1. Angular Momentum in Quantum Mechanics Other ways of thinking about tranformations 2-d rotations in QM Angular Momentum in 3-d: Oscillator Basis Wave functions of Angular Momentum

2. Representations In QM so far we have stressed the structural aspects of the ' theory...that there is always a Hilbert space on which we aim to find linear (as in 'a matrix') “representations' of physical quantities of interest. It is often useful to determine a basis of the Hilbert space relative to some of these quantities... the so-called eigenbasis. We recall a familiar example of this in the linear momentum operators;... Where the Hilbert space in this case is, essentially, the space of functions in x and y.

3. Representations But we can (and often do need to) find new ways of representing physical observables. For example, if we find one that has no classical antecedent OR if we wish to describe an observable one on some truncated Hilbert space....this is more than just a convenience...it can really help us think about what are the relevant degrees of freedom quantum mechanically. A useful case in point is again the linear momentum operators. Since they have a continuous spectrum, one would think that is all there is to that....but, as in the case of the particle in the box, it is really most useful (in that it captures the essential physical point) to consider a infinite but discrete subspace of the full Hilbert space in which the linear momentum operator takes on values. +/- m/L

4. Representations We can think of an example of how this arises in the case of a particle moving in a central potential in 2-d. Rotations in 2-d are a symmetry of such a problem and the connection between 2-d angular momentum and complex numbers is our starting point. Recall that we often have complex phases in QM. Note Euler's relation. Recall that for any x; The Euler identity extends this formula to complex arguments

5. Where the symbol satisfies the central identity; and we recognize the expansion of the trig functions, As an abstraction is nothing other than a place holder But we can ask, can we find a real representation for We can with matricies; for example, in 2-d take the real matrix = This is a real matrix that squares to (negative unit matrix)

6. And note that this carries over to functions of For the exponential for example, So Which for our choice of real reduces to and which we identify as rotations in the plane, the 2-d space over which acts.

7. In summary, what we have done is represent the action of a mathematical notion (complex numbers) as an activity (rotation, and scaling) on a real vector space. For those of you who have studied complex numbers you will recognize this as the complex plane. In the language of the last chapter, we think of i as the generator of the rotation group in the plane and the exponent (as before) is then the finite transformation, rotation through an angle x. We didn't have to limit ourselves to this interpretation of though...if we were interested in representing complex number as actions on higher dimensional -complex- vector space (like Hilbert space!!) we have access to other choices.. even in just two dimensions. Ex:...and others...

8. Rotations in QM in 2-d From the previous we see the connection between matrix representations of a rotation on a real space and an associated linear transformation in an complex vector space (like Hilbert space). Thus, we can talk about states in the co-ordinate basis, for which the generator matrix can be written Note that this implies for the x-operator and p-operator Which follow from the fundamental CCR and the definition of above.

9. Rotations in QM in 2-d And thus, representing the momentum operators in the co-ordinate basis, we have This is a form that we will use later in the 3-d context. However, here we specialize to 2-d where it is most revealing to re-write this in polar co-ordinates If the Hamiltonian is that of a central force in 2-d, it will commute with this operator...

10. Rotations in QM in 2-d So that H and can be simultaneously diagonalized. Said in equation form, one can find vectors |E,l> Or in co-ordinate basis, we write these |E,l> as;

11. We solve the angular momentum part first, Which we integrate to find for R a yet undetermined function. Note now that the fact that must be hermitian means for the matrix element we should get the same value whether the acts on the bra or the ket. Of course, we can relate those two computations via integration by parts on the interval But this leads to differences of on the

12. endpoints of the integration. Thus, the hermiticity of implies that the wave functions themselves must be periodic on the interval This is just the particle on the ring (very similar but not quite the same as the particle in the box...can you tell me what the difference is?) This means that the spectrum of l is discrete: with So l =

13. Note also the orthogonality relation: OK...simple enough...now let us return to the other eigenvalue equation, that of the energy: For H and to commute the Hamiltonian cannot have just any form...but the following is an example of a Hamiltonian operator for a central potential: For any function V. Not all functions V actually can make this make sense, but that is a story for later....assume for

14. now that this V is a reasonable (not too singular and finite in range, basically). Our eigenvalue equation when we substitute in Then yields an equation for R; = E And finding solutions of this equation with physical asymptotics (such as vanishing at infinity for a bound state for example) gives a condition on E, the so-called quantization condition. -

15. Note that, just as in the classical case, the equation for R; = E Has the interpretation that in selecting states with angular momentum We note the appearance of a centrifugal - Barrier term, the term. This means that only those states with can get to, in complete analogy the classical case.

