VI Angular momentum Up to this point we have been dealing primarily with one dimensional systems In practice of course most of the systems we deal with live in three dimensions and 1D quantum mechanics is at best a useful model In this section we will focus in particular on the quantum mechanics of 3D systems Many of the elements we discovered for one dimensional problems will carry over directly to higher dimensions however we will encounter certain effects that are qualitatively new and we will spend most of our time exploring these new phenomena The first change comes in how we associate operators with classical observables In one dimension we had q q p p i q In three dimensions the position and momentum are vectors and so we must substitute vector calculus for the single variable results r r ix jy kz p p ip jp kp y z x Where r is the position vector and vector quantities will always be indicated in bold face Note that the operators that correspond to different axes i e p x and z commute with one another while the position and momentum along a given axis i e p x and x obey the normal commutation relation We can summarize this in a few equations p i p j 0 r i r j 0 r i p j iZ ij where i and j can take the value 1 2 or 3 to indicate the x y and z components of each vector a Rotations The first difference between 3D and 1D is the possibility of performing a rotation of our system about one of the three axes Let us denote a rotation of an angle about a unit vector n by R n Clearly R n is a matrix it transforms vectors to vectors Further it is clear that R n R m R m R n rotations about different axes do not commute Note that this has nothing to do with quantum mechanics and everything to do with geometry It is easy to verify that the rotation operators associated with the three Cartesian axes are 0 0 1 R x 0 cos sin 0 sin cos cos 0 sin R y 0 1 0 sin 0 cos cos sin 0 R z sin cos 0 0 0 1 Note that the rotation matrices for x and y can be obtained from the z matrix by the cyclic permutation x y y z z x This must always be the case because our labeling of the x y and z axes is totally arbitrary The only thing we must be careful of is that the triple product z x y is always 1 This defines the handedness or chirality of our coordinates Cyclic permutations preserve the handedness while a simple interchange of two axes i e x y will flip the sign of the triple product reverse the handedness of our coordinates and give us the wrong answer try it and see This cyclic invariance is very important because it reduces the work we need to do by a factor of 3 but we must be careful to apply it correctly In the future we can therefore state the result for the z axis and then infer the results for x and y by cyclic permutations These rotations are unitary i e R T R 1 and like many unitary transformations they can be written in the form e i J Z where J is called a generator For example the generator of rotations about the z axis is 0 i 0 J z Z i 0 0 0 0 0 This can be verified by actually computing ei J z Z and checking that it gives the rotation operator discussed earlier b Commutation Relations We now wish to compute the commutator between J x and J y 0 0 i 0 0 i 0 i 0 0 0 0 0 0 0 0 0 J x J y Z2 0 0 0 i 0 J y J x Z 2 0 i 0 0 0 0 0 i i 0 0 Z2 1 0 0 0 0 i Z2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 J x J y iZJ z As discussed previously all other commutators between the elementary generators of rotations can be deduced from the above relation by cyclic permutations of the indices These are the fundamental commutation relations for angular momentum In fact they are so fundamental that we will use them to define angular momentum any three transformations that obey these commutation relations will be associated with some form of angular momentum It is also useful to define the vector J J x i J y j J z k and the scalar J 2 J J J x2 J y2 J z2 It is easy to show that while the elementary generators do not commute with J they do commute with J 2 J z J 2 J z J x2 J z J y2 J z J x J x J x J z J x J z J y J y J y J z J y iZJ y J x J x iZJ y iZJ x J y J y iZJ x 0 Note that these J matrices are not quantum operators they are simply transformations of 3D space However we can use this as a definition of angular momentum in the quantum case Specifically we assume that the quantum operators which act in Hilbert space obey the same commutation rules as the classical transformations which act in real space Hence quantum angular momentum operators obey J x J y iZJ z and cyclic permutations thereof This seems strange at first but momentarily we will show that this rule for associating operators with classical variables is consistent with our definitions of r and p which strongly supports the new quantization rule Further we will later see that the same commutation rules apply to a particle s intrinsic spin angular momentum which cannot be described as some function of r and p Hence the commutation relation above actually generalizes the standard quantization rules Classically angular momentum is given by L r p Using our standard prescription this means the corresponding quantum operator should be L r p We proceed to verify that the components of L obey the expected commutation relations L x L y y p z z p y z p x x p z y p z z p x z p y z p x y p z x p z z p y x p z L y L x z p x x p z y p z z p y z p x y p z z p x z p y x p z y p z x p z z p y L x L y y p x p z z x p y z p z i Zy p x iZx p y iZL z c Eigenstates Since J 2 and J z commute they share common eigenstates We …