 3. The generating function allows one to derive useful properties of the Legendre polynomials. For example, in case of x = holds w, t = = t + t t = l=0 t l Comparision with 5.56 yields P l =, l = 0,,, see, e.g., Classical Electrodynamics, nd Ed. by J.D. Jackson John Wiley, New York, 975, pp. 9 3 To prove this property see, for example, pp. 45 of Special Functions and their Applications by N.N. Lebedev PrenticeHall, Englewood Cliffs, N.J., 965 which is an excellent compendium on special functions employed in physics.
22 8 Theory of Angular Momentum and Spin For wx, t holds lnwx, t t = x t xt + t Using lnw/ t = /w w/ t and multiplying 5.59 by w xt + t leads to the differential equation obeyed by wx, t xt + t w t + t x w = Employing 5.56 and / t l P lxt l = l lp lxt l this differential equation is equivalent to xt + t lp l x t l + t x P l x t l = l=0 Collecting coefficients with equal powers t l yields 0 = P x x P 0 x [l + P l+ x l + x P l x + l P l x ] t l l= The coefficients for all powers t l must vanish individually. Accordingly, holds l=0 P x = x P 0 x 5.63 l + P l+ x = l + x P l x l P l x 5.64 which, using P 0 x = [c.f. 5.5] allows one to determine P l x for l =,,.... Inversion Symmetry of Y lm θ, φ Under inversion at the origin vectors r are replaced by r. If the spherical coordinates of r are r, θ, φ, then the coordinates of r are r, π θ, π + φ. Accordingly, under inversion Y lm θ, φ goes over to Y lm π θ, π + φ. Due to cosπ θ = cos θ the replacement r r alters P lm x into P lm x. Inspection of 5.4 allows one to conclude P lm x = l+m P lm x 5.65 since n / x n = n n / x n. Noting exp[imπ + φ] = m expimφ we determine Properties of Y lm θ, φ Y lm π θ, π + φ = l Y lm θ, φ We want to summarize the properties of the spherical harmonics Y lm θ, φ derived above.. The spherical harmonics are eigenfunctions of the angular momentum operators J Y lm θ, φ = ll + Y lm θ, φ 5.67 J 3 Y lm θ, φ = m Y lm θ, φ. 5.68
23 5.4: Angular Momentum Eigenstates 9. The spherical harmonics form an orthonormal basis of the space C S of normalizable, uniquely defined functions over the unit sphere S which are infinitely often differentiable π 0 sin θdθ π 0 dφy l m θ, φ Y lmθ, φ = δ l lδ m m The spherical harmonics form, in fact, a complete basis of C S, i.e., for any fθ, φ C S holds fθ, φ = C lm = l l=0 m= l π 0 sin θdθ C lm Y lm θ, φ 5.70 π 0 dφ Ylm θ, φfθ, φ The spherical harmonics obey the recursion relationships where J ± = J ± ij. J + Y lm θ, φ = l + m + l m Y lm+ θ, φ 5.7 J Y lm+ θ, φ = l + m + l m Y lm θ, φ The spherical harmonics are given by the formula l + l m! m Y lm θ, φ = 4π l + m! l sin m θ 5.74 l! m P l cos θ e imφ, m 0 cos θ where l =,,... P 0 x =, P x = x, 5.75 P l+ x = l + [ l + x P lx l P l x ] 5.76 are the Legendre polynomials. The spherical harmonics for m < 0 are given by Y l m θ, φ = m Ylm θ, φ Note that the spherical harmonics are real, except for the factor expimφ. 6. The spherical harmonics Y l0 are Y l0 θ, φ = l + 4π P lcos θ For the Legendre polynomials holds the orthogonality property + dx P l x P l x = l + δ ll. 5.79
24 0 Theory of Angular Momentum and Spin 8. The spherical harmonics for θ = 0 are Y lm θ = 0, φ = δ m0 l + 4π The spherical harmonics obey the inversion symmetry 0. The spherical harmonics for l = 0,, are given by Y lm θ, φ = l Y lm π θ, π + φ. 5.8 Y 00 = 4π 5.8 Y = 3 8π sin θ eiφ 5.83 Y 0 = 3 cos θ 4π 5.84 Y = 5 4 π sin θ e iφ 5.85 Y = 5 8π sin θ cos θ eiφ 5.86 Y 0 = 5 3 4π cos θ 5.87 together with For the Laplacian holds hr Y lm θ, φ = [ ] ll + r r r r hr Y lm θ, φ Exercise 5.4.: Derive expressions 5.86, Exercise 5.4.: Derive the orthogonality property for the Legendre polynomials 5.79 using the generating function wx, t stated in For this purpose start from the identity + dx wx, t = l,l =0 + t l+l dxp l x P l x, 5.89 evaluate the integral on the l.h.s., expand the result in powers of t and equate the resulting powers to those arising on the r.h.s. of Exercise 5.4.3: Construct all spherical harmonics Y 3m θ, φ.
