Theory of Angular Momentum

Chapter 3 Theory of Angular Momentum HW#3: 3.1, 3., 3.5, 3.9, 3.10, 3.17, 3.0, 3.1, 3.4, 3.6, 3.7 3.1 Transformations of vectors Clearly the choice of the reference frame or coordinates system i.e. origin, axes, handedness, etc is arbitrary, and we want the laws of physics not to depend on this choice. In other words if we make predictions of how a given system should behave in one coordinate system then we should have a rule how to make predictions in another coordinate system. For that we need a rule how to transform different quantities from one system to another. In fact all of the quantities such as scalars, vectors, tensors, ket-vectors etc are distinguished from other by the way they transform under changes of coordinates. What are the possible transformations in Euclidean three dimensional space denoted by 3D? There are: 3translationsor shifts along ˆx, ŷ and ẑ directions 3rotationsfrom ˆx to ŷ,fromŷ to ẑ and from ẑ to ˆx. These are linearly independent transformations i.e. there are no non-zero linear combinations of these six transformations which leaves the system untransformed, but one can produce other linearly dependent transformations by forming linear combinations of these six transformations e.g. shift by -5 meters along ŷ,rotatebyπ/5 fromẑ to ˆx and then rotate by π/7 fromˆx to ŷ. Exercise: How many linearly independent transformations in 1D? D? 4D? nd? In n dimensions there are n translations and as rotation many 30

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 31 rotations as there are distinct pairs of axis rotations from x to y, fromx toz, etc. nn 1 n 1 + n +... ++1=. 3.1 Thus there are nn 1 nn +1 n + = 3. independent transformations. Linear combination of translations can be described by a translation vector ˆT =T x,t y,t z 3.3 of the old coordinate system x, y, z to new coordinate system x,y,z. Note that the translation vector is also expressed in the old unprimed coordinates. Then scalars e.g. A and vectors e.g. A = A x,a y,a z transforms into A and A =A x,a y,a z suchthat and A = A 3.4 A = A. 3.5 This is just a statement of the fact that vectors parallel transported in the Euclidean space do not change. The most general rotations can be described by a rotation matrix R xx R xy R xz R yx R yy R yz R zx R zy R zz cos φ 1 sin φ 1 0 sin φ 1 cos φ 1 0 0 0 1 3.6 1 0 0 0 cosφ sin φ 0 sin φ cos φ cos φ 3 0 sin φ 3 0 1 0 sin φ 3 0 cosφ 3 for some angles φ 1, φ and φ 3. Then scalars and vector transform respectively as. and A = A 3.7 3 A i = R ij A j 3.8 j=1

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 3 where it is assumed that i =1,, 3 and it stands correspondently for either x, y, z. Or using the Einstein summation convention always sum over repeated indices A i = R ija j. 3.9 For more complicated objects such as tensors the transformation law would be written as M ij = 3 R ik R jl M kl 3.10 k,l=1 or with Einstein summation convention simply M ij = R ikr jl M kl. 3.11 Clearly, the simplest transformation rule is for scalar quantities - they do not change under coordinate transformations. In Newtonian physics the three spatial dimensions x, y and z are connected by coordinate transformations, yet the time coordinate t was always treated separately. For example one can rotate the Cartesian coordinate system without altering the distance s = x + y + z = x + y + z. 3.1 There are six linearly independent transformations of space: three shifts and three rotations. However, until special theory of relativity there were no useful definitions of invariant distance in space-time described by all four coordinates t, x, y and z. We knew that the time coordinate must be different from space, but the connection was not clear until the following proposal for the invariant distance was made s = c t + x + y + z = c t + x + y + z. 3.13 It is more convenient to set c = 1 and to use the upper indices notation t x 0 3.14 x x 1 3.15 y x 3.16 z x 3. 3.17 Note that upper indices should not to be confused with exponentiating! Then with the Einstein summation convention i.e. always sum over

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 33 repeated upper and lower indices we obtain s = η µν x µ x ν η µν x µ x ν, 3.18 where µ,ν=0,1,,3 η µν = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1. 3.19 What are the transformation which leave the distance in space-time invariant? For example if we change x µ x µ + a µ 3.0 the invariant distance defined by 3.0 would not change. These are the shifts in space a 1,a,a 3 and in time a 0. What about other transformations analogous to rotations? Consider x µ x µ =Λ µ ν xν 3.1 where Λ is some 4 4 matrix such that in matrix notation x =Λx. Forthe distance 3.18 to be invariant we must have and thus, or s = x T η x = x T η x = x T Λ T ηλ x 3. η =Λ T ηλ 3.3 η ρσ =Λ µ ρλ ν σ η µ ν. 3.4 The new ingredient in special relativity that you must have seen before are the boosts which correspond to changing coordinates to a moving frame, e.g. Λ= cosh φ sinh φ 0 0 sinh φ cosh φ 0 0 0 0 1 0 0 0 0 1 From 3.1 the transformed coordinates are t = t cosh φ x sinh φ 3.5 x = t sinh φ + x cosh φ. 3.6

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 34 Thus, in the original coordinate system the point corresponding to x =0or t sinh φ + x cosh φ = 0 3.7 is moving with velocity v x t =tanhφ. 3.8 Then the transformation laws 3.5 and3.6 can be written in a more familiar form t = t cosh tanh 1 v x sinh tanh 1 v = γ t vx 3.9 x = t sinh tanh 1 v + x cosh tanh 1 v = γx vt 3.30 where γ =1/ 1 v, but you should check this. It is also easy to see what the boost transformation do to your axes x =0 and y = 0 using the space-time diagram. The transformed axes x =0 and t = 0 in terms of the old coordinates are described by lines x = t tanh φ 3.31 x = t tanh φ. 3.3 The only paths that are the same in both coordinates systems are the ones described by light rays, i.e. x = ±t is the same as x = ±t. The set of all light rays emitted from a given space-time point is called the light cone. This set divides all of the points into light-like or null separated, space-like separated, future time-like separated and past time-like separated. 3. Transformations of ket-vectors Consider infinitesimal translations T x ε = T y ε = T z ε = 1+ε 0 0 0 1 0 0 0 1 1 0 0 0 1+ε 0 0 0 1 1 0 0 0 1 0 0 0 1+ε 3.33

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 35 and rotations up to the second order in ε R x ε = R y ε = R z ε = 1 0 0 0 1 ε ε 0 ε 1 ε 1 ε 0 ε 0 1 0 ε 0 1 ε 1 ε ε 0 ε 1 ε 0 0 0 1. 3.34 Then it is easy to check that once again, up to the second order in ε R x εr y ε = and R y εr x ε = 1 0 0 0 1 ε ε 0 ε 1 ε 1 ε 0 ε 0 1 0 ε 0 1 ε Then commutator is given by 1 ε 0 ε 0 1 0 ε 0 1 ε 1 0 0 0 1 ε ε 0 ε 1 ε 1 ε 0 ε ε 1 ε ε ε ε 1 ε 3.35 1 ε ε ε 0 1 ε ε ε ε 1 ε 3.36. [R x ε,r y ε] = R x εr y ε R y εr x ε =R z ε I 3.37 Now the idea is study how ket-vectors should transform, i.e. ψ = ˆD ψ. 3.38 under these infinitesimal rotations or translations space and also time. ε dφ ε dx ε dt 3.39 Let us denote by ˆD n,ε, ˆD n,εand ˆDdt operator on ket-vectors which is due to respectively spacial rotations about n, translations in the direction of n and time translations. Since the transformation must take physical state

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 36 to a physical state it must be unitary and also become identity in the limit of ε, i.e. ˆD = Î iĝε 3.40 where Ĝ is some Hermitian matrix. For spacial and time translations the appropriate operators can be defined from the operators that we have already seen: and Ĝn,ε= ˆp n = ˆp x n x +ˆp y n y +ˆp z n z 3.41 Ĝε =Ĥ 3.4 where n is a unit three-vector. One can expect that for rotations there also exist operators Ĵx, Ĵy and Ĵz we call them angular momentum operators such that Ĵ n Ĝn,ε= = Ĵx n x + Ĵy n y + Ĵz n z. 3.43 Before we go on and discuss angular momentum operators Ĵx, Ĵy and Ĵz in any particular representation, it is useful to derive commutation relations between them. The guiding principle will be that operators ˆDR behave in exactly the same way as operators R. In particular we have already derived a commutation relations [R x ε,r y ε] = R x εr y ε R y εr x ε =R z ε I 3.44 and thus we must have up to the second order in ε [ ˆDx ε, ˆD y ε] = ˆD z ε I 3.45 or [ Î iĵxε Ĵ x ε, Î iĵyε Ĵ ] y ε = Î iĵzε Ĵ z ε I 3.46 which gives us [Ĵx, Ĵy] =iĵz. 3.47 One can also find all of the other commutation relations between angular momentum operators which can be written compactly as [Ĵi, Ĵj] =iϵ ijk Ĵ k 3.48

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 37 where the so-called Levi-Civita or completely antisymmetric tensor is defined as +1 if i, j, k is 1,, 3, 3, 1, or, 3, 1 ϵ ijk = 1 if i, j, k is 1, 3,,, 1, 3 or 3,, 1 3.49 0 if i = j, j = k, or i = k. Note that even if we only define infinitesimal rotations of ket-vectors any finite rotation can be preformed using an infinite number of infinitesimal transformations. For example, finite rotations on angle φ about z axis are given by ˆD = lim N [ 1 i Ĵz ] N φ N = 1 iĵz φ Ĵ z φ +... =exp 3.3 Two-component formalism iĵzφ. 3.50 The lowest dimensional representation of angular momentum operators Ĵx, Ĵy and Ĵz which satisfy commutation relations 3.48 is not three as one might have naively guessed, but two. It is easy to check that spin matrices defined as satisfy S 1 z+ z + z z + S i z+ z + i z z + S 3 z+ z + z z 3.51 [Ŝi, Ŝj] =iϵ ijk Ŝ k 3.5 as desired See Eq. 3.48. Thus if we set angular momenta operators Ĵ i = Ŝi 3.53 then according to 3.50 theketvectorsaretransformedas Ŝ ψ i n φ =exp ψ. 3.54

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 38 Note that by defining the angular momentum operators as in Eq. 3.51 we already work in Pauli s two-component formalism where the ket vectors are represented by two-component column vectors or spinors, 1 z+ = = χ 0 + 0 z = = χ 1 and the bra-vectors are represented as z + = 1 0 = χ + z = 0 1 = χ and the angular momentum operators are given 0 1 S 1 1 0 0 i S S 3 i 0 1 0 0 1 These matrices are known as Pauli matrices sometime denotes as ˆσ 1 ˆX 0 1 1 0 ˆσ Ŷ 0 i i 0 ˆσ 3 Ẑ 1 0 0 1. 3.55 It is easy to check that Pauli matrices satisfy desired commutation relations which can be compactly written [ ˆX,Ŷ ] = iẑ [Ŷ,Ẑ] = i ˆX [Ẑ, ˆX] = iŷ 3.56 [ˆσ i, ˆσ j ]=iϵ ijkˆσ k 3.57

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 39 3.4 Rotation of spin 1/ systems Consider rotation by an arbitrary angle φ around z-axis. According to 3.54 the most general ket-vector ψ = z + ψ z+ + z ψ z = ψ + z+ + ψ z = ψ + χ + + ψ χ ψ+ = ψ is transformed as Ŝ ψ i n φ = exp ψ iŝzφ = exp ψ + z+ + ψ z iφ iφ = exp ψ + z+ +exp ψ z 3.58 Note that ψ + z+ + ψ z φ=π ψ + z+ ψ z ψ + z+ + ψ z φ=4π ψ + z+ + ψ z 3.59 This is why we call such systems as half-spin: you have to rotate by angle φ =70 for the ket vector to come back to the original state. Equation for transformation of operators a la Heisenberg picture can be derived from transformations of ket-vectors a la Schrodinger picture, i.e. Ŝ ψ i n Ô ψ = ψ exp φ Ô exp Ŝ i n φ ψ = ψ Ô ψ and thus 3.60 Ŝ Ŝ Ô i n φ =exp Ô exp i n φ. 3.61

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 40 Exercise: Use commutation relations to calculate angular momentum operators Ŝ i s after rotation by angle φ around z-axis in terms of angular momentum operators before rotation Ŝi s, i.e. Starting from 3.61 we can Taylor expand both exponential and then use commutation relations 3.5 to obtain: Ŝ k iŝ3φ iŝ3φ = exp Ŝ k exp iŝ3φ = Î + + 1 iŝ3φ +... Ŝ k Î + = Ŝk + iφ = Ŝk [Ŝ3, Ŝk] 1 φ! +... [Ŝ3, [Ŝ3 Ŝk]], + 1 iφ! ϵ 3kj Ŝ j φ φ3 3! +... + 1 3! iŝ3φ + 1 iŝ3φ [Ŝ3, [Ŝ3, [Ŝ3 Ŝk]]], +... +... = Ŝk cos φ ϵ 3kj Ŝ j sin φ 3.6 To study rotations around arbitrary axis it is useful to note that ˆσ n n = { 1 for n even ˆσ n for n odd 3.63 and then exp i ˆσ n φ = 1 1 φ ˆσ n + 1 4 4 φ ˆσ n... +! 4! φ + i ˆσ n + i 3 3 φ ˆσ n... 3! φ φ = Î cos i ˆσ n sin. 3.64 Also note that ket-vectors transform as Ŝ ψ i n φ =exp ψ 3.65

