Rotations and Angular momentum

Transcription

1 apr_-8.nb: 5//04::8:0:54 Rotations and Angular momentum Intro The material here may be found in Saurai Chap : -, (5-6), 7, (9-0) Merzbacher Chap, 7. Chapter of Merzbacher concentrates on orbital angular momentum. Saurai, and Ch 7 of Merzbacher focus on angular momentum in relation to the group of rotations. Just as linear momentum is related to the translation group, angular momentum operators are generators of rotations. The goal is to present the basics in 5 lectures focusing on. J as the generator of rotations.. Representations of SO. Addition of angular momentum 4. Orbital angular momentum and Y lm ' s 5. Tensor operators. Rotations & SO() ü Rotations of vectors Begin with a discussion of rotations applied to a -dimensional real vectorspace. The vectors are described by three real v x numbers, e.g. v = v y. The transpose of a vector is vt = v x, v y, v z. There is an inner product defined between two v z vectors by u T ÿ v = v T ÿ u=uv cos f, where f is the angle between the two vectors. Under a rotation the inner product between any two vectors is preserved, i.e. the length of any vector and the angle between any two vectors doesn't change. A rotation can be described by a äreal orthogonal matrix R which operates on a vector by the usual rules of matrix multiplication v' x v' y = R v' z v x v y v z and v' x, v' y, v' z = v x, v y, v z R T To preserve the inner product, it is requird that R T ÿ R = u' ÿ v' = ur T ÿ Rv = uv = u ÿ v As an example, a rotation by f around the z-axis (or in the xy-plane) is given by

2 apr_-8.nb: 5//04::8:0:54 R z f = cos f -sin f 0 sin f cos f The sign conventions are appropriate for a right handed coordinate system: put the thumb of right hand along z-axis, extend fingers along x-axis, and curl fingers in direction of y-axis. z y x The direction of rotation for f is counter-clocwise when looing down from the +zdirection, i.e. rotate the x-axis into the y-axis. Similarly the rotations around the x and y axes are R y f = -sin f cos f 0 sin f cos f and R xf = cos f -sin f 0 sin f cos f The sign of sin f in R y is related to the handed-ness of the coordinate system and the sense of rotation. For -dimensions, it is equivalent to tal about rotations around the z-axis, or rotations in the xy-plane. In any other number of dimensions, the correct language is to tal about rotations in the x i x j -plane, where x i defines one of the coordinate directions of the vector space. Thus, while for -dimensions there are independent rotations, in N -dimensions, there will be ÅÅÅ NN- independent rotations. ü Direction ets To mae the correspondence to quantum states, just as a translation was defined by its action on position eigenets, a rotation around the origin also can act on position eigenets by Sx' = R è Sx = SR x where a distinction has been made between the operator R è, which acts on the state, and the rotation matrix R which acts on the coordinates. Since the rotations don't change the length of the vector, it is possible to define spherical coordinates, r, q, f, and spherical position ets, Sx Ø Sr Sǹ, where r determines the radial position, and ǹ indicates the direction from the origin. The rotations act only on the Sǹ degrees of freedom. R è Sr Sǹ = Sr SR ÿ ǹ Direction ets will be used more extensively in the discussion of orbital angular momentum and spherical harmonics, but for now they are useful for illustrating the set of rotations. The set of all direction ets Sǹ can be visualized by the surface of a sphere, and the rotations are the set of all possible ways to reorient that sphere.

3 apr_-8.nb: 5//04::8:0:54 ü Orthognal group SO() The set of all possible rotations form a group. Consider the four properties: closure, identity, inverse and associativity. Using the picture of rotations as reorientations of a sphere, one can construct visualizations to illustrate each property. With greater mathematical rigor, the set of all possible rotations form the group SO(), where O Ø orthogonal, Ø dimensions, and S Ø special, which in this case means the matrix has a determinant of. The rotations are described by three continuous, but bounded, parameters. From the matrix point of view, a ämatrix has nine degrees of freedom. The constraint that the matrix is orthogonal, RT i j R j = d i yields 6 conditions, i.e. three for i = and three for i. The properties of a group are obeyed: closure: For any two orthogonal matrices R and R, the product R = R R, is also orthogonal. The combination of two rotations is also a rotation. identity: The ä unit matrix acts as an identity element for the group. R = R = R inverse: Each element has an inverse R - = R T, R T R = R R T = associativity: R R R = R R R Unlie the Translation group, SO() is not abelian, i.e. in general R R R R. The significance of the S-condition, Det R =, is that reflections are not included in the group, i.e. for three dimensions one cannot turn a right-handed object into a left-handed object by doing a rotation. If we allowed reflections, e.g Then, the group would be O() instead of SO(). O() is called "disconnected" since not all elements of the group can be reached by a succession of infinitesimal transformations. SO() is connected. The rotation matrices R are just one "representation" of the group SO(). For two different representations, there has to be a mapping of the elements of one representation to the other. The mapping has to preserve the combination law. Consider two representations R and S. Label a rotation by a subscript which represents the three parameters to define a rotation, and identify R a S a, If R = R R, then we must have S = S S to preserve the combinatin law. ü Full set of rotations: ǹ, f There are two common methods for parameterizing rotations. The first is to choose an axis for rotation and then perform a rotation by an angle between 0 and p. The axis of rotation can be chosen anywhere on the sphere. Why not 0 p? Then rotations with poles on opposite sides of the sphere would be redundant. An explicit form for Rǹ, f will be given after developing the language of infinitesimal rotations. Draw your own picture showing the rotation of a sphere around an off axis pole. The sphere represents the set of states, n +, i.e. the set of direction ets. The rotation reorients the sphere.

4 apr_-8.nb: 5//04::8:0:54 4 ü Euler angles The second parameterization is to give Euler angles. In this method one describes where the "north pole" moves to under a rotation, and the orientation of the sphere after the pole has been moved. The location of the pole is determined by first choosing a longitude by rotating around the z-axis, then a latitude by rotating around the new y-axis. Finally, the orientation of the sphere is given by a final rotation around the new z-axis. Pictorially, Draw more pictures, showing the sequence of rotations to move the pole, and then reorient the sphere around the new pole. In the Euler parameterization the range of angles is a = 0, p, b = 0, p, g = 0, p and an arbitrary rotation is given by Ra, b, g = R z' g R y' b R z a Note that the z' and y' rotations are not defined with respect to the original coordinate axes, but rather with respect to where those axes have moved with the reorientation of the sphere. Later it will be shown that R z' g R y' b R z a = R z a R y b R z g where the order has been reversed, but now all rotations are conveniently defined around the axes of the original coordinate system. ü Equivalency of the two parameterizations The two parameterizations may not seem equivalent, but they are, as can be seen by a pictorial mapping of ǹ, f to a, b, g. Observe that there are two ways to produce the same set of Euler angles consistent with the restriction of f to 0, p. This picture didn't mae it into the classroom presentation. it's a bit of wor The two techniques have different uses. Euler angles tend to be more useful for building up actual rotation matrices in a calculation. This is because R z and R y are generally fairly easy to construct for a representation, and the matrix multiplication is straightforward. The ǹ, f notation has advantages in some analytic manipulations, as we will see below.

5 apr_-8.nb: 5//04::8:0:54 5 ü J as the generator of infinitesimal rotations. In analogy to the discussion of translations and time evolution, it is useful to build up the finite rotations from generators of infinitesimal rotations. Recognizing that we are eventually interested in a quantum mechanical formulation, it is useful to develop this formalism in a way that realizes the rotations as unitary operations. For example, an infinitesimal rotation around the z-axis is given by R z d = - i d J z where J z is the generator of infinitesimal rotations around the z-axis. Since R is unitary (note: orthogonal matrices are unitary), J must be Hermitian. In the present case, to leading order in d -d 0 0 -i 0 R z d = d 0 or J z = i Similarly, for this representation i J x = 0 0 -i, J y = i 0 -i 0 0 Note that since this is still a classical discussion I haven't put in any factors of. ü Commutation relations The generators obey the commutation relations J i, J j = i e i j J where e i j =, if i j is an even permutation of x y z = -, if i j is an odd permutation of x y z = 0, if any two of i j are equal as a useful aside e i j e lm = d il d jm - d im d jl. The commutation relations are a property of the group, not just a particular representation. ü Finite rotations For rotations around a particular axis, it should be clear that we can build up an arbitrary rotation by a sequence of infinitesimal rotations, similar to the procedure for building up a finte translation as the limiting product of a large number of infinitesimals. Rf = Lim nø P i R z f I n = Lim nø - i J z f I n n = e -i J z f

6 apr_-8.nb: 5//04::8:0:54 6 It is now possible to give a form for an arbitrary rotation in the ǹ, f parameterization. The infinitesimal rotation around the ǹ-axis is given by Rǹ, d = - i d ǹ ÿ J There is no concern about which component of J is operated on first, since the effects of commutation amongst the different genreators shows up at second order in the infinitesimal d. The finite rotation is then given by Rǹ, f = e-i f ǹÿj Representations of SO() ü relation between D and R - notational Rotations can act to change a wide variety of objects, e.g. classical position vectors, position eigenstates Sx, operators such as X, P or L, angular momentum states Slm etc. In principle, the notation R to denote a rotation operator can be used for all of these applications, if a sufficiently detailed definition is supplied for the case at hand. In practice, a common convention is to use R when the object in question has the properties of a position vector, but to use a notation D when operating on angular momentum states or objects with similar characteristics. For example, to operate on a classical vector use x' i = R i j x j or to operate on a position eigenstate Sx' = R Sx = SR i j x j In contrast, to perform a rotation on a state Sa which is an angular momentum state Sa = Slm one would write a' = DR lm where DR is an operator that depends on orbital angular momentum l. One would still write D in the exponential form D ǹ, f = e - i Jÿǹ f but the form of the generators would be specific to the set of states, or representation, which is the object of the rotation. The set of possible representations is quite large. To simplify the discussion as much as possible, one defines the irreducible representations of SO which is the subject of the next section.

7 apr_-8.nb: 5//04::8:0:54 7 ü Irreducible representations ü Casimir operators and maximal set of commuting operators. The first item of business is to determine a maximal set of commuting operators that can be simultaneaously diagonalized. For SO() or SU() this would be J and one other component of J, typically taen to be J z. J is an example of what is nown as a Casimir operator. Casimir operators commute with all operators within the algebra of the group. Other groups may have more than one Casimir. The number of Casimir operators is equal to the "ran" of the group. SO(4) and SU() for example have two Casimir operators and are ran. Generally, the most useful Casimir is the quadratic operator C = i O i where the O i are the infinitesimal generators of the group. J is such a quadratic casimir operator. In groups with more than one casimir operator, they all commute with each other. In addition to the casimir operators, one can choose a set of operators equal in number to the ran from the group algebra that also commute with each other. For example in SU() one can find two generators that are diagonal. So in general, the dimension of a maximal set of commuting operators with which to define the representations of a group is twice the ran. For completeness, J, J z = J x + J y + J z, J z = J x + J y, J z = J x J x, J z + J x, J z J x + J y J y, J z + J y, J z J y = -ij x J y + J y J x + ij y J x + J x J y = 0 ü Labeling of states Since J, J z can be diagonalized simultaneously, we can specifiy the states by S j, m, where m is the eignevalue when J z operates on the state and the operation of J yields a j J z j, m = m j, m J j, m = a j j, m where the eigenvalue of J is not yet determined. Note that the labeling of the states is rather arbitrary. In the case of J z, it is convenient to use the eigenvalue directly. We will also tae the states S j, m to be normalized. ü j-representations as irreducible representations As J commutes with all the generators of the group, it also commutes with any function of those generators, and in particular J, D = 0. As with other commutation relations, this implies that the rotation operators don't change the eigenvalue of J, J D S jm = DJ S jm = Da j S jm = a j D S jm

8 apr_-8.nb: 5//04::8:0:54 8 On the other hand J z, D 0, and so the rotations mix states of different m-value but not of different j-value. In this case we say that all the states of a given j-value, taen together, form a representation of the group. The dimension of the representation is equal to the number of distinct basis states which may be chosen for the same j-value. In the case of j-representations we say that the representation is irreducible, i.e. it is not possible to brea the representation down into two subspaces that don't mix under the action of rotations. Thus, the result of performing a rotation on a state S jm is given by a linear combination of all states S jm' j D S jm = S D mm' S jm' m' j where the exact value of the coefficients D mm' depends on the parameters describing the rotation. ü Spin The notes below should include a discussion of the j = representation, beginning with the observation that the s-matricies obey the angular momentum algebra, and thus explicitly form a -dimensional representation of the angular momentum algebra. ü spin j=/ representation ü ways to get rotation matrices ü SU() ü Eigenvalue problem using the algebra of the generators for the group We have already discussed the eigenvalue problem in terms of orbital angular momentum. That discussion was couched in the language of solutions to partial differential equations with boundary conditions. In this discussion, we'll derive the eigenstate spectrum algebraically. This discussion is given in Merzbacher, chapter. ü Raising and lowering (ladder) operators J ± It is useful to define the raising and lowering operators J = J x i J y In general, after choosing a maximal set of commuting operators, it is possible to define linear combinations of the remaining generators which act as pairs of ladder operators. In the case of SO() there is just one such pair. Since they are linear combinations of generators, one has J, J = 0. The commutator with J z is given by

9 apr_-8.nb: 5//04::8:0:54 9 J z, J = J z, J x i J y = J z, J x i J z, J y = i J y i -i J x = J And, J +, J - = J x + i J y, J x - i J y = ij y, J x - J x, J y = J z Similar relations hold for other groups. ü Effect of ladder operators on states First, one can show that operating on a state j, m with a ladder operator does not change the value of j. For example, 0 = J, J + j, m = J J + - J + J j, m = J - a j J + j, m J + j, m = 0 or J + j, m is an eigenstate of J with eigenvalue a j. Next, observe that when acting on a state j, m ladder operators act to raise or lower the value of m. For example, consider the operation of J + 0 = J z, J + - J + j, m = J z J + - J + J z - J + j, m = J z - m - J + j, m It follows that either J + j, m = 0 or J + j, m = c + jm j, m + i.e. either J + annihilates the state, or J + creates an eigenstate with the same value of j, but with an eigenvalue for J z which is increased by. Similarly, J - lowers the eigenvalue of J z J - j, m = 0 or J - j, m = c - jm j, m The coefficients c jm and c jm are as yet undetermined. From this discussion one may see that starting from a particular state j, m one can generate a sequence of states with the same eigenvalue of J, and values of m that differ by integers. ü Representations are finite dimensional The next step is to show that the j-representations are finite dimensional. Consider

10 apr_-8.nb: 5//04::8:0:54 0 J - J z = J x + J y = J + J - + J - J + = J + J + + J + J + where at the end, both terms are positive definite since j, m J + J + j, m( = j, m + c + jm * c + jm j, m + = c + jm It follows that J - J z j, m = a j - m j, m > 0 or a j - m > 0. Since a j does not change through out the sequence, it follows that for each j-representation m has both an upper bound m max and a lower bound m min. ü Determination of a j Consider J - J + = J x - i J y J x + i J y = J x + J y + ij x, J y = J - J z - J z so, when operating on the highest state of the representation J - J + j, m max = J - J z - J z j, m max = a j - m max - m max j, m max = 0 since J + j, m max = 0. It follows that a j = m max m max +. Similarly, operating with J + J - j, m min = J - J z + J z j, m min = a j - m min + m min j, m min = 0 since J - j, m min = 0. It follows that a j = m min m min -. There is also the constraint that m max = m min + n, where n is some non-negative integer. These relations for a j, m max, m min can only be satisfied if m min = -m max. This, in turn, implies that m max - m min = m max = n, or m max = n This gives the desired result. The maximum value of m is integer or half-integer. Conventionally we label this quantum number j. The representations of SU() are labeled by j. They have dimension j +. The states are labeled by a second quantum number m, which runs from - j to j. The eigenvalue equations are then J z j, m = m j, m J j, m = j j + j, m