25 5.5: Irreducible Representations 5.5 Irreducible Representations We will consider now the effect of the rotational transformations introduced in 5.4, exp i ϑ J, on functions fθ, φ C S. We denote the image of a function fθ, φ under such rotational transformation by fθ, φ, i.e., fθ, φ = exp i ϑ J fθ, φ Since the spherical harmonics Y lm θ, φ provide a complete, orthonormal basis for the function space C S one can expand fθ, φ = l,m c lm = One can also represent C S where [ { of the set. c lm Y lm θ, φ 5.9 S dω Y lm θ, φ exp i ϑ J fθ, φ 5.9 C S = [ {Y lm, l = 0,,..., m = l, l +,..., l} ] 5.93 } ] denotes closure of a set by taking all possible linear combinations of the elements The transformations exp i ϑ J, therefore, are characterized completely if one specifies the transformation of any Y lm θ, φ Ỹ lm θ, φ = [D ] ϑ Y lmθ, φ 5.94 l,m l m ; lm [ D ϑ ] = l m ; lm S dω Y l m θ, φ exp i ϑ J Y lm θ, φ, 5.95 i.e., if one specifies the functional form of the coefficients [Dϑ] l m ; lm. These coefficients can be considered the elements of an infinitedimensional matrix which provides the representation of exp i ϑ J in the basis {Y lm, l = 0,,..., m = l, l +,..., l}. We like to argue that the matrix representing the rotational transformations of the function space C S, with elements given by 5.95, assumes a particularly simple form, called the irreducible representation. For this purpose we consider the subspaces of C S Comparision with 5.93 shows The subspaces X l exp i ϑ J, i.e., exp X l = [ {Y lm, m = l, l +,..., l} ], l = 0,,, C S = l= X l have the important property that i they are invariant under rotations i ϑ J X l = X l, and ii they form the lowest dimensional sets X l obeying C S = l X l which have this invariance property.
26 Theory of Angular Momentum and Spin The invariance of the subspaces X l under rotations follows from a Taylor expansion of the exponential operator exp i ϑ J. One proceeds by expanding first ϑ J in terms of J +, J, and J 3 ϑ J = ϑ J + + ϑ +J + ϑ 3 J where ϑ ± = ϑ ϑ. The exponential operator exp i ϑ J can be expanded in powers ϑ J + + ϑ +J + ϑ 3 J 3 n. One employs then the commutation properties [J +, J ] = J 3, [J 3, J ± ] = ± J ± 5.99 which follow readily from 5.56, Application of 5.99 allows one to express exp i ϑ J = c n,n,n 3 J+ n J n J n n,n,n 3 Accordingly, one needs to consider the action of J n + J n J n 3 3 on the subspaces X l. Since the Y lm are eigenfunctions of J 3 the action of J n 3 3 reproduces the basis functions of X l, i.e., leaves X l invariant. The action of J on Y lm produces Y lm or, in case m = l, produces 0, i.e., J n also leaves X l invariant. Similarly, one can conclude that J n + leaves X l invariant. Since any additive term in 5.00 leaves X l invariant, the transformation exp i ϑ J leaves X l invariant as well. One can conclude then that in the expansion 5.94 only spherical harmonics with l = l contribute or, equivalently, one concludes for the matrix elements 5.95 [Dϑ] l m ; lm = δ l l S dω Y lm θ, φ exp i ϑ J Y lm θ, φ. 5.0 From expression 5.00 of the rotational transformations one can also conclude that the X l are the smallest invariant subspaces of C S. If one orders the basis according to the partitoning 5.97 of C S the matrix representation of Dϑ assumes a blockdiagonal form, with blocks of dimension, 3, 5,..., i.e., Dϑ = 5.0 l + l +...
27 5.6: Wigner Rotation Matrices 3 This representation of Dϑ is the one wich has blocks of the lowest dimensions possible. The representation is referred to as the irreducible representation, other representations are termed reducible representations. Exercise 5.5.: Representations of the group SO SO is the set of all matrices R which are orthogonal, i.e. for which holds R T R = R R T =, and for which detr =. a Show that SO with the group operation defined as matrix multiplication, is a group. b Prove that the elements of SO can be completely characterized through a single parameter. c Show that the map Rϕ x y = x y = cosϕ sinϕ sinϕ cosϕ x y = Rϕ defines a representation of SO. d Show that the similarity transformation T ϕ defined in the space of nonsingular, real matrices O through O = T ϕo = RϕOR ϕ with Rϕ as in c is also a representation of SO. e In the space C of infinitely often differentiable, and periodic functions fα, i.e. fα + π = fα, the map ρϕ defined through fα ρ gϕ = fα ϕ = ρϕfα also defines a representation of SO. Determine the generator of ρϕ in analogy to 5., f The transformation ρϕ as defined in e leaves the following subspace of functions considered in e X m = { fα = A e imα, A C, m N} invariant. Give the corresponding expression of ρϕ. x y 5.6 Wigner Rotation Matrices We will now take an important first step to determine the functional form of the matrix elements of Dϑ. This step reconsiders the parametrization of rotations by the vector ϑ assumed sofar. The three components ϑ k, k =,, 3 certainly allow one to describe any rotation around the origin. However, this parametrization, though seemingly natural, does not provide the simplest mathematical description of rotations. A more suitable parametrization had been suggested by Euler: every rotation exp i ϑ J can be represented also uniquely by three consecutive rotations: i a first rotation around the original x 3 axis by an angle α, ii a second rotation around the new x axis by an angle β, iii a third rotation around the new x 3 axis by an angle γ.
