Chapter 3: Theory of Angular Momentum


Summary of this chapter in "Modern Quantum Mechanics" (J.J. Sakurai), by Patrick Van Esch
Last revision November, 14, 2003.

1. Rotations and angular momentum commutation relations.

Finite versus Infinitesimal Rotations

It is pointed out that finite rotations around different axes do not commute in 3-dimensional Euclidean space.  The finite rotation matrix for an active rotation around the z-axis is written down in equation (3.1.3), and the infinitesimal rotations around the z, x and y axes are written down next (up to order 2).  Manipulation of these inifinitesimal rotations leads to a very important commutation relation given in equation (3.1.9).  Although this may seem bizarre, this equation contains all the information needed about what we call "rotations".

Infinitesimal rotations in Quantum Mechanics

In equation (3.1.10) a most important concept is introduced: to a rotation in 3-dimensional Euclidean space must correspond an operator (will turn out to be unitary) acting on the state kets that does to the states what is considered a rotation in space.  The same structure as for translations in space and time evolution is used: an infinitesimal change corresponds to a hermitian operator that is the generator for the finite change.  For translations in space, this generator was momentum, for time evolution, it was the hamiltonian, and we now define angular momentum as the generator of rotations in 3-dim space.  As in those other cases, this is just a matter of definition, and the correspondence with classical variables will turn out to be the case (or not !  In this case, orbital angular momentum will, and spin angular momentum will not, correspond to the classical concept).  To go from the infinitesimal generator (3.1.15) to the finite rotation operator, we use operator exponentiation in (3.1.16).  The rotation operator in Hilbert space is a representation of the rotations in 3-dim Euclidean space.  Hence, the commutation relations for infinitesimal operators should be respected as worked out in equation (3.1.18).  This leads to the commutation relations (3.1.20).  They are called the fundamental commutation relations of angular momentum because they specify completely the algebra of angular momentum.

2. Spin 1/2 systems and finite rotations

Rotation Operator for Spin 1/2

In the two-dimensional Hilbert space of spin-1/2 particles, the operators defined in (3.2.1) satisfy the fundamental commutation relations of angular momentum.  If we postulate that they are the angular momentum, then we have the rotation operators, which work out to several interesting relations.   Equation (3.2.6) confirms for example the property we expect of a rotation operator.  However, equation (3.2.15) is a surprise: a rotation over 2 pi gives minus the state ! 

Spin Precession Revisited

It is noted in equation (3.2.18) that the time evolution operator is nothing else but a rotation about the axis of the  field with the rotation angle being proportional to time.

Neutron Interferometry Experiment to Study 2 pi rotations

In order to expose this sign flip of the state of a spin-1/2 particle after a 2 pi rotation, neutron interferometry has been applied: we compare the phase of a particle with the phase of the same particle when it went through a magnetic field.  It turns out that we need indeed a rotation of 4 pi to get back to the same interference pattern, confirming the sign flip after 2 pi.

Pauli two-component Formalism

The two-dimensional Hilbert space for spin-1/2 particles is written out in matrix form.  The elements are complex 2-tuples, and the Pauli matrices (3.2.32) play an important role.  Several properties of these matrices are worked out.

Rotations in the two-component formalism

Taking S in the 2-component system as the angular momentum, the rotation operator follows, as written down in equation (3.2.42).  Working this out in the form of a unit vector along the rotation axis, together with the rotation angle, one obtains equation (3.2.44), which is explicitly given in equation (3.2.45) as a 2x2 matrix.
It is argued that the expression in (3.2.47) behaves as a 3-vector under rotations.   Looking for the eigenvector with eigenvalue +1 of the rotation operator we expect this to correspond to the state with spin up in the rotation axis direction.  Working this out, one finds the result in (3.2.52).

3 O(3), SU(2) and Euler rotations

Orthogonal Group

The orthogonal group is the group represented by the 3x3 real, orthogonal matrices.  It's called O(3).
I'm a bit confused here: I thought the group we're talking about is SO(3) !  What happens with the disconnected piece ?