11 apr_-8.nb: 5//04::8:0:54 ü Other matrix elements From above j, m J + J + j, m = j, m + c + jm * c + j jm j, m + = c +m whereas, we can also use the relation J - J + = J + J + = J - J z - J z j, m J + J + j, m( = j, m J - J z - J z j, m or = j j + - mm + = j - m j + m + c + jm = H j - m j + m + Similarly c - jm = H j + m j - m + ü Explicit matrix form It is often convenient to show the angular momentum operators in explicit matrix form. The form of the matrix depends on the representation. For example, the figure shows matrices corresponding to J for the j =,, representations J j= = , J j= = , J j= = 5 Å ÅÅ Å 4 0 The matrices operate on a j + -dimensional state vector, which defines a state of the j-representation as a linear combination of the S jmstates. Since J is a Casimir operator, it is diagonal and proportional to the identity matrix in each representation, but with a different eigenvalue. J z can be simultaneously diagonalized with J, but has different eigenvalues for each state. The matrix form for J z for the j =,, representations is J z j= = , J z j= = , J z j= = The other generators, chosen to be either J x, J y or J +, J -, are not diagonal in the basis where J z is diagonal, but the matrix entries are easily determined by the c +, c - coefficients given above. For example, the matrix form for J + for the j =,, representations is 5 Å 4

12 apr_-8.nb: 5//04::8:0:54 J + j= = 0 0 0, J + j= = 0 H 0 H , J + j= = 0 H H J - is similar, but on the lower off-diagonal. J x = J + + J -, J y = - i J + - J - are determined by adding the matrix components, entry by entry. Angular momentum addition Suppose one has two particles and it is nown that one has angular momentum j and the second has angular momentum j. It is a common queston to as, "What are the possible angular momentum states for the combined system. Or it may be that one nows the orbital angular momentum and spin of a particle, but it is needed to now the total angular momentum. A third case is that one nows the spin of two particles, but wants to now the total spin for the system. All three cases are mathematically equivalent. Perhaps even more important than forming a state from two angular momentum degrees of freedom, is determining the resulting angular momentum when operating on a state with a "spherical-tensor" operator, such as occurs when calculating the rates and selection rules for radiative transitions. Tensor operators will be addressed in the following section, but the algebraic concepts are essentially identicle to those developed for the addition of angular momentum presented here. The notes below consider the general case of adding two angular momentum degrees of freedom, described by the operators J and J. ü Product of two spaces As usual, when there are two degrees of freedom, one can describe the full set of states for a system as the direct product of the states for the two subspaces. In this case, the states of the J operator are given by j m and those of J by j m. Where j i describes the eigenstate with respect to the casimir operator J i and m i is the eigenvalue of J i z. The two operators commute with each other J, J = 0, so a full set of commuting opertors is J, J, J z, J z. A state for the full system is then given by the direct product j m j m = j m j m If it is nown that the first particle has j and the second j, then there are n = j + possible states for the first, n = j + possible states for the second, and the full system has n = n n = j + j + possible states, with varying values of m and m. ü total J It often happens that one is interested in the total angular momentum. The total angular momentum operator is given by J = J + J.

13 apr_-8.nb: 5//04::8:0:54 where the vector nature of the operators has been emphasized. The states of total angular momentum are defined by the operators J and J z J = J + J + J ÿ J J z = J z + J z There are many cases where the Hamiltonian includes a term of the form J ÿ J, for example the spin-orbit coupling is proportional to L ÿ S. In this case the relation J ÿ J = J - J + J is useful. It is also possible to write J ÿ J in terms of raising and lowering operators J ÿ J = J z J z + J + J - + J - J + ü two alternative basis sets for the j, j representation The last relation maes it clear that because J contains pieces that raise and lower the states of J z and J z, eigenstates S j m j m, are not generally eigenstates of J. Such states are, however, eigenstates of J z. The effect of J + J - + J - J + is to leave the total m = m + m unchanged, since if m is raised m is lowered, and vice versa. It appears that there are two ways one can describe the angular momentum states of the system. First, in the J z J z scheme one chooses 4 commuting operators J, J, J z, and J z. In this case the states are labeled by S j m j m. Alternatively, in the J, J z scheme a different set of four operators J, J, J, and J z define the states. It is straightforward to show that J, J i = J z, J i = 0 for i =,. In this case the states are labeled by S j m j j. The values of j and j are common to both schemes. Accordingly, if the values of j and j are well defined, it is common to list them first or to omit them entirely. In the J z J z scheme one has, S j j m m, or uses the short hand Sm, m. Corespondingly, in the J J z scheme one has S j j j m or S j m. ü definition of clebsch gordon coefficients Since j and j are valid in either representation, it follows that the n = j + j + states present in the Sm, m scheme must be rearranged into the same number of states in the S jm scheme, although the allowed values of j and m are still undetermined. It follows that the j j jm states can be written as linear combinations of S j j m m states, S j j jm = S S j j m m j j m m j j jm m m j = S c j jm,m m m m S j j m m where the last line defines the Clebsch-Gordon coefficients.

