Symmetries and Conservation Laws (Shankar Chpt. 11)Symmetries are of fundamental importance in our understanding of the universe, and play a key role in virtually every branch of physics. Here we investigate the role of symmetries in the quantum-mechanical domain.
The generator of translations:The translation operator can be understood better by considering infinitesimal translations. For this purpose, we let ( small), and write .
Since , we expect(the factor of is put in for notational convenience.)
(This is why the generator is called the generator! -- once you know how to describe an infinitesimal change (here, translation), you can readily describe finite changes too. This idea is formalized in the theory of Lie groups.)
Some additional remarks: The translation operator is unitary. (For classical mechanics buffs, we note that the unitarity of the translation operator in quantum mechanics is illustrative of a more general principle: canonical transformations in classical mechanics correspond to unitary operators in quantum mechanics.)
For multi-particle systems, the generator of translations is simply the total momentum operator
For reasons that will become apparent shortly, operators like the translation operator are often called symmetry operators.
Translational invariance of a physical systemHaving defined the translation operator, we can now talk about the notion of translational invariance: Suppose we have some isolated physical system. If we assume that space is homogeneous, then if we translate that physical system to some new location in space, any experimental results we obtain at the new location should be identical in all respects to those at the original location. This is what we mean by translational invariance (also called translation symmetry).Every known physical interaction (e.g.,gravitational, weak, electromagnetic, and strong interactions) exhibits translational invariance!
Equivalently, this is the same as demanding that if satisfies the Shroedinger equation
then so should the translated state :
Conservation law associated with translation symmetry:In classical physics, anytime you have a system with a continuous symmetry (e.g., translations by an arbitrary amount, like weve been considering), then Noethers theorem declares that there is some conserved quantity associated with the symmetry. An analogous situation is found in quantum mechanics. Lets investigate
So substituting into Ehrenfests theorem the generator of the symmetry, , and noting that it commutes with the hamiltonian (and that it has no explicit time dependence) yields: This is the law of momentum conservation in quantum mechanics! Summarizing: If you have an isolated physical system, then the homogeneity of space dictates that its behavior will be invariant under translations of the system as a whole. This translation symmetry means that the hamiltonian must be invariant under infinitesimal translations, and so must commute with the generator of the symmetry group (i.e., the momentum operator). This in turn means that the expectation value of momentum doesnt change in time, which is the quantum-mechanical law of energy conservation!(In classical mechanics, it is also true that the homogeneity of space leads to the law of momentum conservation.)That was the first of several symmetries well be discussing. Now on to the next!
B. Time-translation InvarianceOverview: Just as homogeneity of space means that performing the same experiment at different spatial locations should yield equivalent results, homogeneity of time dictates that performing the same experiment at different moments in time should also yield equivalent results.Moreover, just as we saw that invariance under spatial translations led to the law of momentum conservation, here well see that invariance under time-translation leads to the law of energy conservation!
A conservation law associated with time-translation invariance:In the case of time translations, the generator of the group (being the hamiltonian itself!) obviously commutes with the hamiltonian, so Ehnrenfests theorem applied to the generator yieldsSo if the hamiltonian doesnt explicitly depend on time, then the expectation value of the hamiltonian i.e., the average energy doesnt change in time.
This is the law of energy conservation!
Summary: Conservation of energy (in an isolated system) results from the invariance of the Hamiltonian with respect to time translations. This time-translation invariance (which implies the hamiltonian contains no explicit time dependence) is associated with the homogeneity of time.
where the wavefunction also satisfies the Schroedinger eqn.The interpretation of this wavefunction is straightforward: The exponential factor that appears has the form of a free-particle wavefunction traveling to the left. This makes sense because in the primed frame (which moves to the right relative to the unprimed frame), the system appears to move leftward.
From this definition of the parity operator, we can determine how it acts on an arbitrary ket:We can also check how it acts on momentum eigenkets:
A key feature of the parity operator:Lets find the eigenstates of the parity operator (in the x-representation):
We can define even and odd operators as follows:On can check that both the position and momentum operators are odd.
e) Time-reversal symmetryIn classical physics, the fundamental microscopic law governing behavior (Newtons law) is invariant under time reversal t -t. In other words, if x(t) is a solution to
Then so is the time-reversed state, x(-t) (presuming that the forces depend only on position, not time or velocity). Moreover, since v=dx/dt, it follows that under time reversal, velocity, and hence momentum, get reversed: Likewise, angular momentum also gets flipped. Now on to the quantum case
1) since the time-reversal operator involves complex conjugation, it is not a linear operator (unlike all the other operators weve considered thus far) it is anti-linear.2) while most hamiltonians are invariant under time reversal, we note that those associated with the weak interaction are not.Other remarks: