Chapter 4: Symmetry in Quantum Mechanics


Summary of this chapter in "Modern Quantum Mechanics" (J.J. Sakurai), by Patrick Van Esch
Last revision November, 22, 2003.

There are some brilliant parts in this chapter (for example, the part on time inversion), and there are some extremely poor explanations (the lattice symmetry).

1. Symmetries, conservation laws and degeneracies.

Symmetries in Classical Physics

Symmetry in classical physics is here displayed in the case we have managed to have one of the configuration space coordinates to be the symmetry, as given in (4.1.1).  Although this may seem to be a very special case, it isn't, because any continuous symmetry can be formatted in that way using canonical transformations (at least, in principle !).  This is probably the most simplistic derivation of Noether's theorem that exists!  In essence, it states that to any single-parameter symmetry of the dynamics, corresponds a conserved quantity during the time evolution.

Symmetry in Quantum Mechanics

A symmetry of a quantum system, or a candidate symmetry, depending on a single parameter, corresponds to  a unitary transformation of Hilbert space.  (well, we'll see that there  is also another possibility called an anti-unitary transformation).  It is pointed out that to any continuous parametrisation of a unitary transformation (we're considering a one-parameter family of unitary transformations), we can associate a Hermitian generator of the family, which can be extracted by looking at an infinitesimal value of the parameter.  Although we can play that game with just any one-parameter family of unitary transformations, the name of symmetry is deserved in the case the generator commutes with the Hamiltonian.  In that case, the generator corresponds to an observable which is a constant of motion.  Note that the infinitesimal transformation determined by the generator determines the entire parametrized set of transformations by exponentiation.

Degeneracies

A very basic observation is made: if a system has a certain symmetry, then this doesn't mean that every "solution", for example, energy eigenstate, of that system, possesses that symmetry: only the set of solutions has to posses the symmetry.  In the case of energy eigenkets, if there is more than one (up to a factor) eigenket for a given value, we call that energy value degenerate, and we have a linear subspace of kets that are eigenkets of this value.  This means that the symmetry operator applied to any ket of this set has to be (another or the same) ket of this set.  The example of rotational symmetry and the set of spin-l eigenkets is given.

2. Discrete symmetries, parity or space inversion

Apart from symmetries related to a continuous parameter (such as rotations, translations etc..), there can also be a set of discrete symmetries, which belong to a finite, or a countable set of operations.  In this section, we study the parity or space inversion operation.    We start to get used to the way of implementing a symmetry in quantum mechanics: to the abstract idea of symmetry corresponds a unitary operator acting in Hilbert space, and we have to find it.
As was the case with other symmetry operators, like translation and rotation, we have to start from the interpretation of certain observables, and define what we mean by the symmetry operation we want to implement, and what it should do on expectation values.  That gives us then mathematical contraints on the form the unitary operator corresponding to the symmetry.  If we have enough constraints, it is uniquely specified, but usually we need to put in some extra ad hoc "phase conventions".  This can be simple or tricky !
In the case of a Hilbert space where a position operator for a particle is defined (so our quantum system is a spinless single particle in 3-dim Euclidean space) we say that the action of parity changes the sign of the expectation value of the position operator for any state.  From this can be deduced equation (4.2.4).  But that doesn't fix the operator pi completely.  Indeed, the best we can do is equation (4.2.5), but that still leaves an arbitrary phase factor to choose.  There is then an extra convention, that this phase factor equals one.  From that follows that pi-squared equals one, and that the eigenvalues of pi can only be 1 or -1.  Note how tricky this is!  A basic property, namely (4.2.7), follows from what seems to be a convention. The reason is of course that we take it to be fundamental for parity to be such that twice applying a parity operation brings us back to the original state.  This is a stronger statement than rotation over 360 degrees for spin-1/2 particles, where we were back to the original state with a minus sign, but that's because for spin-1/2 particles, we were mathematically forced to do so (representation of SU(2) is simply that way).
From the known properties of parity working on position, is deduced how parity behaves with momentum.  The defining property of momentum, namely that it is the generator for space translations, is used to find this out. 
The effect of parity on angular momentum is worked out for orbital angular momentum from the properties of position and momentum, but that has nothing fundamental.  For angular momentum in general, it is postulated that parity should commute with the rotation operator (and hence with angular momentum).  Note that this is extra input, because we cannot use the properties deduced from what happens to a featureless point particle to uniquely fix what should happen to particles with spin.

Parity is a discrete symmetry that gets its defining action from something that happens in 3-dim Euclidean space, just as rotations (but that was a continuous symmetry).  We can hence consider the combined group of symmetries, with rotations and parity.  It turns out that quantities that behaved in a vectorial or scalar way under rotations, now can have an additional property: namely they can be odd, or they can be even under parity.  This is what makes the difference between scalars, and pseudoscalars, and vectors and pseudovectors.

Wave Functions under Parity

The wave function is nothing else but the representation of kets in the position basis.  Parity eigenkets then have even or odd wavefunctions.  The parity of angular momentum eigenkets is discussed.
A theorem (which will come back under many different forms) is stated for parity.  It is a variant of a more general theorem which states that non-degenerate eigenkets of a hamiltonian that satisfies a certain symmetry (that commutes with a certain symmetry operator) are also eigenkets of the symmetry.  In this specific case, the symmetry is the parity operator.
We already pointed out that if there is degeneracy, then this theorem doesn't hold, and only the set of eigenkets (and not the individual eigenkets) satisfies the symmetry.

Symmetrical double well potential

A very instructive example is qualitatively discussed, and shows how coupling (interaction in the broad sense) of two systems (the individual wells) leads to breaking of degeneracy (spitting of energy levels) on one hand, and gives an idea of perturbative interactions on the other hand.  Let's make this clearer.
If the individual wells are completely separated, then we have of course degeneracy: the particle can be in the left well, or in the right well, and these solutions are equivalent.  If there is interaction (the barrier is finite), then the energy eigenstates are not degenerate (but unlocalized).  If we work in localized states, then these are not stationary, and the oscillation frequency between both localized states is given by Bohr's formula (4.2.39).

Parity Selection Rule

It is quite simple to establish that the matrix element of the position operator between two parity eigenstates of same parity vanishes.  This is the parity extension of the Wigner-Eckart theorem if one wants to.  This property will play a role in radiative transitions, and will suppress certain radiative transitions.  As such, it gives rise to a selection rule (known before the advent of quantum mechanics) called Laporte's rule.

Parity non conservation

Parity is an important operation because it often is a symmetry.  Until 1956, it was thought that the fundamental interactions in nature were symmetric under parity  But then it was established that weak nuclear interactions are not symmetrical under parity.

3. Lattice translation as a discrete symmetry

A symmetry that is extremely important in solid-state physics, is a lattice symmetry.  In true solid-state physics, this has to be a 3-dimensional lattice of course, but we will limit ourselves here to a toy model in 1 dimension.  The hamiltonian is supposed to commute with translations over the lattice distance a. 
Then there comes a confusing discussion.  I think that Sakurai's aim is to deduce Bloch's theorem (for mathematicians: Flouquet's theorem).  The first part is understandable, when the ket labeled with n corresponds to a localized state and there is no "coupling" between adjacent lattice sites due to an infinite potential barrier.  The combination given in equation (4.3.6) and the fact that this is an eigenket of the translation operator over the lattice spacing, is fundamental.  Here, each ket with label n is to be understood as coming from a set of kets that obeys equation (4.3.5).  In this first part, this set is well-defined: it are the eigenkets of the Hamiltonian, given infinite barriers.  But what comes next is totally confused in my opinion: once you lower the barriers, what is now the definition of our set of kets with label n ?  Clearly, they are not eigenkets of the hamiltonian.  Also, they are not completely localized.  So what are they ?
Of course, one can try to understand what Sakurai was at: he was probably thinking of the state of an electron in a free atom at a lattice position, and taking |n) as that solution at lattice point n, and |n+1) as the same solution, but shifted to lattice point n+1.   Taking into account then the potentials from the other atoms, at other lattice positions, as a perturbation, is exactly what the tight binding method is about.  But personally, I'm really sorry he's confusing this specific technique in solid state physics with the more general property derivable from the lattice symmetry. 