14 apr_-8.nb: 5//04::8:0:54 4 ü Reduce product of j, j to irreducible representations of J The next step is to determine which representations of total angular momentum are found in the product of j j. The ey to this is to consider a) the multiplicity of states with a particular value of m, and b) to realize that the representations of j must be complete, implying that all states from m = - j to j must be present. Without any loss of generality, tae j j and define D j = j - j. Then the multiplicity of states with eigenvalue m is given by N m = j + j + - m m D j j + m D j The highest m-state has m = j + j. The increase in multiplicity with decreasing m corresponds to the different ways in which J - and J - can be applied to decrease the total m. The increase stops when m has been decreased by j + steps, since the number of times J - can be applied is limited. The multiplicity holds steady until a similar constraint applies to J - at which point the multiplicity decreases until m = - j + j. The multiplicity is symmetric under m Ø -m. The highest m-state is unique. Since the representations must be complete, the highest m-state must be accompanied by a set of states S j m for j = j + j with j + values of m, - j < m < j, included. Specifically, there are two states with m = j + j -, and one of these belongs to the j = j + j representation. The other heads a representation with j = j + j -. Similarly, there are three states with m = j + j -, one of which heads a new representation with j = j + j -. All told this procedure results in complete representations with j + j j j - j. The total number of states in these representations is N = j= j + j S j + j= j - j i= j = S D j + i + i=0 i= j = D j + j + + S i i=0 = D j + j + + j j + = j - j + + j j + = j + j + which is equal to the number of states in the product representation S j j m m. ü Procedure for calculating CG coeffs It remains to find the Clebsch-Gordon coefficients. This is done iteratively by repeated use of the operator J - = J - + J -. The process begins with the observation that there is only one state with maximal total m, and it belongs to the representation j = j + j, S j, j, j = j + j, m = j + j = S j, j, m = j, m = j Proceeding with the application of J -

15 apr_-8.nb: 5//04::8:0:54 5 J - S j, j, j = j + j, m = j + j = J - + J - S j, j, m = j, m = j c- j + j, j + j S j, j, j = j + j, m = j + j - = - - c j, j S j, j, m = j -, m = j + c j, j S j, j, m = j, m = j - where the c - jm = H j + m j - m + were defined above. Dividing through by c- j + j, j + j, we obtain the second highest m-state of the j = j + j representation, S j, j, j = j + j, m = j + j - = c- j, j - ÅÅÅ S j, j, m = j -, m = j + c j, j - ÅÅÅ S j, j, m = j, m = j - c j + j, j + j - c j + j, j + j Comparing to the definition of the CG coefficient, for the case j = j + j, m = j + j -, m = j -, m = j j c j j + j, j + j -; j -, j = - c j, j c- ÅÅÅ = j Å j + j, j + j j + j and similarly j c j j + j, j + j -; j, j - = - c j, j ÅÅÅ c- = j Å j + j, j + j j + j In response to a question in class, if j 5 j the j = j max representation is predominately composed of the larger of the j and j representations. At least in this case it is clear that the CG coefficients are normalized so that j S Sc j jm;m,m-m m W The other m = j + j - state, which plays the role of the head of the j = j + j - representation is given by S j, j, j = j + j -, m = j + j - = c- j, j - ÅÅÅ S j, j, m = j -, m = j - c j, j - ÅÅÅ S j, j, m = j, m = j - c j + j, j + j and is explicitly orthogonal to the j = j + j state. - c j + j, j + j This procedure can now be repeated applying J - to the two states above to fill out the j = j + j and j = j + j - representations. A third state may be constructed, orthogonal to the first two, which will head the j = j + j - representation. It may appear a bit awward to create the new orthogonal state, but one can use the following construction. Suppose one has an n-dimensional vector space spanned by an orthonormal basis i, and n - linear combinations of these which are orthogonal and normalized, which can be labeled by index Sa = S i a i Si. Then an n th state can be added to the a-basis which will be orthogonal to the others by construction Sa n = e i i i n n- a i a i a in- Si n ü Example of two spin / reps As an explicit example of angular momentum addition, consider the case of adding two spins of s = s =. Since s and s are specified I'll adopt the shorter notation where they are suppressed. Ssm sm = Sm m m m In the m m -basis there are four states, (,, - (, -, (, and -, - (

16 apr_-8.nb: 5//04::8:0:54 6 In the s -basis there are representations extending from s max = s + s to s min = s - s. In this case, there are just two representations s max = and s min = 0. The highest m-state has m = and belongs to the highest s-representation s = S sm = T ( m m Acting with S - = S - + S - on both sides gives the algebraic S - S sm = S - + S - T c - S0 sm = c - T-, H S0 sm = T- S0 sm =, H T-,, ( m m ( + c m m - T ( m m + T, - ) + H m m S0 sm = S c 0;m m Sm, m * m T, - ( m m, - ) m m ) m m where in the third line, the result c - sm = H s + m s - m + has been used. In general, it is noted that for the highest m-state c - j j = H j. In the last line, the sum over m-states defines the Clebsch-Gordon coefficients. Evidently c 0; = and c = 0;-,-, This result could have been written down from the general result above for the CG coefficients of the m = j + j - state of the j = j + j representation. Having determined the S0 sm state, it remains to determine the orthogonal m = 0 linear combination, which in this case is S00 sm = H S-, - m m H S, - This result aslo defines the CG coefficients c 00; = - and c = 0;-,-, m m Note that there is an arbitrariness to which of these CG coefficients gets the minus sign. The next step is to apply S - again to determine the m = j max - states. S - S0 sm = S - + S - H S-, S - S- m m + S - + S - H c- 0 S, - sm = H 0 + c - S-, - m m + H c H S, - sm = H, - + m m H, - S- S, - sm = S-, - m m S- m m, - m m, - m m + 0 which maes sense, since the lowest m-state should also be defined uniquely. As a chec, applying S - to the S00 sm state