28 4 Theory of Angular Momentum and Spin The angles α, β, γ will be referred to as Euler angles. The axis x is defined in the coordinate frame which is related to the original frame by rotation i, the axis x 3 is defined in the coordinate frame which is related to the original frame by the consecutive rotations i and ii. The Euler rotation replaces exp i ϑ J by e i γj 3 e i βj e i αj For any ϑ R 3 one can find Euler angles α, β, γ R such that exp i ϑ J = e i γj 3 e i βj e i αj is satisfied. Accordingly, one can replace 5.94, 5.95 by Ỹ lm θ, φ = l,m [Dα, β, γ] l m ; lm Y lmθ, φ 5.05 [Dα, β, γ] l m ; lm = dω Yl m θ, φ e i γj 3 e i βj e i αj 3 Y lm θ, φ, S 5.06 The expression 5.03 has the disadvantage that it employs rotations defined with respect to three different frames of reference. We will demonstrate that 5.03 can be expressed, however, in terms of rotations defined with respect to the original frame. For this purpose we notice that J can be expressed through the similarity transformation J = e i αj 3 J e i αj which replaces J by the inverse transformation i, i.e., the transformation from the rotated frame to the original frame, followed by J in the original frame, followed by transformation i, i.e., the transformation from the original frame to the rotated frame. Obviously, the l.h.s. and the r.h.s. of 5.4 are equivalent. For any similarity transformation involving operators A and S holds Accordingly, we can write e S A S = S e A S e i βj = e i αj 3 e i βj e i αj The first two rotations in 5.03, i.e., i and ii, can then be written The third rotation in 5.03, in analogy to 5.09, is Using 5.09 in this expression one obtains e i βj e i αj 3 = e i αj 3 e i βj 5.0 e i γj 3 = e i βj e i αj 3 e i γj 3 e i αj 3 e i βj 5. e i γj 3 = e i αj 3 e i βj e i αj 3 e i αj 3 e i γj 3 e i αj 3 e i αj 3 e i βj e i αj 3 = e i γj 3 = e i αj 3 e i βj e i γj 3 e i βj e i αj 3 5.
29 5.7: Spin and the group SU 5 Multiplication from the left with 5.0 yields the simple result e i γj 3 e i βj e i αj 3 = e i αj 3 e i βj e i γj 3, 5.3 i.e., to redefine the three rotations in 5.03 with respect to the original unprimed frame one simply needs to reverse the order of the rotations. This allows one to express the rotational trasnformation 5.05, 5.06 by Ỹ lm θ, φ = l,m [Dα, β, γ] l m ; lm Y lmθ, φ 5.4 [Dα, β, γ] l m ; lm = S dω Y l m θ, φ e i αj 3 e i βj e i γj 3 Y lm θ, φ. 5.5 The evaluation of the matrix elements 5.5 benefits from the choice of Euler angles for the parametrization of rotational transformations. The eigenvalue property 5.64 yields dω fθ, φ e i γj 3 Y lm θ, φ = S dω fθ, φ Y lm θ, φ e imγ. S 5.6 Using, in addition, the selfadjointness of the operator J 3 one can state dω Ylm αj 3 fθ, φ = e im α S dω Ylm θ, φ fθ, φ. S 5.7 Accordingly, one can write [Dα, β, γ] lm; l m = e iαm dω Ylm βj Y lm θ, φ S e iγm δ ll 5.8 Defining the socalled Wigner rotation matrix d l mm β = dω Y lm θ, φ e i βj Y lm θ, φ 5.9 S one can express the rotation matrices [Dα, β, γ] lm; l m = e iαm d l m m β e iγm δ ll. 5.0 We will derive below [see Eqs , 5.30] an explicit expression for the Wigner rotation matrix Spin and the group SU The spin describes a basic and fascinating property of matter. Best known is the spin of the electron, but many other elementary components of matter are endowed with spinlike properties. Examples are the other five members of the lepton family to which the electron belongs, the electron e and the electron neutrino ν e of the first generation, the muon µ and its neutrino ν µ of the second
30 6 Theory of Angular Momentum and Spin generation, the tau τ and its neutrino ν τ of the third generation and their antiparticles carry a spin. So do the three generations of six quarks, two of which in certain linear combinations make up the vector mesons which carry spin, and three of which make up the baryons which carry spin and spin 3. There are also the mediators, the gluon, the photon, the two W± and the Z o, particles which mediate the strong and the electroweak interactions, and which carry spin. The particles mentioned, e.g. the quarks, carry other spinlike properties which together, however, have properties beyond those of single spins. In any case, there is nothing more elementary to matter than the spin property. The presence of this property permeates matter also at larger scales than those of the elementary particles mentioned, leaving its imprint on the properties of nuclei, atoms and molecules; in fact, the spin of the electron