Tensor Operators Michael Fowler,2/3/12 Introduction: Cartesian Vectors and Tensors Physics is full of vectors: x, L, S and so on Classically, a (three-diensional) vector is defined by its properties under rotation: the three coponents corresponding to the Cartesian x, y, and z axes transfor as V R V, i i with the usual rotation atrix, for exaple R z cos sin 0 ( ) sin cos 0 0 0 1 for rotation about the z-axis (We ll use,, and,, x y z x x x interchangeably) 1 2 3 A tensor is a generalization of a such a vector to an obect with ore than one suffix, such as, for exaple, T or T (having 9 and 27 coponents respectively in three diensions) with the i i reuireent that these coponents ix aong theselves under rotation by each individual suffix following the vector rule, for exaple T R R R T i il n ln where R is the sae rotation atrix that transfors a vector Tensors written in this way are called Cartesian tensors (since the suffixes refer to Cartesian axes) The nuber of suffixes is the ran of the Cartesian tensor, a ran n tensor has of course 3 n coponents Tensors are coon in physics: they are essential in describing stress, distortion and flow in solids and liuids Tensor forces play an iportant role in the dynaics of the deuteron, and in fact tensors arise for any charge distribution ore coplicated than a dipole Going to four diensions, and generalizing fro rotations to Lorentz transforations, Maxwell s euations are ost naturally expressed in tensor for, and tensors are central to General Relativity To get bac to non-relativistic physics, since the defining property of a tensor is its behavior under rotations, spherical polar coordinates are soeties a ore natural basis than Cartesian coordinates In fact, in that basis tensors (called spherical tensors) have rotational properties closely related to those of angular oentu eigenstates, as will becoe clear in the following sections
2 The Rotation Operator in Angular Moentu Eigenet Space As a preliinary to discussing general tensors in uantu echanics, we briefly review the rotation operator and uantu vector operators (A full treatent is given in y 751 lecture) Recall that the rotation operator turning a et through an angle (the vector direction denotes the axis of rotation, its agnitude the angle turned through) is i J e U R Since J coutes with the al angular oentu suared J 1, we can restrict 2 2 our attention to a given al angular oentu, having as usual an orthonoral basis set,, or for short, with 2 + 1 coponents, a general et in this space is then: Rotating this et, i J e Putting in a coplete set of states, and using the standard notation for atrix eleents of the rotation operator, e i J,, D R i J e i J D e is standard notation (see the earlier lecture) So the et rotation transforation is D, or D with the usual atrix-ultiplication rules Rotating a Basis Ket Now suppose we apply the rotation operator to one of the basis ets,, what is the result?
3 ij ij e,,, e,, D R Note the reversal of, copared with the operation on the set of coponent coefficients of the general et (You ay be thining: wait a inute,, is a et in the space it can be written, with, so we could use the previous rule D to get D D D Reassuringly, this leads to the sae result we ust found) Rotating an Operator, Scalar Operators Just as in the Schrödinger versus Heisenberg forulations, we can either apply the rotation operator to the ets and leave the operators alone, or we can leave the ets alone, and rotate the operators: ij ij A e Ae U AU which will yield the sae atrix eleents, so the sae physics A scalar operator is an operator which is invariant under rotations, for exaple the Hailtonian of a particle in a spherically syetric potential (There are any less trivial exaples of scalar operators, such as the dot product of two vector operators, as in a spin-orbit coupling) The transforation of an operator under an infinitesial rotation is given by: fro which ( ) ( ) with ( ) 1 i S U R SU R U R J i J S S, S It follows that a scalar operator S, which does not change at all, ust coute with all the coponents of the angular oentu operator, and hence ust have a coon set of eigenets with, say, J 2 and J z
4 Vector Operators: Definition and Coutation Properties A uantu echanical vector operator V is defined by reuiring that the expectation values of its three coponents in any state transfor lie the coponents of a classical vector under rotation It follows fro this that the operator itself ust transfor vectorially, V U R VU R R V i i i To see what this iplies, it is easiest to loo at a siple case For an infinitesial rotation about the z-axis, the vector transfors R z 1 0 ( ) 1 0 0 0 1 Vx 1 0Vx Vx Vy Vy 1 0 Vy Vy Vx V z 0 0 1V z V z The unitary Hilbert space operator U corresponding to this rotation U R 1 / 1 / U VU i i J z Vi i J z i V J, V i z i i J so z z 1, The reuireent that the two transforations above, the infinitesial classical rotation generated by Rz ( ) and the infinitesial unitary transforation U RVU i R, are in fact the sae thing yields the coutation relations of a vector operator with angular oentu: i J, V V z x y i J z, Vy Vx Fro this result and its cyclic euivalents, the coponents of any vector operator V ust satisfy: Vi, J ii V