Unitary unimodular group

The unitary unimodular group is the group of unitary 2x2 matrices with determinant 1.  A correspondence between two complex numbers satisfying the condition (3.3.8) and the group elements is established.  Next, these complex numbers are put into correspondence with the explicit rotation matrix in the 2-dimensional Hilbert space (which is also a unitary unimodular matrix in the basis in which (3.2.45) was worked out.  Working things backward, we can associate a direction (in 3-dim Euclidean space) and a rotation angle to this matrix (in 2-dim Hilbert space).  The relations are worked out in equation (3.3.10).  This relationship between rotations in Euclidean space in 3 dimensions, and two complex numbers, was in fact already known long before any quantum mechanical treatment, in classical mechanics.  Interpreted as such, the two complex numbers are called the Cayley-Klein parameters of the rotation.  The group (as an abstract entity, whether we take them as 2x2 matrices, Cayley-Klein parameters, or other) is called SU(2).
It is pointed out that there is a 2 to 1 relationship

Euler Rotations

An arbitrary rotation in Euclidean 3-dim space can be resolved in 3 rotations about special axes, and these rotations are called Euler rotations.  This is well known from classical mechanics (used a lot in rigid body motion).  However, there is no unique convention about what exactly are Euler rotations !  In classical mechanics, it is usually taken to be rotations around axes fixed to the rigid body we're rotating.  In this part, it is shown that there is an equivalence with rotations around axes fixed in space (the coordinate axes z and y).  In fact, the rotation angles in the two cases are the same, but the order is opposite. 
One should carefully read what is written in this part to follow the argument.  Once formula (3.3.19) is established, writing a general rotation (specified by "mechanical" Euler angles) as a product of rotations around space-fixed axes, the representation of this property by the rotation operators in Hilbert space is written down in (3.3.20).  This is easily translated in the case of 2-dim Hilbert space, and results in the 2x2 matrix (3.3.21).

4. Density Operators and pure versus mixed ensembles

Polarized Versus Unpolarized beams

And now for something completely different !   I have to say that I don't understand why this important item has been set out in the "angular momentum" section, but that doesn't really matter.
It is first explained that _every_ state in the 2-dimensional Hilbert space corresponds to a preferred direction in 3-dim Euclidean space, namely the one that has this state as an "up" state. If we want to say that there "is no preferred direction" we don't have a ray in Hilbert space corresponding to that.  In fact, it turns out that we do not have to take a bigger Hilbert space to do this, we need to take a statistical ensemble of Hilbert space states to express this idea.  What is introduced here cannot be of more importance.  It turns out that what we called "states" up to now were special kinds of systems, called "pure states".  We are now considering more general kinds of states called "mixtures".  At first sight, mixtures are bunches of "pure states" mixed together with "classical probability".  No big deal, one would say.  We have the same in classical physics: individual points in phase space correspond to pure states, but we can of course define a density function over phase space, and then we have a statistical mixture.  In a certain way, what we are doing here is the quantum equivalent of that.  But it will turn out that these quantum mixtures are much more intertwined with the quantum formalism than the classical notion of a density function over phase space, and this for two reasons.  First of all, in the statistical interpretation (Copenhagen School, which we take here) of quantum mechanics, we always need an ensemble of states to be able to define a probability.  In mixtures we have a combination of quantum probabilities and statistical mixture probabilities.  But second, where a statistical mixture of different classical states in phase space defined uniquely the weight of each state in the mixture, in quantum mixtures we have the choice !  Mixing different Hilbert rays together with different weights can give rise to the same mixtures, which are indistinguishable in every sense.  So from the mixture we cannot go back to the precise composition, as there are many different compositions giving the same result.  All this makes that "quantum mixtures" are much more intertwined with quantum mechanics than is classical statistical mechanics with classical physics.