is likely the most important property in Chemistry. We may finally mention that the spin is at the heart of many properties of condensed matter systems, like superconductivity and magnetism. It appears to be rather impossible for a Physicist not to be enamored with the spin property. We will find that the spin in its transformation behaviour is closely related to angular momentum states, a relationship, which might be the reason why consideration of rotational symmetry is so often fruitful in the study of matter. We will consider first only the socalled spin, generalizing then further below. Spin systems can be related to two states which we denote by χ + and χ. Such systems can also assume any linear combination c + χ + + c χ, c ± C, as long as c + + c =. If we identify the states χ + and χ with the basis of a Hilbert space, in which we define the scalar product between any state = a + χ + + a χ and = b + χ + + b χ as = a +b + + a b, then allowed symmetry transformations of spin states are described by matrices U with complexvalued matrix elements U jk. Conservation of the scalar product under symmetry transformations requires the property U U = U U = where U denotes the adjoint matrix with elements [ U ] jk = U kj. We will specify for the transformations considered detu =. This specification implies that we consider transformations save for overall factors e iφ since such factors are known not to affect any observable properties. The transformation matrices are then elements of the set SU = { complex matrices U; U U = U U =, detu = } 5. One can show readily that this set forms a group with the groups binary operation being matrix multiplication. How can the elements of SU be parametrized. As complex matrices one needs, in principle, eight real numbers to specify the matrix elements, four real and four imaginary parts of U jk, j, k =,. Because of the unitarity condition U U = which are really four equations in terms of real quantities, one for each matrix element of U U, and because of detu =, there are together five conditions in terms of real quantities to be met by the matrix elements and, hence, the degrees of freedom of the matrices U are three real quantities. The important feature is that all U SU can be parametrized by an exponential operator U = exp i ϑ S 5. where the vector S has three components, each component S k, k =,, 3 representing a matrix. One can show that the unitarity condition requires these matrices to be hermitian, i.e. [S k ] mn = [S k ] nm, and the condition detu = requires the S k to have vanishing trace. There
31 5.7: Spin and the group SU 7 exist three such linear independent matrices, the simplest choice being S = σ, S = σ, S 3 = σ where σ k, k =,, 3 are the Pauli matrices 0 σ =, σ 0 = 0 i i 0, σ 3 = Algebraic Properties of the Pauli Matrices The Pauli matrices 5.4 provide a basis in terms of which all traceless, hermitian matrices A can be expandend. Any such A can be expressed in terms of three real parameters x, y, z z x iy A =, x, y, z, R. 5.5 x + iy z In fact, it holds, A = x σ + y σ + z σ The Pauli matrices satisfy very special commutation and anticommutation relationships. One can readily verify the commutation property σ j σ k σ k σ j = [σ j, σ k ] = iɛ jkl σ l 5.7 which is essentially identical to the Lie algebra 5.3 of the group SO3. We will show below that a  homomorphic mapping exists between SU and SO3 which establishes the close relationship between the two groups. The Pauli matrices obey the following anticommutation properties σ j σ k + σ k σ j = [σ j, σ k ] + = δ jk, 5.8 i.e. σ j = ; σ j σ k = σ k σ j for j k 5.9 which can also be readily verified. According to this property the Pauli matrices generate a 3 dimensional Clifford algebra C 3. Clifford algebras play an important role in the mathematical structure of physics, e.g. they are associated with the important fermion property of matter. At this point we will state a useful property, namely, σ a σ b = a b + i σ a b 5.30 where a, b are vectors commuting with σ, but their comonents must not necessarily commute with each other, i.e., it might hold a j b k b k a j 0. The proof of this relation rests on the commutation relationship and the anticommutation relationship 5.7, 5.9 and avoids commuting the components a j and b k. In fact, one obtains 3 j= σj a j b j σ a σ b = 3 j,k= σj σ k a j b k = + 3 j,k= j>k σ j σ k a j b k + 3 j,k= j<k σ j σ k a j b k. 5.3
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