16. Angular Momentum in 3-d We now generalize the discussion to include the notion of angular momentum in 3-dimensions. First some framing remarks: in 2-d, the group of rotations was abelian (commutative); that led to all the representations being one dimensional i.e. associated with one vector direction (the There each representation vector was labeled by a single integer, m. In 3-d, as you know, the group of rotations is not abelian. But we still find representation vectors...they will be more than one-dimensional, and so will not be labeled by a single integer...but two will suffice ! We explain below...

17. Angular Momentum in 3-d To start, note that in the co-ordinate basis the generators of rotations in 3-d are : Which satisfy the algebra (coming from the CCR of p and x),...and others which we can write compactly as

18. Angular Momentum in 3-d “Quadratic Casimir” is the fancy name given to an operator quadratic in all the L's that commutes with them all. Note that the vector square of the angular momentum is such an operator For which the forgoing commutation relations imply: The goal is to represent these operators on a Hilbert space. To that end it is most convenient to note that the algebra of the L's can be built out of something more simple...the harmonic oscillator!

19. Harmonic Oscillator and Angular Momentum algebra Reminder: The QHO had the operator algebra And the structure of the Hilbert space is a ladder. 0 1 2 3 n=

20. Harmonic Oscillator and Angular Momentum algebra Now take two QHO's..think of this as two non-interacting particles in the same potential. = And we can take the Hilbert space as the product of the single particle QHO Hilbert space ladders. Where the states in the full Hilbert space are now labeled by two integerts i, and j as |i,j>. The activity of the raising and lowering operators is then;

21. i j Our Hilbert space: states are the grid points.

22. i j |0,0> Each one of these states are not 'moved' by either or But they are rescaled by them (eig-states!)

23. Connection to the Angular Momentum Algebra Take : Then these satisfy the angular momentum algebra as a consequence of the QHO commutators on the previous page. This gives us a way of constructing the representations of the L's rather than deducing them as described in the book... Finally note that the has a simple form in a's and b's With

24. Connection to the Angular Momentum Algebra It is also very useful to define 'raising and lowering' combinations of the angular momentum algebra... and Where And = The reason these are so 'nice' to define is that from the point of view of the QHO algebra they are simple translations -diagonally- between the grid points of states (the Hilbert space basis vectors!)

25. i j Connection to the Angular Momentum Algebra On this state it is zero! wipes out this state.

26. i j Connection to the Angular Momentum Algebra And note that and do not move the state but have the following level sets as shown. both =0 =-1 =1 =4 =-2 Where units are

27. i j Connection to the Angular Momentum Algebra And note that and do not move the state but have the following level sets as shown. both =03/4 2 M = 0 1/2 1 5/2 35/4

28. The finite dimensional representations of the Angular Momentum Algebra So we have found a representation of the algebra in terms of 2 QHO's. Closer inspection of the Hilbert space indicates that since L + and L - and L z all act inside just one diagonal slice of this Hilbert space, we can find (all) finite dimensional representations of the angular momentum algebra this way! |0,0> is a 1-dimensional representation. The L + and L - are trivial (i.e. 0) here, this is called the scalar or 'S' (from the first letter of a German word used in spectroscopy) state. |1,0> and |0,1> furnish a 2-d representation. This called the 'spinor' representation. Let's look at this representation in more detail!

29. The finite dimensional representations of the Angular Momentum Algebra The Spinor Representation of the ang mom algebra: L - |0,1> = 0 L - |1,0> = s|0,1> With s = So as a matrix on the two-dim space with basis vectors |0,1> and |1,0> the L - matrix is; By similar reasoning, or using gives, L - = L+=L+=

30. The finite dimensional representations of the Angular Momentum Algebra The Spinor Representation of the ang mom algebra: Following the formula for the L z in terms of the oscillator ops then also gives; And L 2 is L z = L2=L2=

31. The finite dimensional representations of the Angular Momentum Algebra So the M we identify as the largest element in the L z matrix for that representation. These are called the weights of the representation and vectors with such a L z are called highest (or lowest!) weight vectors. Note that the number of states (i.e. The dimension of the representation) is just 2M+1. They are always |M,0> and |0,M> So, from this we can constructively build any finite dimensional representation of the ang mom alg. Some more examples....

32. i j Connection to the Angular Momentum Algebra =12 3 M = 0 1/2 1 5/2 6dim scalar spinor Spin 1 Spin 5/2