17 apr_-8.nb: 5//04::8:0:54 7 S - S00 sm = S - + S - H S- H S-, m m - S - + S - H S, - 0 = H 0 + c - S-, - - m m H c - S-, - m m =, - m m - H, - m m = 0 S- yields the null state on both sides. m m Another chec is to apply S = S + S + S ÿ S = S + S + S z S z + S + S - + S - S + to all four states. For the triplet states S =, so S + S + S z S z + S + S - + S - S + T ( = T ( = T S + S + S z S z + S + S - + S - S + T- - ( = T- and for the m = 0state S + S + S z S z + S + S - + S - S + T, - ( + T T, - ( + T-,, ( = ( = T, - ( + T-, ( ( - ( = T- where the term S + S - + S - S + acts to exchange the two spins, which yields (+) for a symetric state. - ( For the singlet state, S = 0. Comparing to the S0 calculation, only the exchange term changes sign, which results in = 0, so everything checs. A few comments: Comparing to the discussion of identicle particles, the S = 0 state is antisymmetric and is a singlet, whereas the S = states are symmetric and form a triplet. In general, when adding angular momentum for the case j = j, the j max representation is a symmetric under exchange of m m, the j = j max - representaion will be anti-symmetric, and the lower j representations alternate accordingly. Orbital angular momentum The orbital angular momentum operators can be realized as generators of rotations operating on the space of direction ets, as differential operators acting on spherical harmonics, or as algebraic operators acting on integral representations of SO(), typically denoted by lm. Historically, the differential operators were first to be discussed, but for many practical purposes algebraic techniques are more valuable. This discussion will be fairly brief, and will be focussed on connecting the different descriptions just mentioned.. still in transcription ü Direction ets The direction ets Sǹ developed earlier.

18 apr_-8.nb: 5//04::8:0:54 8 ü L = xâ p ü L as a differential operator ü Spherical Harmonics as inner product between state basis and position basis ü Orthogonality of SH Tensor operators The discussion so far has focussed on the effect of rotations on states. In a position basis, the states are transformed in an obvious way: the coordinates describing the position et are transformed. However, since position ets are not typically eigenstates of the Hamiltonian, it is typical to consider the eigenstates of angular momentum, described by the quantum numbers j, m. Under rotations the quantum number j does not change, but m does. One may say that the set of states with the same j constitutes a representation consisting of j + components. The components are mixed up by rotations, but the representations are not. As usual in quantum mechanics, transformations may affect operators as well as states. For rotations, the operators may be charactrized either in a coordinate representation or in terms of spherical tensor operators which are conveniently described in the same language used for angular momentum states. A spherical tensor operator of ran consists of + components, each identified by an azimuthal number q. The relation between identifiers for the states and operators is j and q m. The use of, q is purely conventional to cue the association with an operator as opposed to a state. One may thin of acting with a spherical tensor operator on a state as "adding" angular momentum to the state, and the algebra for this angular momentum addition is similar to that for combining the angular momentum content of two states, as described above. Similarly, spherical tensor operators may be combined to form new operators with different angular momentum content. The material below proceeds in two steps. The first discusses properties of rotations on simple operators: a) a general discussion of the effect of rotations on operators, b) an explicit application to the behavior of rotations on themselves and c) specializing to the effect of rotations on the generators of rotations J i leading to d) a more general discussion of vector operators under rotations, and e) isolating that discussion to the effect of infinitesimal rotations on a vector operator. Then the second step begins with f) a generalization of vector operators to a definition of tensor operators, g) their transformations under infinitesimal rotations, h) their addition properties, i) the result of operating on states with tensor opertors, and culminating with j) a discussion of the Wigner-Ecart theorem. ü General remars To focus on the effects of rotations, consider the position operator X = X, Y, Z and the set of position ets Sx, which may be used as a basis set to specify both the state Sa of a system and the operators acting on those states. In terms of the position ets, one may expand

19 apr_-8.nb: 5//04::8:0:54 9 Sa = x Sx x a X i = x Sx x i x W In this form the position operator is diagonal, and has been written in component form. The expectation for X i in a particular state is X i = a X i a At this point it is important to distinguish the effects of rotations in three distinct scenarios. First, one may rotate the states Sa Ø Sa' = R Sa. This does not involve changing the position basis ets, but it does imply a transformation of the expansion coefficients x a. Second, one may rotate the position operators X i Ø X ' i = R i j X j = x Sx R i j x j x W. Again, this does not involve a change of the basis ets, merely a different linear combination of the operators X, Y, Z. The new operators are still diagonal, but with different eigenvalues. If one leaves the states unchanged, but rotates the operators, then the expectation values change as a classical position vector would change under rotations X i Ø R i j X j. Third, one may change coordinate systems. In this case, the state of the system Sa is unchanged, but the basis ets for describing the system are changed: x Sx x Ø x Sx ' x ', where Sx ' = R Sx. Note that the integration is still over all position ets. In addition, the position operators are also transformed so that, for example, the x` -direction transforms to the x` -direction in the new basis. A change of viewer should not change the expectation value for an observable, and indeed X i = x Sx x i x W Ø x Sx ' R i j x j x 'W = x ' a Sx ' x i ' x 'W a = x a Sx x i x W a = X i where in the second line one uses the equivalance of the integration volumes x = x ', and in the third line a change of variables from x Ø x' is performed. The result of these considerations is that the effect of a rotation on the position operator can be described either by a transformation of the basis states, X i ' = x Sx ' x i x 'W = RX i R - or by a transformation of the operators X i ' = x Sx ' R i j x j x 'W = R i j X j Since the symbol R is used in different fashions (either to act on the basis states or in the space of operators) there is some danger of confusion! Fortunately, the D-notation is established to describe more generally how rotations operate on different repesentations, so one may use the notation X i ' = DX i D - instead. Further, although convenient to use the position basis for pedagogical purposes, any complete set of basis ets would do, so one may replace x Ø S jm. In addition, since rotations do not mix states from different representations, the D-operators can be chosen to be finite dimensional, so as to apply to a single irreducible representation at a time.