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 41 but expectation values of the angular momentum operators transforms as three-vectors, e.g. ψ Ŝ1 ψ iŝφ iŝφ = ψ exp Ŝ 1 exp ψ φ = ψ Î cos + i φ φ Ŝ sin Ŝ 1 Î cos i φ Ŝ sin φ φ φ φ = ψ Ŝ1 cos sin ψ + ψ Ŝ3 sin cos ψ ψ = ψ Ŝ1 ψ cosφ+ ψ Ŝ3 ψ sinφ 3.66 For general rotations S k = l R kl S l 3.67 and for general angular momentum operators J k = l R kl J l. 3.68 3.5 Group theory and representations We can now summarize the groups whose elements described transformations of physical quantities. Orthogonal group. The first group was the orthogonal group O3 or its subgroup special orthogonal group SO3 which describes transformations of three-vectors, i.e. 3 V i = R ij V j 3.69 j=1 where the elements R of the group are represented by 3 3 real matrices, i.e. R xx R xy R xz R = R yx R yy R yz. 3.70 R zx R zy R zz The condition on the elements of the group is RR T = I 3.71 But since 3.71 also implies RR T = R T R = 1 it gives us only 6 constraints leaving us with only continues three parameters. An additional condition detr =1. 3.7

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 4 reduces all possible orthogonal transformations O3 to a subgroup of special orthogonal SO3 which does not include inversions. However 3.7 doesnot reduce the number of parameters. So the space of special orthogonal matrices SO3 can be described by thee variables, e.g. R = cos φ 1 sin φ 1 0 sin φ 1 cos φ 1 0 0 0 1 1 0 0 0 cosφ sin φ 0 sin φ cos φ cos φ 3 0 sin φ 3 0 1 0 sin φ 3 0 cosφ 3 3.73 where φ 1, φ and φ 3 are the Euler angles widely used in rotational problems in classical mechanics. Lorentz group. The second group was the Lorentz group O1, 3 or proper Lorentz group SO1, 3 which describes transformations of four-vectors, i.e. 3 V µ = R µν V ν 3.74 ν=0 where the elements R of the group are represented by 4 4 real matrices, i.e. R tt R tx R ty R tz R = R xt R xx R xy R xz R yt R yx R yy R yz. 3.75 R zt R zx R zy R zz The condition on the elements of the group is where η = η = R T ηr 3.76 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1. 3.77 But since 3.76 also implies RηR T = R T ηr = 1 it gives us only 10 constraints leaving us with only continues six parameters. An additional condition detr =1. 3.78 reduces all possible orthogonal transformations O1, 3 to special orthogonal SO1, 3 which once again does not reduce the number of parameters. So the space of special orthogonal matrices SO1, 3 can be described by six.

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 43 variables. In other words there are six generators of O3,1: the three usual rotations and three Lorentz boosts. For example, 1 0 0 0 Λ= 0 cosθ sin θ 0 0 sin θ cos θ 0 3.79 0 0 0 1 describes rotation where the angle θ [0,π] Λ= cosh φ sinh φ 0 0 sinh φ cosh φ 0 0 0 0 1 0 0 0 0 1. 3.80 describes boosts where the parameter φ, +. Nevertheless, it is still useful to think of boosts as rotations between space and time. Altogether how many linearly independent continuous transformations leave the distance 3.18 invariant? one time shift + three space shifts + three rotations + three boosts = ten! the group of such transformation is a non-abelian group known as the Poincare group. There are also discrete reflections of each of four coordinates, but they are not a part of the Poincare group. Special Unitary group. The last group that we considered was a special unitary group SU which describes transformations of ket-vectors known as two-component spinors, i.e. V i = 3 R ij V j 3.81 j=1 where the elements R of the group are represented by complex matrices, i.e. a b R =. 3.8 c d The condition on the elements of the group is R R = I 3.83 Which gives us four constraints and leaves us with four generators of U. An additional condition detr =1. 3.84 reduces all possible unitary transformations U to a subgroup of special unitary transformation U. So the space of special unitary matrices SU

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 44 can be described with three parameters or with two complex parameters and a constraint, e.g. a b R = b a 3.85 and a + b =1. 3.86 The parameters a and b are known as Cayley-Klein parameters. Exercise: Derive an expression for the two complex Cayley-Klein parameters in terms of the three real Euler angles. This will tell you how a spinor must transform under an arbitrary rotation. Check that 3.86is satisfied. 3.6 Density matrix If the exact knowledge of the quantum microstate is not available, the system is said to be in a not pure, butmixed state. Such states are not specified by a unique vector in Hilbert space, but by a collection of vector { ϕ α } with relative probabilities {p α },suchthat p α =1. 3.87 And the entropy of a mixed state is defined as α S α p α log p α. 3.88 Then, the ensemble average of a given operator Ô is given by Ô = α p α ϕ α Ô ϕ α. 3.89