Ensemble Averages and Density Operator

For starters, we consider classical ensembles of quantum states, meaning: we consider states (rays in Hilbert space) with classical probabilities which sum up to 1.  If we take the ensemble average of any operator, we first calculate the expectation value of that operator with each of the quantum states present in the ensemble, and then weight it with the classical weight.  That's expressed in equation (3.4.6).  Note that we can put as many or as few quantum states in the ensemble as we want.  The states do not have to be independent, and hence we can have more states than dimensions of the Hilbert space.  The case of only one single state in the ensemble (with probability 1) is then called a "pure state".  And then something remarkable happens: in equation (3.4.7), we work in an explicit basis (the b kets), and it turns out that we can separate the part containing on one hand the classical probabilities and the Hilbert state kets, and on the other hand the representation of the operator we're taking the expectation of.  The first part is represented by a construction, called the density operator, as given in (3.4.8).  The formula for the expectation values then reduces to (3.4.10).  So all ensemble information is included in the density operator, because all physical quantities we'll ever measure can be expressed as expectation values of operators.  So two ensembles with identical density operators are in all respects physically equal.  The fact that total probability has to be equal to 1 is expressed by the trace condition (3.4.11).
We stress again that the remarkable fact is that different compositions of hilbert space vectors and weights can give rise to the same density operator, and are hence physically equal situations.
Next one analyses what is the density operator of a pure state.  Some examples in the 2-dimensional hilbert space are given.

Time evolution of Ensembles

One can think of the evolution of a statistical ensemble of states if one keeps the classical probabilities the same, and evolves the states in time (Schroedinger picture).  Of course, because different mixtures give rise to the same density operator, we'd better find that they evolve in the same way too !  But it turns out that the evolution of the density operator can be written in equation (3.4.29) in a way that is only dependent on the density operator and the hamiltonian.  So the evolution is not dependent on the precise mixture we took to build the density operator.  Note, that the time evolution of the density operator looks a lot like the Heisenberg equation of motion.  But there are two things wrong: first we're in the Schroedinger picture and second, there's this minus sign.  In fact, both remarks are equivalent if one thinks about it !
There's also a striking equivalence with the classical evolution equation of the density function in phase space, as written in equation (3.4.30).

Continuum generalizations

There's nothing special here, we already get used to the idea that when we work in a continuous basis, sums become integrals, so the trace becomes an integral too.  In fact, most of the work of mathematical physics is to make these heuristic formulations rigorous.  It then turns out that the results are in the overwhelming majority the same as with the heuristic techniques used here.

Quantum Statistical mechanics

The density operator opens the gate to statistical physics in quantum mechanics.  It actually turns out that quantum statistical physics is a lot easier and more coherent than classical statistical physics. 
The first point is the introduction of a quantity which is the entropy ; a quite mysterious quantity in classical physics, it turns out to have a very simple and precise definition in quantum physics: equation (3.4.35).  There's actually still a unit conversion factor needed, called Boltzmann's constant, so entropy is given by equation (3.4.41).  In thermodynamical equilibrium, the density operator doesn't change in time anymore (so we can diagonalise the density operator and the Hamiltonian together).  The internal energy is the expectation value of the hamiltonian.  We can now maximise entropy with the constraint of constant internal energy (using a langrange multiplier) and then we find the equation (3.4.48) for the elements of the density matrix in the energy eigenbasis.  This is the canonical ensemble.
The partition function can be written as given in equation (3.4.51).  As it is known from statistical physics that from the partition function one can deduce a lot of interesting thermodynamic quantities.
All this is illlustrated by a simple example in 2-dim Hilbert space, using spins subjected to a magnetic field and a finite temperature.

5 Eigenvalues and Eigenstates of Angular Momentum

This is a very calculational part.  We only give a brief overview of the ideas.

Commutation Relations and the Ladder Operators

It is worked out that J^2 commutes with the three components of angular momentum, so we can work in a basis where both J^2 and one of the components (Jz) are diagonal.  Two helper operators are introduced in equation (3.5.5), and lots of interesting stuff is derived concerning those.

Eigenvalues of J2 and Jz

It is first worked out that there must be kets which obey (3.5.17) and (3.5.23).  From these extremal states on, the whole finite set of independent eigenstates of Jz is worked out, climbing the ladder.  The result is summarized in equations (3.5.33) and (3.5.34).

Matrix Elements of Angular Momentum Operators

In the above basis of J2 and Jz, these operators take on of course quite simple representations (diagonal).  Next, the representations in that basis of the other angular momentum operators is worked out.

