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

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

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' ofphysical 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.

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 linearmomentum operator takes on values. +/- m/L

Representations

We can think of an example of how this arises in the case of aparticle moving in a central potential in 2-d. Rotations in 2-dare 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

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)

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.

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 thegenerator 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- vectorspace (like Hilbert space!!) we have access to other choices..even in just two dimensions.

Ex: ...and others...

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 statesin 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 definitionof above.


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...


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;

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 thosetwo computations via integration by parts on the interval

But this leads to differences of on the

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 =

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 makethis make sense, but that is a story for later....assume for

now that this V is a reasonable (not too singular and finitein 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.

-

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.

Angular Momentum in 3-d

We now generalize the discussion to include the notionof angular momentum in 3-dimensions.

First some framing remarks: in 2-d, the group of rotationswas abelian (commutative); that led to all the representations being one dimensional i.e. associated withone 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 morethan one-dimensional, and so will not be labeled by a single integer...but two will suffice ! We explain below...


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


“Quadratic Casimir” is the fancy name given to an operatorquadratic in all the L's that commutes with them all.

Note that the vector square of the angular momentum issuch 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 moresimple...the harmonic oscillator!

Harmonic Oscillator and Angular Momentum algebra

Reminder: The QHO had the operator algebra

And the structure of the Hilbert space is a ladder.

01

23

n=

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 thesingle 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;

i

j

Our Hilbert space: states are the grid points.

i

j

Our Hilbert space: states are the grid points.

|0,0>

Each one of these states are not 'moved' by either

or

But they are rescaledby them (eig-states!)

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


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!)

i

j


On this state it is zero!

wipes out this state.

i

j


And note that

and

do not move the statebut have the followinglevel sets as shown.

both

=0

=-1

=1 =4

=-2

Where units are

i

j


And note that

and

do not move the statebut have the followinglevel sets as shown.

both

=0 3/4 2

M = 0 1/2 1 5/2

35/4

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 spaceindicates that since L+ and L- and Lz all act inside justone diagonal slice of this Hilbert space, we can find (all)finite dimensional representations of the angular momentumalgebra 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 calledthe 'spinor' representation. Let's look at this representation in more detail!


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+=


The Spinor Representation of the ang mom algebra:

Following the formula for the Lz in terms of the oscillatorops then also gives;

And L2 is

Lz =

L2=


So the M we identify as the largest element in the Lz

matrix for that representation. These are called the weights of the representation and vectors with such a Lz are calledhighest (or lowest!) weight vectors.

Note that the number of states (i.e. The dimension of therepresentation) 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....

i

j


=1 2 3

M = 0 1/2 1 5/2

6dim

scalar

spinor

Spin 1

Spin 5/2

Documents

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