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 45 In a given set of orthonormal basis, ψ n, the above expression takes the following form Ô = p α ϕ α ψ n ψ n Ô ψ m ψ m ϕ α = α,n,m = n,m p α ψ m ϕ α ϕ α ψ n ψ n Ô ψ m = α = n,m ψ m ˆρ ψ n ψ n Ô ψ m = = n,m ψ m ˆρÔ ψ m = = TrˆρÔ 3.90 where the so-called density matrix is defined as ˆρ α p α ϕ α ϕ α. 3.91 This is the Hermitian operator which replaces the probability distribution function in classical phase space. Let ψ n be the energy eigenstates, then ψ n i t ˆρt ψ m = i p α ψ n ϕ α t ϕ α t ψ m t α = p α ψ n i t ϕ α ϕ α ψ m + ψ n ϕ α i t ϕ α ψ m α = p α ψ n Ĥ ϕ α ϕ α ψ m + ψ n ϕ α i t ψ m ϕ α α = α p α E n ψ n ϕ α ϕ α ψ m E m ψ n ϕ α ψ m ϕ α = ψ n ˆρ E n E m ψ m = ψ n Ĥ ˆρ ˆρĤ ψ m 3.9 Thus, independently of basis we get the Von Neumann equation: i ˆρt =[Ĥ, ˆρ], 3.93 t which is a quantum version of Liouville s equation obtained once again by a formal substitution of Poisson brackets with commutator, i.e. {, } i [, ].

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 46 3.7 Eigenstates of angular momentum operators The commutation relation for angular momentum operators is given by If we define a new operator then one can show that [Ĵ, Ĵx] = [Ĵ x + Ĵ y + Ĵ z, Ĵx] Similarly = [Ĵ y, Ĵx]+[Ĵ z, Ĵx] [Ĵi, Ĵj] =iϵ ijk Ĵ k. 3.94 Ĵ = Ĵ x + Ĵ y + Ĵ z 3.95 = Ĵy[Ĵy, Ĵx]+[Ĵy, Ĵx]Ĵy + Ĵz[Ĵz, Ĵx]+[Ĵz, Ĵx]Ĵz = Ĵy iĵz + iĵz Ĵy + Ĵz iĵy + iĵy Ĵz =0. 3.96 [Ĵ, Ĵx] =[Ĵ, Ĵy] =[Ĵ, Ĵz] =0. 3.97 We can now look of simultaneous eigenkets of Ĵ and, for example, Ĵ z, i.e. Ĵ a, b = a a, b Ĵ z a, b = b a, b. 3.98 To construct the set of these eigenkets we will use raising and lowering operators which turn out to be Ĵ ± Ĵx ± iĵy. 3.99 Exercise. Derive the following commutation relations [Ĵ+, Ĵ ] = Ĵz [Ĵz, Ĵ±] = ±Ĵ± [Ĵ, Ĵ±] = 0. 3.100 Give these relations is easy to show that is a, b is an eigenket of Ĵ z with eigenvalue b, thenĵ± a, b is also an eigenket of Ĵz with eigenvalue b ± : Ĵ z Ĵ± a, b = Ĵ±Ĵz a, b ±Ĵ± a, b = Ĵ±b a, b ±Ĵ± a, b = b ± Ĵ± a, b. 3.101

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 47 On the other hand the eigenvalue of Ĵ stays the same, i.e. Ĵ Ĵ± a, b = Ĵ±Ĵ a, b Therefore = Ĵ±a a, b = a Ĵ± a, b. 3.10 Ĵ ± a, b = c ± a, b ± 3.103 where c ± will be determined from normalization of eigenkets. The next step is to determine possible values of a and b, or a b = a, b Ĵ Ĵ z a, b = a, b Ĵ x + Ĵ y a, b = a, b 1 Ĵ+ Ĵ + Ĵ Ĵ+ a, b = 1 a, b Ĵ Ĵ + Ĵ +Ĵ+ a, b 0 3.104 a b 3.105 This means that b cannot be arbitrary large small or in other words which implies that Ĵ + a, b max = 0 Ĵ a, b min = 0 3.106 0 = Ĵ Ĵ+ a, b max = Ĵ x + Ĵ y Ĵy i Ĵ x ĴxĴy a, b max Ĵ = Ĵ z Ĵz a, b max = a b max b max a, bmax 3.107 and similarly 0 = a b min + b min a, bmin 3.108 or a = b max b max + a = b min b min 3.109

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 48 The above equations are satisfied only if but we also know that for some integer n and thus we conclude It is a convention to work with b max = b min 3.110 b max = b min + n 3.111 b max = n b min = n 3.11 and j b max m b 3.113 3.114 instead of aand b and to denote eigenkets a, b as i, m. Then eigenvalues are given by where and Ĵ j, m = jj +1 j, m Ĵ z j, m = m j, m. 3.115 a = jj +1 3.116 m max = j = n m min = j = n 3.117 or m { j, j +1,...j 1,j}. 3.118 We can also find eigenvalues c ± of Ĵ± operators Ĵ ± a, b = c ± a, b ± 3.119 by contracting the ket-vector with itself c ± = j, m Ĵ ±Ĵ± j, m = j, m Ĵ Ĵ z Ĵz j, m = jj +1 m m = j mj ± m + 1 3.10