Representations of the Rotation Operator

Once we have the representation of the angular momentum operators in a certain basis, by exponentiation, we can find of course the rotation operator representation that goes with it.  This can be done using polar representation of the rotation, or using Euler angles.  The case j=1 is worked out for the y-rotation.

6. Orbital Angular Momentum

Orbital angular momentum is defined by the vector product of operators r x p.  This is a priori different from the generator of rotations, which is the definition of angular momentum, but on functions depending on position (wave functions in wave mechanics), it turns out to be equivalent.

Orbital Angular Momentum as Rotation Generator

One can easily verify that the three operators of r x p satisfy the commutation relations of angular momentum.  That by itself is of course not sufficient to be called "angular momentum", but when a state is characterised only by a point in space (the basis functions in wave mechanics), then one shows,in equation (3.6.5), that these operators _are_ the generators of what is reasonably taken to be a rotation.  We used the fact that p is the generator of translation to find this out.  Once again, this worked because our basis states in Hilbert space were supposed to be completely defined by a point in 3-dim Euclidean space.
Next, the same game is worked out in spherical coordinates, and one finds representations in this basis for the 3 operators of orbital angular momentum, as given in equations (3.6.9), (3.6.11) and (3.6.12).  Also the ladder operators are worked out (equation 3.6.13) and the L2 operator, in (3.6.14).  It is then observed that this L2 operator corresponds to the angular part of the Laplacian operator in 3-dim vector analysis, expressed in spherical coordinates.  Because this is so funny, this is worked out in two different ways.

Spherical Harmonics

If we can assume that the functions on which we operate (the wave functions) factorise as written down in equation (3.6.22), which is known to be the case if we have a spinless particle in a spherically symmetrical potential, then the angular part always takes on the form of spherical harmonics.  An equivalence between the eigenstates (abstract) of Lz and L2 as we worked them out above, and the spherical harmonics is established.  From the representation by spherical harmonics of the eigenstates of Lz and L2 on one hand, and the explicit representations of the L-operators in terms of spherical coordinates on the other hand, one builds up the explicit functional expressions of the spherical harmonics.
Because spherical harmonics have to be single valued functions in 3-dim Euclidean space, it is next argued that only integer values of  l can appear.
To give a short overview of what happened here: in the case of orbital angular momentum - which only makes sense if we have a Hilbert space in which there is a basis of position kets in 3-dim Euclidean space - and its associated operators can be written in that basis ; actually even in the angular part of that basis.  The eigenkets of L2 and Lz then become spherical harmonics, and the operators take on a representation as worked out.

Spherical Harmonics as Rotation Matrices

The link between the rotation operator in Hilbert space, and spherical harmonics is worked out here.  Perspectives get complicated if one doesn't pay attention.  We want to write the rotation operator as represented in the basis of the eigenstates of L2 and Lz.  As such, we don't care whether the original angular momentum is "orbital angular momentum" or not.  But in the case of orbital angular momentum, we can use a trick to sneak in the 3-dim coordinates (which normally don't have anything to do with the rotation operator in Hilbert space).  We can consider the rotation of the ket representing the z-axis in 3-dim Euclidean space onto the ket representing just any direction n.  With Euler angles, this can always be achieved as given in (3.6.47).  Inserting a complete set, and projecting onto an eigenket of L2 and Lz, we find a relationship between two spherical harmonics (the l,m projections of the kets corresponding to the z and the n direction respectively), and elements of the rotation operator in the l,m basis.  Given the fact that the l,m representation of a rotation operator has nothing to do with the origin of the angular momentum (whether this is coming from orbital or other angular momentum), we find the universal relationship in (3.6.52).
The calculation is not difficult, the subtlety resides in keeping distinct what is valid in which case...