If we want to progress, I think we have to see this tight binding approximation as an example.  We take, in equation 4.3.12, a general state which obeys the lattice symmetry.  He then derives, from that, Bloch's theorem, stated in 4.3.18.  We also know that, because of the commutation between the hamiltonian and the lattice translation operator, we can find a set of simultaneous eigenkets.  So we look for the energy eigenkets which take on the form specified in Bloch's theorem.  Now Bloch's theorem is interesting, because in it appears a "modulated" plane wave with wave number k, and this k will have a lot of properties that really make us think of an actual wave number and associated momentum, but that's not worked out here.

Going back to our example of tight binding, we can establish a kind of "dispersion relation" where we have the energy eigenvalue as a function of k.  It turns out that not all k-values show up: the values that do show up are called the Brillouin zone (and not, as Sakurai says, the energy level!).  Well, in fact, this is much more complicated in true 3-dim solid state physics !  The first Brillouin zone is the Wigner-Seitz primitive cell of the reciprocal lattice, defined as the geometrical place of points that can be reached from a reciprocal lattice point without crossing a single bragg plane.  The second brillouin zone is similar, except that you have to cross exactly one Bragg plane etc...  Reminder: the reciprocal space is the space of k-vectors (up to factors of 2 pi...).

Anyway, I think this section in Sakurai is a bit confusing, but the main result is that a lattice symmetry gives rise to eigenstates that take on the form specified by Bloch's theorem.

4. The time-reversal discrete symmetry

Contrary to the previous section, this one is in my opinion brilliant !
Time reversal is a very subtle symmetry, and plays in fact a much bigger role in quantum field theory than in NRQM, but it is, as a preparation, a very good idea to treat it here, and can be very illuminating for the rest.
As Sakurai explains, one should better use the term "reversal of motion".  The importance of this operation is explained in classical physics first.  For a potential depending on position, Newton's equation of motion satisfies the symmetry of reversal of motion.  Next, this symmetry is also explained to be valid in classical electromagnetism, although at first sight the magnetic part doesn't seem to obey this symmetry.  If the reversal of the currents generating the field is taken into account, however, it is clear that classical electrodynamics is symmetrical under reversal of motion.
Then, an important point is raised: when looking at the Schrodinger equation, it is seen that the time-reversed solution is not a solution, but its complex conjugate is.  So reversal of motion and complex conjugation seem to have something to do with eachother in quantum mechanics.  That's why we need the following discussion.

Digression on Symmetry Operations

The problem is tackled what form needs to take on the operator in Hilbert space that represents a symmetry.  Before delving into what can be a symmetry of a specific system, we should ask ourselves what are the formal isomorphisms within the Hilbert space that conserve the formalism of quantum mechanics.  By this, I mean: we have attached physical meanings to all the states in a Hilbert space: to the ket |x) corresponds the meaning Mx.  Now, the question is, what kind of transformation T on Hilbert space can be performed, such that we could associate this time T|x) with meaning Mx, and keep the same formalism.  A simple example is the multiplication of all kets with a phase factor exp(i a).  It is clear that we can do that, and no physics changes.  If you think about it, the only measurable thing that comes out of a quantum mechanical theory are probabilities, which are always taken to be absolute values of brackets.  So a transformation T that conserves the absolute value of brackets is ok.  We know that unitary transformations conserve values of the brackets, so these are already ok.  But that's too severe.  It turns out that also anti-unitary operators (which are not linear operators) conserve the absolute value of brackets.
So transformations T which are unitary or anti unitary possess the property that quantum mechanics as a physical theory gives the same predictions whether we associate |x) or T|x) to the meaning Mx.  But that doesn't mean that any such T is a symmetry of the theory (= the dynamics).  In order for an operator S to be a symmetry, we do the following.  Interpretation Mx is associated to ket |x), and interpretation Sx, corresponding to the symmetry applied to situation Mx, is associated to the ket S|x).  If S is a true symmetry of the dynamics, then the evolution of S|x) after a time t comes down to S of the situation which corresponds to the evolution of |x) after time t. 
Maybe a small discussion can be added here.  The candidate symmetries S here are NOT symmetries of the physical situation, in the sense that the physical situation Mx and the physical situation Sx are physically distinct situations (which, if S turns out to be a symmetry, have similar behaviour under the dynamics).  One shouldn't confuse this with something like gauge symmetry.  Here, in the formalism, we talk about different kets that correspond to exactly the same physical situation.  In fact, this is an abuse of the quantum formalism, because to each physical situation should only correspond one ray.