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 49 or if we choose to define c ± as positive real numbers then c ± = j mj ± m +1. 3.11 Now that we have constructed all of the orthonormal eigenkets we can represent operators as matricies j,m Ĵ± j, m = j mj ± m +1δ j jδ m m j,m Ĵz j, m = mδ j jδ m m j,m Ĵ j, m = jj +1 δ j jδ m m. 3.1 and study matrix representation of rotations of ketvectors ˆD R j,m ;j,m = j,m exp iĵ n φ j, m. 3.13 for arbitrary spatial rotations R. Note that these matrix elements are known as Winger function vanish for j j since Ĵ ˆD R j, m = ˆD R Ĵ j, m = jj +1 ˆD R j, m 3.14 and thus the value of j cannot change under rotations i.e. spin of a particle cannot change. For this reason ˆD R j,m ;j,m can be thought of as a block diagonal matrix with blocks of size j +1 j +1 known as j + 1dimensional irreducible representations of the rotation operator ˆD R j,m ;j,m. For a fixed j rotation matrices ˆD R j,m ;j,m form a subgroup under multiplication defined as j+1 m =1 and inverse defined as ˆD R j,m;j,m ˆD Rj,m ;j,m = ˆD R j,m;j,m 3.15 ˆD R 1 j,m ;j,m ˆD R j,m ;j,m = ˆD R 1 j,m ;j,m. 3.16 Then if one starts with a particle in an eigenstate j, m then after rotation ˆD R j, m 3.17 the amplitudes to find a particle in one of the eigenket states is given by ˆD R j,m ;j,m. 3.18

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 50 Exercise. Why? Hint: Insert a completeness relation into 3.17. An arbitrary rotation expressed in terms of Euler angles α, β, γ would rotate eigenkets as ˆD α, β, γ j,m ;j,m = j, m exp iĵzα = exp im α + mγ j, m exp exp iĵyβ exp iĵyβ iĵzγ j, m Exercise: Find a matrix representation of rotations of spin j = 1 and j =1particles in terms of the three Euler angles. 3.8 Orbital angular momentum Up to this point the rotation operator ˆD generated by angular momentum operator Ĵ was transforming ketvectors that did not have any positional degrees of freedom. these degrees of freedom we had previously represented in terms of wave functions and our next task is to study how these wave-functions transform. For such transformations we define new generators known as orbital angular momentum operators defined as j, m ˆL = ˆx ˆp 3.19 where ˆx and ˆp are the position and momentum operators respectively. Exercise. Check that the orbital angular momentum operators ˆL satisfy the same commutation relation as the total angular momentum operator Ĵ,i.e. [ˆL i, ˆL j ]=iϵ ijk ˆLk. 3.130 It is now convenient to switch from cartesian coordinates with line element to spherical coordinates ds = dx + dy + dz 3.131 x = r sin θ cos φ y = r sin θ sin φ z = r cos θ 3.13 with line element ds =dr + r dθ + r sin θdφ. 3.133

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 51 Exercise. Use 3.13 to show 3.133. Then the position generalized ket vectors and corresponding wave functions can be denotes by spherical variables, i.e. and rx, y, z,θx, y, z,φx, y, z x, y, z = x 3.134 ψr, θ, φ = r, θ, φ ψ. 3.135 Under transformations generated by orbital angular momentum operators these ket vectors would transform as r, θ, φ 1 iϵ ˆL z ψ = r, θ, φ ϵ ψ = r, θ, φ ψ ϵ r, θ, φ ψ 3.136 φ or Similarly, x, y, z 1 iϵ ˆL x x, y, z x ˆL z ψ = i x ψ. 3.137 φ ψ = x, y + zϵ, z yϵ ψ = x ψ + zϵ y x ψ yϵ z x ψ 1 iϵ ˆL y ψ = x zϵ, y, z + xϵ ψ = x ψ zϵ x which gives us x ˆL x ψ = i zϵ But since x ˆL y ψ = i y x ψ yϵ z x ψ zϵ x ψ + xϵ x z x ψ x ψ + xϵ z 3.138 x ψ 3.139 r = x + y + z θ = cos 1 z x + y + z y φ = tan 1 x 3.140 one can show that x ˆL x ψ = i sin φ cot θ cos φ θ φ x ˆL y ψ = i cos φ cot θ sin φ θ φ x ψ x ψ. 3.141