7. Addition of Angular Momenta

Simple Examples of Angular Momentum Addition

An implicitly introduced but very important concept here is the direct product of Hilbert spaces.  For finite dimensional complex vector spaces, its construction is simple: take a basis in each space, construct (set) product of the two bases, and build a new complex vector space with as a basis this product.  So if the first one is n-dimensional, and the second is m-dimensional, the direct product space is n.m dimensional.  The basis vectors of this new space are then noted as the "direct product" of the basis vector of the first and the basis vector of the second space that gave rise to it.  This "direct product" notation is bilinear, in that you can write in general a direct product of a vector of the first space, and a vector of the second space.  However, most elements in the product space cannot be written in that way (but only as a superposition of vectors that can be written that way).  One shouldn't confuse the direct product with the direct sum of two complex vector spaces.  The direct sum can be handled in a similar matter, except that the basis of the new space is now the union (and not the set product) of the two bases, so the dimension of the direct sum space is n + m.
For infinite-dimensional spaces (although this is not rigorous), the product space is given by the direct product of individual kets from each of the spaces, assuming bilinearity, and we also consider all possible linear superpositions of these products to be an element of the product space.
When we consider the "whole" of two quantum systems, each of them having their Hilbert space, then the Hilbert space of the "whole" is the direct product of those hilbert spaces.  If you think about it, this is the logical way to consider "taking the respective degrees of freedom together".
All this to explain what's written down in equation (3.7.1): the hilbert space of  a featureless point particle in 3-dim Euclidean space (spanned by the position eigenkets) is multiplied with the 2-dim hilbert space of the spin state of a spin-1/2 particle, to give us the hilbert space of a spin-1/2 particle in 3-dim euclidean space.  Product states are then a general ket of the hilbert space of featureless point particles and a spin-1/2 ket.  But the most general ket in this space is not to be written as such a product state, but as a linear superposition of such states.  So the most general state of a spin-1/2 particle is NOT a wave function in space and a spin-1/2 state !  That's a very peculiar case.
Operators acting in one of the Hilbert spaces being a factor in the direct product have their natural extension in the following way: imagine we have an operator that acts on the first space.  In that case, we define its action on a product state as the product state which is the product of the action of the original operator on the first part, and simply a copy of the second part.  Now, a general ket of the product space cannot be written as a product state !  So we have to expand this general state into a superposition of product states, and apply our natural extension of the operator to each of the terms individually, making the superposition afterwards (which is no problem because we're still dealing with a linear operator).  This is what one tries to explain in equation (3.7.3).
The other example worked out in this part is the composition of two spin-1/2 systems.  This is an interesting example, because it is the easiest example of the addition of angular momentum, which will be worked out in all generality in the next section.

Formal Theory of Angular Momentum Addition

One really needs to sit down, take pencil and paper and follow the calculations, at least once in ones life, in order to understand this.  So we limit ourselves here to some very general overview.  The product of the Hilbert space of a spin-j1 system and the Hilbert space of a spin-j2 system can have two natural bases.  We can work with bases, because the Hilbert spaces are finite dimensional.  One way of taking a basis is by the product states of the eigenstates of the angular momentum in each of the two component spaces ; but the only thing that matters there is the eigenvalues of J1z and J2z, because J1^2 and J2^2 are fixed (j1 and j2).  So we have a basis which is determined by m1 and m2, as written in equations (3.7.27x).  The other possible basis is the eigenvalues of J^2 and Jz, which are based on the sum angular momentum (the sum of the natural extensions of the angular momenta in the two component spaces).  Here, J^2 is NOT fixed, so this basis is determined by j and m, as written down in equations (3.7.30x).
We are looking for the unitary transformation from one basis into the other.  All the elements of this unitary transformation can be determined (up to a few arbitrary sign conventions) by using the identies of the ladder operators of the sum being the sum of the ladder operators, which gives recursion relations between the different basis kets.  The elements of these unitary transformation matrix are called the Clebsch-Gordan coefficients.

Recursion Relations for the Clebsch-Gordan Coefficients

The fundamental (but trivial) relation (3.7.45), together with the relations giving us the effect of a ladder operator on an angular momentum base ket, are all we need to establish the Clebsch-Gordan coefficients.  They are written down explicitly in equation (3.7.49).  The nitty-gritty details should be worked out with pencil and paper...