5 Exercise: verify that the coponents of x, L, S do in fact satisfy these coutation relations (Note: Confusingly, there is a slightly different situation in which we need to rotate an operator, and it gives an opposite result Suppose an operator T acts on a et to give the et T For ets and to go to U and U respectively under a rotation U, T 1 itself ust transfor as T UTU (recall U U ) The point is that this is a Schrödinger rather than a Heisenberg-type transforation: we re rotating the ets, not the operators) Warning: Does a vector operator transfor lie the coponents of a vector or lie the basis ets of the space? You ll see it written both ways, so watch out! We ve already defined it as transforing lie the coponents: V U R VU R R V i i i but if we now tae the opposite rotation, the unitary atrix URis replaced by its inverse U R and vice versa Reeber also that the ordinary spatial rotation atrix R is orthogonal, so its inverse is its transpose, and the above euation is euivalent to V U R VU R R V i i i This definition of a vector operator is that its eleents transfor ust as do the basis ets of the space so it s crucial to loo carefully at the euation to figure out which is the rotation atrix, and which is its inverse! This second for of the euation is the one in coon use Cartesian Tensor Operators Fro the definition given earlier, under rotation the eleents of a ran two Cartesian tensor transfor as: T T R R T i i ii i where R i is the rotation atrix for a vector It is illuinating to consider a particular exaple of a second-ran tensor, Ti UiV, where U and V are ordinary three-diensional vectors
6 The proble with this tensor is that it is reducible, using the word in the sae sense as in our discussion of group representations is discussing addition of angular oenta That is to say, cobinations of the eleents can be arranged in sets such that rotations operate only within these sets This is ade evident by writing: U V UiV U Vi UiV U Vi U V UV i i i 3 2 2 3 The first ter, the dot product of the two vectors, is clearly a scalar under rotation, the second ter, which is an antisyetric tensor has three independent coponents which are the vector coponents of the vector product U V, and the third ter is a syetric traceless tensor, which has five independent coponents Altogether, then, there are 1 + 3 + 5 = 9 coponents, as reuired Spherical Tensors Notice the nubers of eleents of these irreducible subgroups: 1, 3, 5 These are exactly the nubers of eleents of angular oenta representations for = 0, 1, 2! This is of course no coincidence: as we shall ae ore explicit below, a three-diensional vector is atheatically isoorphic to a uantu spin one, the tensor we have written is therefore a direct product of two spins one, so, exactly as we argues in discussing addition of angular oenta, it will be a reducible representation of the rotation group, and will be a su of representations corresponding to the possible al angular oenta fro adding two spins one, that is, = 0, 1, 2 As discussed earlier, the atrix eleents of the rotation operator within a definite subspace are written U R e i J i J,, D R e so under rotation operator a basis state, transfors as: ij ij e,,, e,, D R The essential point is that these irreducible subgroups into which Cartesian tensors decopose under rotation (generalizing fro our one exaple) for a ore natural basis set of tensors for probles with rotational syetries
7 Definition: We define a spherical tensor of ran as a set of 2 + 1 operators T,, 1,, such that under rotation they transfor aong theselves with exactly the sae atrix of coefficients as that for the 2 + 1 angular oentu eigenets for =, that is, U R T U R D T To see the properties of these spherical tensors, it is useful to evaluate the above euation for infinitesial rotations, for which D I i J i J, /,, /, (The atrix eleent, J /, is ust the failiar Clebsch Gordan coefficient in changed notation: the ran corresponds to the usual, and to the agnetic uantu nuber ) Specifically, consider an infinitesial rotation J J (Strictly speaing, this is not a real rotation, but the foralis doesn t care, and the result we derive can be confired by rotation about the x and y directions and adding appropriate ters) The euation is and euating ters linear in, 1 / 1 /, /, i J T i J i J T 1 J, T 1 T J z, T T Saurai observes that this set of coutation relations could be taen as the definition of the spherical tensors Notational note: we have followed Shanar here in having the ran as a subscript, the agnetic uantu nuber as a superscript, the sae convention used for the spherical haronics (but not for the D atrices!) Saurai, Bay and others have the ran above, usually in parentheses, and the agnetic nuber below Fortunately, all use for ran and for agnetic uantu nuber A Spherical Vector The = 1 angular oentu eigenets are ust the failiar spherical haronics Y 3 z 3, x Y iy 4 r 4 2r 0 1 1 1