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 5 Then the ladder operators are defined as x ˆL ± ψ x ˆL x ± iˆl y ψ = ie ±iφ ±i θ cot θ x ψ 3.14 φ and x ˆL ψ x ˆL z + 1 ˆL+ ˆL ˆL ˆL+ ψ [ = e φ + R iφ i θ cot θ φ = [ = [ 1 sin θ φ + θ +cot θ φ + 1 sin θ θ φ +cotθ θ sin θ θ e iφ i θ cot θ φ ] x ψ ] x ψ ] x ψ. 3.143 Moreover one can show that actions of Laplacian operator on a wave-function can be decomposed into radial and angular parts, [ x ˆp ψ = r x ψ + r r x ψ 1 ] r x ˆL ψ 3.144 3.9 Schrodinger equation for central potential We are now in a position of solving Schrodinger equation for central potential with Hamiltonian given by Ĥ = ˆp m + V ˆr = ˆp x +ˆp y +ˆp z m + V ˆx +ŷ +ẑ 3.145 For such systems the orbital angular momentum operator commutes is conserved because it commutes with Hamiltonian [ˆL, ˆp ] = [ˆL, ˆr] =0 [ˆL, Ĥ] =0. 3.146 The commutation relation also suggest the following set of simultaneous eigenkets Ĥ E,l,m = E E,l,m ˆL E,l,m = ll +1 E,l,m ˆL z E,l,m = m E,l,m 3.147

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 53 where for orbital angular momentum operators we used 3.115 since it must satisfy the same commutation relations. For spherically symmetric problems the wave-function corresponding to these simultaneous eigenkets can be decomposed into r, θ, φ E,l,m = R El ry m l θ, φ, 3.148 where Yl m θ, φ are known as spherical harmonics. Then, for example, the amplitude to find a particle in a particular direction is given by dr r, θ, φ r, l, m = Yl m θ, φ 3.149 where equality is due to the fact that spherical harmonics are orthonormal. Also from 3.147and 3.137 we can get [ 1 sin θ with solution φ + 1 sin θ and a differential equation [ 1 sin θ φ + 1 sin θ sin θ θ θ θ i φ Y l m θ, φ ] = myl m θ, φ Yl m θ, φ = ll +1 Yl m θ, 3.150 φ Y m l θ, φ expimφ 3.151 sin θ ] + ll +1 Yl m θ, φ = 0 3.15 θ which is sometime use to define spherical harmonics. To solve for radial dependence we use 3.144, ˆp r, θ, φ m + V ˆr E E,l,m = = m r + + V ˆr E r, θ, φ E,l,m + r r r r, θ, φ ˆL E,l,m to get r + V ˆr E + ll +1 r, θ, φ E,l,m =0. mr r r mr 3.153 And using ansatz 3.148 we get the radial component of Schrodinger equation d r d + V r E + ll +1 R mr dr dr mr El r =0. 3.154

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 54 which may or may not have exact or perturbative analytic solutions depending on V r, but can always be solved numerically. It is convenient to substitute R El r = u Elr r 3.155 into 3.154 to get an equation for u El, i.e. m d u El r dr + [V r E + ll +1 mr ] u El r =0. 3.156 and then from normalization of ket-vectors and spherical harmonics we get 1= r drrelrr El r = r dr u Elru El r. 3.157 Thus u El r must satisfy one- dimensional Schrodinger equation with potential W r =V r+ ll + 1 3.158 mr 3.10 Addition of angular momentum Consider a system of two particles which two sets of angular momentum operators Ĵ1 and Ĵ satisfying the usual commutation relation between operators in the same set, [Ĵ1i, Ĵ1j] = iε ijk Ĵ 1k [Ĵ1i, Ĵ1j] = iε ijk Ĵ 1k. 3.159 Fort any two operators from different sets to commute we can form tensor product operators, i.e. [Ĵ1i Î, Î1 Ĵj] =0. 3.160 where Î1 and Î the identity operators acting on the first and second particle respectively. Then the most general infinitesimal rotation is given by Î 1 i Ĵ1 n ϵ Î i Ĵ n ϵ = Î1 Î i Ĵ1 Î + Î1 Ĵ n ϵ 3.161

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 55 and the finite rotation is ˆD 1 R ˆD R =exp i Ĵ1 n ϵ exp Î i Ĵ n ϵ. 3.16 It is easy to check that the total angular momentum operator Ĵ = Ĵ 1 Î + Î1 Ĵ 3.163 satisfies the usual commutation relations [Ĵi, Ĵj] = iε ijk Ĵ k. 3.164 Because of this the combined system of two particles is described by eigenkets of Ĵ and Ĵz operators and thus to construct all these eigenkets we can use creation an annihilation operators Ĵ ± = Ĵ1x Î + Î1 Ĵx ± i Ĵ1y Î + Î1 Ĵy. 3.165 However if we are interested in construction of eigenkets of the combined system from eigenkets of the individual particles then we can use the following maximal set of commuting operators with simultaneous eigenkets defined by Alternatively one can use {Ĵ 1, Ĵ, Ĵ1z, Ĵz} 3.166 Ĵ 1 j 1,j,m 1,m = j 1 j 1 +1 j 1,j,m 1,m Ĵ j 1,j,m 1,m = j j +1 j 1,j,m 1,m Ĵ 1z j 1,j,m 1,m = m 1 j 1,j,m 1,m Ĵ z j 1,j,m 1,m = m j 1,j,m 1,m 3.167 with simultaneous eigenkets defined by {Ĵ 1, Ĵ, Ĵ, Ĵz} 3.168 Ĵ 1 j 1,j,j,m = j 1 j 1 +1 j 1,j,j,m Ĵ j 1,j,j,m = j j +1 j 1,j,j,m Ĵ j 1,j,j,m = j j +1 j 1,j,j,m Ĵ z j 1,j,j,m = m j 1,j,j,m 3.169