Clebsch-Gordan Coefficients and Rotation Matrices

Here, a new notion is introduced: the direct product of two operators.  It is a natural extension of  the definition of the operator in a factor space to the whole product space: the direct product of two operators, the first one acting on the first factor space, and the second acting on the second factor space, is the operator acting on product states such that the first part is transformed under the first operator and the second part is transformed under the second operator.  For a general ket in the product space, which is not a product state, but a superposition of product states, we have to apply the rule to the composing product states in the superposition.
A direct sum of operators, acting in orthogonal subspaces of the final space, is something else: a general vector of the final space can be written in a unique way as a superposition of vectors in the different subspaces.  The direct sum operator (acting on the final space) is then nothing else but the superposition of the actions of the original operators, each on 'their' component in 'their' subspace.
It should now be clear that the rotation operator in a direct product space is the direct product of the rotation operators in the individual spaces (think about it).  On the other hand, we now also know that this direct product operator is reducible (as a representation of the rotation group), and hence can be put in block-diagonal form using a unitary transformation, which is nothing else but what is written down in equation (3.7.68). This unitary transformation is of course again nothing else but the transformation containing the Clebsch-Gordan coefficients !  The similarity transformation is written out componentwise in equation (3.7.69), and is called the Clebsch-Gordan series.
This is then applied for the case where we can write elements of the rotation matrix as spherical harmonics.

8. Schwinger's oscillator model of Angular Momentum

Angular Momentum and Uncoupled Oscillators

A very remarkable model is worked out here: two uncoupled harmonic oscillators (+ and -, for a name) are shown to span a Hilbert space that can also be seen as the direct sum of one copy of each spin l (integer and half integer).  The link is given in equations (3.8.8) concerning the main operators, and (3.8.14) concerning the basis kets.  Equation (3.8.18) then gives the normalized basis kets of the angular momentum system as a function of the "vacuum" ket.

Explicit Formula for Rotation Matrices

The only non-trivial rotation operator, when considering Euler angles, is given by the rotation around the y-axis, so if we can find a general expression for that operator in the basis of angular momentum, we have an explicit expression for just any rotation operator in that basis (and solved the problem of the representation of the rotation group in arbitrary dimensions).  One starts by applying the rotation operator on both sides of equation (3.8.18) and inserting the identity (rotation and its inverse) in between each operator product.
Applying the Bakers-Haussdorff lemma to the expressions in between brackets, we have the result in (3.8.25) and (3.8.26), and using the binomial theorem to write out the powers of these expressions needed, one arrives at the expression (3.8.29).  Although quite involved, it gives us an explicit expression for the rotated ket j,m as a function of the creation and annihilation operators in Schwinger's model applied to the vacuum.  The last thing to do is to write these creation and annihilation operators applied to the vacuum as basis kets j,m.  That's done and the result is Wigner's formula (3.8.33), which gives the explicit expression for the rotation matrix for arbitrary spin (rotation around y).

It is probably not clear immediately what is the power of Schwinger's model here.  In fact, using the creation and annihilation of different harmonic oscillators (here we used two of them) is such a generic scheme, that it can be used to model almost any quantum system ; in fact it may almost be taken as the definition of quantisation, an oscillator being nothing else but a "counter of quanta".  We can almost forget about the underlying "oscillators".  In quantum field theory, this method is used in general to describe "particles" which turn out to be nothing else but quanta of harmonic oscillators.  Each particle type, with each possible momentum, is associated with an "oscillator" and the quanta are the number of particles of that type and with that momentum.  In that case, the number of oscillators considered is infinite, while in this example, there were only two, so this is a great way to see the mechanism at work in a simple case without being bothered by all the other difficulties (infinities everywhere) of quantum field theory at once.

9. Spin correlation measurements and Bell's Inequality

This part is really worth the effort because it settles an extremely important issue, fixing for ever the weirdness of quantum mechanics.  Long thought to be in the realm of metaphysical considerations, it is about the opinion Einstein (and many others) had about quantum mechanics: is the probabilistic aspect of quantum mechanics due to a kind of underlying "statistical mechanics", with parameters we don't know about, which are distributed according to certain laws, and which give the impression of randomness in the outcomes of quantum mechanical measurements, or is this probabilistic nature intrinsic, with no underlying mechanism ?  It turns out that this question can partly be turned into a scientific question, much to the surprise of people, in the following way: Bell showed that if there is an underlying statistical mechanics, that has to obey certain conditions which seem reasonable, then this imposes conditions on the probabilities of experiments which are not always satisfied by quantum mechanics.  So it is sufficient to place oneself in these conditions (where the predictions of quantum mechanics cannot follow from an underlying statistical mechanics), do the experiment and we know that the outcome then has to decide as to whether quantum mechanics or an underlying mechanism is correct.
We now have already all the tools available to do this, so we will work out such a case explicitly in this section.

