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.
A) Spatial Translations and Translational Invariance In classical physics, a spatial translation of a particle by a distance right) is described by
(The momentum is taken to be unchanged, p p). How do we describe translations in quantum mechanics? Let’s start by considering what we would want a “spatial translation operator” to do to a position eigenket:
i.e., the particle at position x has now been translated (to the right) to position x+a. (Note: had tried
, we would have found that
Let’s see how translation affects an arbitrary state work in the x-representation. Given
, what is
. It’s easiest to ?
In other words,
This is as expected – the wavefunction has simply been translated (rightwards) by distance . Aside: An “active” translation of a state to the right by a distance (as done above) is equivalent to a “passive” translation of the coordinate system to the left by . In this passive view, the state remains the unchanged, but an arbitrary operator is transformed into
The generator of translations: The translation operator can be understood better by considering infinitesimal translations. For this purpose, we let ( “small”), and write .
Since (the factor of
, we expect is put in for notational convenience.)
Now let’s find an explicit expression for
The operator is called the generator. What we have found is that “momentum is the generator of translations” -- i.e., the generator is the operator associated with infinitesimal translations. For those of you who have had advanced classical mechanics, the same finding holds true there. Finite Translations Now let’s determine the operator associated with finite translations. The idea is to repeatedly apply the generator of translations (describing infinitesimal translations) to “build up” a finite translation: Since
translates a system by
this operator N times (where N=
, if we apply
), we will have a translation by
(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 system Having 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! Mathematically, if the physical laws governing a system’s behavior are to be invariant under the translations, then the hamiltonian must satisfy:
i.e., in the passive view, if you shift the coordinate system, the hamiltonian shouldn’t change.
Equivalently, this is the same as demanding that if
Shroedinger equation then so should the translated state
So if a system has translational symmetry, then
Said differently, the translation operator commutes with the hamiltonian
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 we’ve been considering), then Noether’s theorem declares that there is some conserved quantity associated with the symmetry. An analogous situation is found in quantum mechanics. Let’s investigate … Translation invariance in quantum means
Focusing on infinitesimal translations, we have
signifying that the hamiltonian (of a translationally invariant system) must commute with the generator of the symmetry group (in this case, momentum). Now, recall Ehrenfest’s theorem describing the time evolution of expectation values
So substituting into Ehrenfest’s 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 doesn’t 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 we’ll be discussing. Now on to the next!
B. Time-translation Invariance Overview: 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 we’ll see that invariance under time-translation leads to the law of energy conservation! The time-translation operator
The generator of time translations Infinitesimal translation: Since
If we expand both sides of the top equation (for small dt), we get
But replacing the time-derivative on the RHS by yields:
(from Shroed. eqn.)
So the hamiltonian is the generator of time translations!
Finite time translations Building up a finite time translation from a series of infinitesimal time translations (analogous to what we did in the spatial-translation case) yields:
(Note: to get this we had to assume that the hamiltonian contains no explicit time dependence.) [You may recall from earlier in the course that the solution to the Schroedinger equation (assuming no explicit time dependence), given initial state , could be written as perspective.]
. Now we see why from the symmetry
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 Ehnrenfest’s theorem applied to the generator yields
So if the hamiltonian doesn’t explicitly depend on time, then the expectation value of the hamiltonian – i.e., the average energy – doesn’t change in time.
This is the law of energy conservation! Alternatively, we could have argued as follows: if a system is time-translation invariant, then if satisfies the Schroedinger equation
then so must the time-translated state
into the preceding equation yields
But taking the time derivative of the Schroedinger eqn. gives
So substituting the above expression into its predecessor yields
But this is only true if the time derivative (on the LHS) can be passed through the hamiltonian. This is allowed only if the hamiltonian itself is timeindependent.
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.
c) Galilean invariance Nonrelativistic classical physics says that the laws of nature don’t change under a galilean transformation:
What about in the quantum case? The short answer is yes. We won’t study this symmetry in detail, but simply point out the “reasonable” result that if describes the state vector of an isolated physical system as seen by the unprimed observer (where of course satisfies the Schroed. eqn.), then in the reference frame of the moving (primed) observer, the state is described by
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.
d) Parity Classical parity refers to spatial reflection through the origin:
(In a sense, the second relation is redundant, since p=mv=m dx/dt, so once you flip the sign of x, the momentum automatically flips.) Quantum mechanically, the parity operator is defined by its action on position eigenkets:
(Don’t confuse yourself:
From this definition of the parity operator, we can determine how it acts on an arbitrary ket:
Hence, we see that (in the x-representation)
We can also check how it acts on momentum eigenkets:
A key feature of the parity operator:
From this, it’s easy to show that 1) 2) The parity operator is hermitian and unitary
3) The eigenvalues of
Let’s 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.
Parity invariance If the parity operator commutes with the hamiltonian of some physical system (i.e., if the hamiltonian is even), the system is said to be “parity invariant”. We know in this case (from Ehrenfest) that parity is conserved.
Said differently, if , then it is possible to find a common eigenbasis for parity and the hamiltonian. This means that the eigenstates (in the x-representation) of the hamiltonian will be even and odd functions. (As an example, just look back at your notes on the particle in a box!) Most interactions in nature are naturally parity invariant. The weak force is an exception – it is NOT invariant under parity.
e) Time-reversal symmetry In classical physics, the fundamental microscopic law governing behavior (Newton’s 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 …
It’s easiest to work in the x-representation. Let to the Schroedinger equation
denote the solution
In quantum mechanics, what does the time-reversed wavefunction look like? Let’s guess and see if it satisfies the Schroedinger equation. Setting t -t, the Schroedinger equation becomes
is not a solution to Schroedinger’s equation! Bad guess.
However, if we take the complex conjugate of the above equation, we find
So we see that is a solution to Schroedinger’s equation. This represents the appropriate time-reversed state in quantum mechanics!
i.e., if solves the Shroedinger equation, so does the time-reversed wavefunction (the caveat being that the potential must be real and not depend explicitly on either time or velocity).
Stop and verify for yourself: in the p-representation
Other remarks: 1) since the “time-reversal operator” involves complex conjugation, it is not a linear operator (unlike all the other operators we’ve 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.