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
(to the
(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
?
Let’s calculate:
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.)
•
•
(as expected).
(as expected).
•
• 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
satisfies the
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
:
Expanding
and substituting
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
are
.
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
Hence,
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.