8 The rotation operator will transfor (x, y, z) as an ordinary vector in three-space, and this is evidently euivalent to 1, 1, D R It follows that the spherical representation of a three vector V, V, V x y z V 1 x iv y 1 0 0 T1 V1, T1 Vz V1 2 has the for: In line with spherical tensor notation, the coponents T 1 0 1 1, T1, T 1 are denoted T 1 Matrix Eleents of Tensor Operators between Angular Moentu Eigenets By definition, an irreducible tensor operator T transfors under rotation lie an angular oentu eigenet, Therefore, rotating the et T,, UT UT U U D T D 1,,, The product of the two D atrices appearing is precisely the set of coefficients to rotate the direct product of eigenets,, where, is the angular oentu eigenet having =, = We have et this direct product of two angular oentu eigenets before: this is ust a syste having two angular oenta, such as orbital plus spin angular oenta So we see that T acting on, generates a state having al angular oentu the su of (, ) and (, ) To lin up (ore or less) with Shanar s notation: our direct product state,, is the sae as, ;, in the notation 1, 1 ; 2, 2 for a product state of two angular oenta (possibly including spins) Such a state can be written as a su over states of the for, ;, where this denotes a state of al angular oentu, z-direction coponent 1 2, ade up of two spins having al angular oentu 1, 2 respectively This is the standard Clebsch-Gordan su: 1 2, ;,, ;,, ;,, ;, 1 1 2 2 1 2 1 2 1 1 2 2 1 2
9 The sued ters give a unit operator within this (2 1 + 1)(2 2 +1) diensional space, the ter, ;,, ;, is a Clebsch-Gordan coefficient The only nonzero coefficients 1 2 1 1 2 2 have = 1 + 2, and restricted as noted, so for given 1, 2 we ust set = 1 + 2, we don t su over, and the su over begins at Translating into our,, notation, and cleaning up,, ;,, ;,, ;,, ;, We are now able to evaluate the angular coponent of the atrix eleent of a spherical tensor operator between angular oentu eigenets: we see that it will only be nonzero for = 1 + 2, and at least The Wigner-Ecart Theore At this point, we ust bear in ind that these tensor operators are not necessarily ust functions of angle For exaple, the position operator is a spherical vector ultiplied by the radial variable r, and ets specifying atoic eigenstates will include radial uantu nubers as well as angular oentu, so the atrix eleent of a tensor between two states will have the for,, T,,, 2 2 2 1 1 1 where the s and s denote the usual angular oentu eigenstates and the s are nonangular uantu nubers, such as those for radial states The basic point of the Wigner Ecart theore is that the angular dependence of these atrix eleents can be factored out, and it is given by the Clebsch-Gordan coefficients Having factored it out, the reaining dependence, which is only on the al angular oentu in each of the ets, not the relative orientation (and of course on the s), is traditionally written as a bracet with double lines, that is, 2, 2 T 1, 1 2, 2, 2 T 1, 1, 1 2, 2, ; 1, 1 2 1 1 The denoinator is the conventional noralization of the double-bar atrix eleent The proof is given in, for exaple, Saurai (page 239) and is not that difficult The basic strategy is to put the defining identities 1 J, T 1 T J, T T z
10 between,, bras and ets, then get rid of the J and Jz by having the operate on the bra or et This generates a series of linear euations for 2, 2, 2 T 1, 1, 1 atrix eleents with variables differing by one, and in fact this set of linear euations is identical to the set that generates the Clebsch-Gordan coefficients, so we ust conclude that these spherical tensor atrix eleents, ranging over possible and values, are exactly proportional to the Clebsch- Gordan coefficients and that is the theore A Few Hints for Shanar s proble 1533: that first atrix eleent coes fro adding a spin to a spin 1, writing the usual axiu state, applying the lowering operator to both sides to get the al angular oentu + 1, = state, then finding the sae state orthogonal to that, which corresponds to al angular oentu (instead of + 1) For the operator J, the Wigner-Ecart atrix eleent siplifies because J cannot affect, and also it coutes with J 2, so cannot change the al angular oentu So, in the Wigner-Ecart euation, replace result of (1) should follow T on the left-hand side by 0 J 1, which is ust J z The (2) First note that a scalar operator cannot change Since c is independent of A we can tae A = J to find c