20 apr_-8.nb: 5//04::8:0:54 0 ü The effect of rotations on rotations Suppose one has a state Sa and two rotations R and R. Define the state Sa = R Sa, and the state Sa = R Sa = R R Sa. In this picture, R acts to rotate the state Sa. Alternatively, one could view the action of R to be a rotation on the system where R has operated, i.e. R R Sa = R ' Sa' where R ' defines the action of the rotation R as viewed from the coordinate system of R, and Sa' = R Sa. The latter relation can also be written in the inverse form Sa = R - Sa', and so R R Sa = R R R - Sa' Comparing the two results, one finds that the rotations themselves transform as R ' = R R R - ü Alternative Consider the effects of a rotation R on the matrix element b R a. In the picture where the transformation is considered to be a change of coordinate system, the matrix element should be left unchanged by the transformation. Using 's to denote the transformed states and operators, and denoting the transforming rotation by R b R a fl b' R' a' = b R - R' R a one finds R = R - R' R or R' = R RR - as above. ü Application to Euler angles. In the discussion of Euler angles, it was stated that R z' g R y' b R z a = R z a R y b R z g where the 's indicated a rotation around the body axis, for example R y' is the rotation around the y-axis as seen after the rotation by R z a. Given the transformation properties of rotations, one may write R y' b = R z a R y b R z - a Defining the combined rotation Ra, b = R z a R y b

21 apr_-8.nb: 5//04::8:0:54 R z' g R y' b R z a = R z' g R z a R y b R z - a R z a as advertised. = R z' g R z a R y b = R z' g Ra, b = Ra, b R z g R - a, b Ra, b = Ra, b R z g = R z a R y b R z g ü The effect of rotations on the generators of rotation The generators of rotations J i constitute a three component vector, with the property that n ÿ J acts as the generator for rotations around an arbitrary axis ǹ. It seems reasonable that these should transform in a manner similar to the position operators. To do this consider the relation R' = R RR - in the case where R is infinitesimal, R = e -ie nÿj - ie n ÿ J. On the left, one has R' = - ie n ÿ J ' with J i ' = R i j J j, and on the right R RR - = R - ie n ÿ J R -. Equating the two sides, one finds the same relation that was obtained for the position vector J i ' = R i j J j = D J i D - where on the left-hand side R mixes the different operators, and on the right-hand side the D notation has been adopted to emphasize the transformation of the basis ets used to describe the vector space. To see the difference in the two determinations of J i ', consider a particular component, say, the component of J '. On the left, J ' is made up of the components of J, J, and J weighted by R, R, and R respectively. On the right, J ' is made from a linear combination of all 9 components of J, but the components of J, and J are not used. ü Vector operators The behavior of X and J can be generalized to define a vector operator V as a set of three operators V i which under a rotation transform as V i Ø V ' i = R i j V j = RV i R - It is useful to consider the infinitesimal form of this relation. On the left, R i j V j = - ie n ÿ J i j V j = d i j - ie n J i j V j where J i j = -ie i j. On the right, RV i R - = - ie n ÿ J V i - ie n ÿ J = V i - ie n J, V i Note that in this equation J operates on the components of an individual V i as opposed to mixing the compenents of V. Equating the two, the requirement for V to be a vector operator is that the components of V obey the commutator relations J, V i = -ie i j V j

22 apr_-8.nb: 5//04::8:0:54 The general vector operator can be defined by commutation relations in analogy to the angular momentum commutation relations. This can be checed explicitly for the case of orbital angular momentum acting on vector operators X and P. ü Cartesian vectors vs "spherical vector" One may recall that the description of angular momentum states is simplified by introducing the raising and lower operators J, and then choosing J 0, J (as opposed to J x, J y, J z ) to be the basis for the space of angular momentum generators. A similar redefinition can be made for vector operators, but before proceeding, it is useful to introduce a transformation of cartesian coordinates which parallels the transformation from J x, J y, J z to J 0, J. Define coordinates x q with q taing the values of q =, 0, -, by x 0 = z, x = x iy. I don't have a name for these coordinates. In analogy to the spherical tensors to be defined below, I'm tempted to call these coordinates "spherical" coordinates, but that name is already taen. Spherical cartesian? Ugh. In any event, the dot product of two vectors can be rewritten as u ÿ v = u i v i = u q v -q. The new coordinates can be expressed as a transformation of the usual cartesian coordinates, x q = U x i, where the transformation matrix is U = H - H i 0 0 H i H 0 0 The next step is to re-examine the defining relation for rotations in cartesian coordinates x' i = R i j x j = e -ifnÿj i j x j. First, the new coordinates can be used to rewrite the generator: n ÿ J = n -q J è q, where J è = H J is defined to account for the difference in definition between the new coordinates and the conventional definition of J. As written, the rotation is operating on the usual cartesian coordinates, so the generators must be written in that basis as well, for example, J è 0 0 S = H 0 0 -i i 0 Second, the new coordinates can be used as the object of the rotations H x' q = R qq' x q' = e -ifnÿj qq' x q' Again there is the choice of writing n ÿ J as n i J i or n -q J è q. In either case, though, J must be written in the form for operating on x q instead of x i. Following the usual rules for transforming operators J ' i = U J i U or J è ' q = U J è q U. The J ' have the same form as when operating on the j = representation, for example, J ' 0 = J l= z = 0 0 0, Jè ' + = H J + l= = (5//04 - I am not sure I don't have some "typos" still in this section)