Correlations in Spin-Singlet States

The systems which have been studied and can show the above remarkable property of quantum mechanics (namely, that no underlying statistical mechanics can result in the same measurement results) are correlated spin systems, meaning, two particles (of which the motion in space can be considered classical) with correlated spins.  A simple way to prepare these states is by using two spin-1/2 particles that result from a total spin-0 state.  We then know that the individual spins have to make up a singlet state, as written down in (3.9.1).  The particles travel in opposite directions, and after they are a distance apart (and there is no reasonable way to assume they can still interact), one can measure a spin component on each side.  The correlations of the outcomes of measurement will turn out to be the important quantity that will defy "statistical mechanics" explanations within certain conditions.   What is of course typically "quantum" is that we can have correlations between two measurements, for example, the measurement of the x-component of the spin of the first particle and the measurement of the z-component of the spin of the second particle ; it doesn't make sense to speak of correlations between the measurement of the x-component of the first particle and the z-component of the first particle, for example, because these are incompatible measurements.  In order to be able to talk about correlations of measurements (on the same system) we have to be able to measure them simultaneously. 

Einstein's Locality Principle and Bell's Inequality

The way Einstein and his followers saw the probabilistic nature of quantum mechanics, was that what people called a pure state in quantum mechanics (a well-defined state in Hilbert space) corresponded in fact to an ensemble of different systems, and the observables we measure are then drawn from functions over this ensemble.  Einstein's view was that in fact, when the two particles get separated, they constitute by themselves independent entities, and all measurements on them are independent.  The correlation between measurements comes from an initial correlation of the "hidden variables" in each of the independent entities.  So this comes down to saying that every possible measurement on each of the particles is "in advance" determined by a set of hidden variables each of the particles carry with them.  What exactly are these hidden variables is not specified and can be the object of different theories.  The only thing that matters here is that we have an initial ensemble of "couples of hidden variables", each particle taking one of them, in such a way that that hidden variable specifies all possible outcomes of all measurements on that particle, and that the case of spin-0 is respected.  It is then worked out in the case we consider only 3 possible directions of measurement of the spin, which means that each particle has to carry with it in advance the outcome to three binary (up or down) measurements in its hidden variables.  This means that there are 8 categories of particles to be considered ; because of the compatibility with spin-0, if the hidden variables say that particle 1 has spin up in the x direction, then the hidden variables of particle 2 have to say that it has spin down in the x direction.  The ensemble that corresponds to our superposed quantum state (remember that in hidden variable theories, to one pure quantum state corresponds an ensemble of hidden variables) is then specified completely if we know the probabilities of the 8 different cases.  Equation (3.9.6) is then evident, and from it follows inequality (3.9.9).
Let us retrace what we've done: we're NOT working with a quantum theory.  We're making the assumption that each particle has 'hidden variables' that determine in advance what will be the outcome to 3 different spin direction measurements.  We know that we can only perform one of them in quantum theory, but that doesn't matter.  Then there exists an innocent-sounding inequality concerning the combined probabilities of two outcomes (which, on one single particle, cannot be measured).  The trick is of course that these combined probabilities of two outcomes CAN be determined if we look at the two particles.  Indeed, for example, if particle 1 is measured along the x-axis, and particle 2 is measured along the y-axis, then the result of the second measurement, inverted, is what the first particle would have given if we had measured it along the y-axis.  So the two correlated particles allow us in fact to measure spin along two different directions at the same time.