Next, it is worked out what is an antiunitary operator.  It is a bit subtle: in a given basis, an antiunitary operator can be represented by a unitary matrix and the operation "complex conjugation".  However, this splitting up is not geometrical, in that if we choose another basis, then the unitary matrix will not be the one of the unitary operator corresponding to the first unitary matrix after the coordinate transformation.  So there is no unitary operator associated to an antiunitary operator: depending on the choice of basis, it is different.  This makes it a headache to define what it means for an anti-unitary operator to act on a bra.  So we will always consider that it can only act on kets.

Time-Reversal Operator

It is first argued that in order to define an operator that implements motion reversal (usually called the time reversal operator), then this operator cannot be a linear operator.  So it has to be an antiunitary operator.  If time reversal is a symmetry of the dynamics, then equation (4.4.31) has to hold.  The important property (4.4.36) is worked out.  This allows us to relate an observable and its "time reversed" version.  Interesting operators have a certain behaviour under this transformation: they are even or odd.  For example, the momentum operator is expected to be odd under time reversal.  Position is even, and angular momentum is odd.

Wave function

If we take the phase convention (4.4.47), then it is worked out that the wave function of the time-reversed state is the complex conjugate of the original wave function.  This is in the case of a spinless particle.  As with parity, we have the theorem that the non-degenerate eigenstate of the hamiltonian should satisfy the symmetry ; in this case, this comes down to saying that its wave function is real.

Time reversal for a Spin-1/2 System

The aim is to find an explicit form of the time reversal operator in the 2-dim Hilbert space of a spin-1/2 particle.  Angular momentum should flip sign under time reversal.  So applying time reversal to the general spin-up ket in direction n will have to give the spin down ket in that direction, up to a phase factor.  If we now write the time reversal operator as a unitary matrix and a complex conjugation (we have to choose a basis for that to have a meaning, and we take the up and down ket for the z-direction as the basis) we find an expression (4.4.65), again, under the assumption that we are going to write it out in matrix notation in the basis of the spin along the z-axis.  The "symmetry breaking" appearing of S_y is only the result of the fact that we have chosen this basis.  As said before, the splitting-up of an antiunitary operator in a complex conjugation followed by a unitary matrix is not a geometrically well defined action.
A very remarkable property is derived: the square of the time reversal operator is -1 for a spin-1/2 system.  It is easily verified that it is +1 for a spinless system.  This is a similar weirdness as was the case when we rotated a spin-1/2 system over 360 degrees and also found -1.   In fact, both weirdnesses are related !
It is worked out what the time reversal operator does on spin-j systems.  In fact, the reasoning applied to spin-1/2 systems didn't really use the fact that it was spin-1/2 up to the last equal sign in 4.4.65, so we have equation 4.4.73 in all generality, and in equation 4.4.75 we note the link between the -1 in rotation over 360 degrees for half integer spins, and the -1 sign in the square of the time reversal operator.  All this is independent of the phase convention (eta) one takes, because that phase is (due to the antilinearity) taken in absolute square. 
Some remarks on what is a good choice for eta closes this section, together with some properties of tensor operators that have a definite parity under time reversal.

Interactions with Electric and Magnetic Fields: Kramers Degeneracy

Because of the fact that the time reversal operator is anti-unitary, the fact that the Hamiltonian is invariant under time reversal doesn't mean that there is a conserved quantity, as is usually the case with such symmetries.
However, the property remains that the set of eigenkets belonging to an energy eigenvalue has to be symmetric under time reversal.  It is then worked out that in the case of half-integer spin, the fact that the time reversal operator squared equals -1 leads to the conclusion that an energy eigenket, and its time reversal, cannot be related by a simple phase factor (and so have to span a 2-dim space).  So half-integer spin systems with a time reversal symmetry have to have degenerate energy eigenvalues.  That's Kramer's degeneracy.

Chapter overview

In this chapter, discrete changes which can turn out to be (or not) symmetries of the dynamics, are discussed.  The relationship between the physical meaning of the change, and  the form it takes as an operator on Hilbert space, is worked out.  The main results concern parity (space reflection) and time reversal.  In order to introduce time reversal, we needed to discuss anti-unitary (and anti-linear) operators.

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