23 apr_-8.nb: 5//04::8:0:54 ü Spherical vector The definition of a spherical vector operator can now be given in terms of the components of a cartesian vector operator. Specifically, if V i is a vector operator, then the spherical vector with components V 0, V can be defined by V 0 = V z V = H V x iv y Later this object will be identified as a ran- spherical tensor T q where q =, 0, -, and T 0 = V 0, T = V. Expressing the effect of the rotation on V q by its effect on the eigenstates RV q R - = - ie n ÿ J V q - ie n ÿ J = V q - ie n -m J èm, V q where m has been used to distinguish the generator from the spherical vector component. Meanwhile, rotating the operator components yields R qq' V q' = - ie n ÿ J qq' V q' = d qq' - ie n -m J m qq' V q' equating the two results, for each m one has, J èm, V q = J è m qq' Vq' On the left, the exact form of J èm and V q depend on the representation upon which the operators act. On the right, V q is defined relative to the representation but J è m q q' is always the j = representation since it is acting on a vector operator. Explicit forms for J èm are J è0 = , Jè+ = , and Jè- = As an example, suppose V q is the angular momentum operator itself, and that the equation is applied to the j = representation. Then, the V q are 5 ä5matrices, and the J èm in the commutator is also 5 ä5, but the J è m qq' on the right is ä to mix the components of the vector operator. ü Cartesian tensors One can form more complicated tensors. For example, T i j = X i X j is a ran- tensor. A ran-n tensor has n-indicies, T i i i n, and transforms under rotations as T i i i n Ø T ' i i i n = R i j R i j R i n j n T j j j n This ran-n tensor has n components. It may be reducible, since one might be able to find subsets of the components of T which are invariant under rotation. ü Brea cartesian into irreducible tensors For example, one can form a 9-component ran- tensor O i j from two vectors U i and V j

24 apr_-8.nb: 5//04::8:0:54 4 O i j = U i V j but this tensor can be reduced to three irreducible tensors O S, O V, O Q, nown as scalar, vector and quadrupole tensors. Their explicit construction is O S = d i j O i j, O i V = e i j O j, and O i j Q = O i j + O ji - These three tensors are invariant under rotations. In terms of labels, the reduction of U V can be expressed as V ä V = S + V + Q S, V, Q have,, and 5 independent components respectively, so one may express the reduction as ä = OS Lastly, one may identify S, V, Q as operators associated with irreducible representtions of angular momentum corresponding to j = 0,,, respectively. Then the brea down of O = U V can be understood in terms of anglar momentum addition ä = where the labels are j-values. This provides the rational for generalizing to spherical tensors. ü Spherical tensors The spherical vector was defined through its properties under rotations. For an infinitesimal rotation R qq' V q' = - ie n ÿ J qq' V q' = d qq' - ie n -m J m qq' V q' Correspondingly, a spherical tensor of ran- is T with + components T q. T q transforms as T q Ø T ' q = R qq' T q' = - ie n ÿ J qq' T q' = d qq' - ie n -m J m qq' T q' where the label has been added to J m to mae explicit that the representation of J must match the ran of the tensor. Although the term ran- is used, it should be understood that this corresponds to an irreducible representation- of angular momenetum ü Spherical tensor operators Motivated by the definition of a sperical vector operator, one can define a spherical tensor operator T with components T q, by the behavior of the operators when transformed by a rotation. The expectation value of an operator when the system is in a particular state is given by T q a = a T q a Under rotations, T q Ø RT q' R -

25 apr_-8.nb: 5//04::8:0:54 5 A reasonable definition of a spherical tensor operator is that the expectation value transforms under rotations in the same way that a classical tensor would. This should be true for any state a and therefore should be a property of the operator, i.e. T q Ø D qq' T q' ü Definition of tensor operators by their commutation properties By considering infinitesimal rotations one can define the tensor operators in terms of commutation properties with J m. Thus, for an infinitesimal rotation, the form RT q' R - becomes RT q R - Ø T q - ie n -m J èm, T q Similarly D qq' T q' Ø T q - ien -m J m qq' T q', which requires nowledge of J m qq'. These operators are nown from the action of J z, J on angular momentum states in different j-representations, J 0 qq' = d qq' q and J qq' = c q d q,q' The definition of a tensor operator can then be specified in terms of how J z, J act on a tensor operator through the commutation relations J z, T q = qt q J, T q = c q T q where c q = H S q q +, as for when the angular momentum generators act on states S j =, m = q. ü Combining tensor operators Just as angular momentum states can be combined via the procedure for angular momentum addition, spherical tensor operators can be combined with a similar addition formula. Consider two normalized tensor operators U, V, with components U q, Vq. Then the product Uq V q can be expressed as a linear combination of objects with tensor properties U q V q = Sq c q,q q T q (Eq. ) There are + + possible products that can be formed from the components of U and V. Under the set of rotations, the various components of U transform into each other. Similarly for V and the T. It follows that for each T all components T q are included in the set of all possible U q V q products. The next step is to consider the effect of J z J z, U q V q = Uq Jz, V q + Jz, U q Vq = q + q U q V q or, in the sum over, q, it is required that q = q + q. This in turn implies that there is a maximum value for q, namely q max = +. Considering that the tensor operators must be complete, also has a maximum value, max = +. The operator T max max = U V is the unique q = max operator.