Quantum Mechanics and Bell's Inequality

If we now come back to quantum mechanics, then it is not difficult to calculate these probabilities if we choose 3 axes in the plane along which we will perform the measurements, and the result is given in equation (3.9.11) for such a probability.  Filling it in in Bell's inquality, we see that it is not satisfied for certain angles.
This already indicates that quantum mechanics cannot find its probabilistic origin in an underlying statistical mechanics using hidden variables as explained above.  But then, quantum mechanics can be wrong. Experiment has to decide whether quantum mechanics is right or not.  Several experiments indicate that quantum mechanics is right, hence excluding the kind of "statistical hidden variable mechanics" in the way it has been used to deduce Bell's inequalities.

10. Tensor Operators

Vector Operator

In classical physics, a vector is somehow a quantity that has "magnitude and direction".  Analytically, vectors are represented by 3 coordinates in Euclidean space, but that doesn't mean that 3 numbers form a vector.  The essential property of a vector is that it transforms under a rotation just as one expects: the same magnitude, and the direction is "rotated" as required.  Mathematically, this means that the 3 coordinates of a vector before and after rotation have to undergo a transformation, which is nothing else but the matrix multiplication with the 3-dimensional rotation matrix that corresponds to the rotation under consideration.
In quantum mechanics, things are the same.  Only, now a vector is not something that is described with 3 numbers, giving "magnitude and direction", it consists of 3 operators on hilbert space.  But not just any three operators form a vector.  In order for them to form a vector, their transformation properties must be such, that the expectation value of the 3 operators under a rotated state must transform as a classical 3-dim vector from the expectation value of the 3 operators under the unrotated state.  The funny thing here is that the 3 operators stay the same, and it are the state kets that are rotated.  Working the condition out for arbitrary states, one arrives at equation (3.10.3) for a vector operator.  Using the infinitesimal representation of the rotation operator as a function of angular momentum, this condition is equivalent to (3.10.8).

Cartesian Tensors versus Irreducible Tensors

In classical physics, cartesian tensors are nothing else but "multiple index vectors", and nothing stops us from defining the equivalent in quantum mechanics.    However, looking at it from another point of view, we can see that a cartesian tensor of rank r consists then of 3^r operators, and then these operators should form a 3^r dimensional representation of the rotation group if we apply the generalisation of equation (3.10.3).  But we already know all irreducible representations of the rotation group !  It turns out that cartesian tensors (their components) can be written as linear combinations of things that correspond to irreducible representations. This is actually no surprise: all representations of a group are irreducible representations, or can be written as a linear combination of irreducible representations.  This is illustrated in equations (3.10.12) and (3.10.13), for a specific case of a rank 2 cartesian tensor.  We can write it as the sum of a scalar (spin 0), a vector (spin 1) and a thing that will turn out to be a spin-2 representation with 5 components.   So we see that what the group representation properties is concerned, these cartesian tensors are quite messy objects.  We therefore define spherical tensors to be things that are irreducible (fixed l) representations of the rotation group.
First some illustrations are used (such as putting a vector as argument of the set of spherical harmonics of given l) to suggest the definition of a spherical tensor, which is a set of operators. That definition is formulated in equation (3.10.22).  So, in a similar way as for a cartesian tensor, a spherical tensor is defined by its properties under rotation ; it turns out that the components of a spherical tensor under rotation transform in a very similar way as do transform the eigenstates (j,m) (look at equation 3.5.49).  In fact, they transform in the opposite way (inverse rotation).  The infinitesimal condition for a spherical tensor is then given in equation (3.10.25).

Product of Tensors

The rather complicated-sounding theorem in fact says that we can make "products of spherical tensors" by using a similar linear superposition as those that  occur when we combine two sets of spin kets into the "sum of spins".

Matrix Elements of Tensor Operators: The Wigner-Eckart Theorem

Finally, a famous theorem: The Wigner-Eckart theorem.  It states that the matrix representation of a spherical tensor in a set of basis kets of angular momentum takes on a specific form: each block between a set of j' and j (and alpha' and alpha, standing for other operators defining a complete set of commuting operators together with angular momentum) consists of elements that are the "sum of spin" elements as if we added together the spin j with the spin of the tensor to obtain the spin j', and a scale factor which depends on the particularity of the tensor, but which is the same in the entire block.  This theorem is not very surprising, but it is practical when having to calculate matrix representations of tensor operators.

Back to main course page