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 56 To express one set of basis vectors in terms of another set we inset acompleteness relations j 1,j,j,m = j 1,j,m 1,m j 1,j,m 1,m j 1,j,j,m m 1,j 1,m,j = m 1,m j 1,j,m 1,m j 1,j,j,m j 1,j,m 1,m 3.170 where j 1,j,m 1,m j 1,j,j,m are known as Clebshch-Gordon coefficients with following properties: 1. Addition of Ĵz eigenvalues m m 1 + m j 1,j,m 1,m j 1,j,j,m = 0 3.171 Proof: Ĵ z = Ĵ1z Î + Î1 Ĵz Ĵz Ĵz Ĵ1z Î Î1 j 1,j,j,m = 0 j 1,j,m 1,m Ĵz Ĵ1z Î Î1 Ĵz j 1,j,j,m = 0. Addition of Ĵ eigenvalues m = m 1 + m 3.17 j< j 1 j j>j 1 + j j 1,j,m 1,m j 1,j,j,m =0 3.173 Note that the dimensionality of the space of j 1,j,m 1,m vectors with afixedj 1 and j is given by N =j 1 +1j + 1 3.174 and if 3.173 is correct that the dimensionality of the space of j 1,j,j,m vectors with a fixed j 1 and j would be the same j 1 +j N = j +1 j=j 1 j j1 + j j 1 + j +1 = = 4j 1 j +j 1 +j +1 j 1 j 1 j 1 j +j 1 + j j 1 j 1 = j 1 +1j +1. 3.175

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 57 This summation can also be understood in terms of rotation operators ˆD j 1 ˆD j = ˆD j 1 j ˆD j 1 j +1... ˆD j 1+j Using Clebshch-Gordon coefficients we can represent rotation operators of two particle systems in terms of rotation operators of individual particles D j 1 m1 m Dj 1 1 m m = D j 1 D j m 1, m = m ;m 1,m ˆDj 1 ˆD j m 1,m 1,m = m 1,m j, m j, m ˆD j1 ˆD j j,m j,m m 1,m j,m,j,m = m 1,m j, m j, m ˆD j1 j... ˆD j1+j j,m j,m m 1,m j,m,j,m = j 1 +j = j= j 1 j j1 +j j= j 1 j where I dropped j 1 and j indices in all bra and ket vectors. 3.11 Schwinger s oscillator model j,m,m m 1,m j, m D j mm δ jj j,m m 1,m m,m m 1,m j, m D j mm j, m m 1,m 3.176 Consider two uncoupled oscillators described by commuting annihilation operators â + and â.thenwecandefine vacuum ladder operators number operators â + 0, 0 = 0 â 0, 0 = 0 3.177 â + Î n + +1,n = n + +1 n +,n â + Î n + 1,n = n + n +,n Î â n +,n +1 = n +1 n +,n Î â n +,n 1 = n n +,n ˆN + â +â + ˆN â â 3.178

CHAPTER 3. THEORY OF ANGULAR MOMENTUM 58 All of the operators for each of the oscillators satisfying the usual commutation relations [â +, â +] = 1 [ ˆN +, â + ] = â + [ ˆN +, â +] = â + [â, â ] = 1 [ ˆN, â ] = â [ ˆN, â ] = â 3.179 but any two operators acting on different oscillators commute, ] [ ] [ ] [â +, â ]= [â +, â = â +, â = â +, â =0. 3.180 Moreover the simultaneous eigenkets of operators ˆN + and ˆN, i.e. ˆN + n +,n = n + n +,n ˆN n +,n = n n +,n 3.181 can be obtained from a vacuum states using â + Î n+ n Î â 0, 0 = 0. 3.18 n+!n! Exercise. For operators Ĵ + â + â Ĵ â + â Ĵ z â +â + â Î Î â check that 3.1 Bell s inequality [Ĵz, Ĵ±] = ±Ĵ± [Ĵ+, Ĵ ] = Ĵz. 3.13 Tensor operators