PHY431 Lecture 5
1
3/5/2000
8.
Symmetries and Conservation Laws
Symmetries and the conservation laws that follow from them are important tools to constrain the pos-
sible forms that the Hamiltonian (or Lagrangian) of a system can take. For example, the requirement of
Lorentz invariance of the Lagrangian describing a system narrows down the possible options for the
Lagrangian to a limited number of scalar combinations of Lorentz (pseudo-) scalars, (pseudo-) vectors
and tensors. Thus, the empirical observation of conservation laws means that the correlated symmetry
law must be obeyed by the Hamiltonian (or Lagrangian) describing the system, and a lot can be learned
about the possible forms the Hamiltonian is allowed to have. From the derived symmetries, one can
then extrapolate and try to find the underlying guiding principles of the theory describing the system.
In the following we will use the non-relativistic Schrödinger equation for illustrating the procedures for
finding conserved quantities and their associated symmetries (invariances under transformations of the
wave functions in the Schrödinger equation). However, the very same reasoning applies in the fully
relativistic description, which typically uses the Lagrangian formalism, and is decidedly more cumber-
some.
In the Schrödinger picture, the observables are time-independent operators, while the wave functions
depend in general on the time parameter (the reverse is true for the Heisenberg picture).
8.1. Conserved Quantities
The Schrödinger equation with Hamiltonian
H
and wave function
ψ
(
r
,
t
) is:
(,)
it
t
t
ψψ
∂
=
∂
rH
r
(8.1)
Observables
are
Hermitian
operators that act on the wave functions describing the system. Because
an observable
F
(time-independent in the Schrödinger formalism) leads to a measurement value, the
expectation value
⟨
F
⟩
, this expectation value must be real:
()
*
*
*
*
*
*†
†
(
)
,
thus:
(Hermitian!)
F
dV
F
dV
dV
dV
ψ ψ
≡=
=
=
=
=
∫∫
∫
∫
FF
F
F
F
F
(8.2)
In the general case the expectation value
⟨
F
⟩
changes with time. Consider now the condition under
which this expectation value
⟨
F
⟩
is constant, i.e. a conserved quantity:
*
**
dd
d
d
Fd
V
d
V
d
V
dt
dt
dt
dt
==+
∫
F
(8.3)
Using the Schrödinger equation and its complex conjugate equation:
*
*
*
and
ii
tt
∂∂
=−
=
=
HH
H
(8.4)
because
H
, being an observable, is Hermitian. Filling in the derivatives of the wave function:
() ()
(
) []
*
*
,,
d
F
dV i
dV
i
i dV
i dV
dt
=+
−
=
−
=
∫
∫
HF
F H
HF FH
(8.5)
with [
H
,
F
]
≡
HF-FH
the
commutator
of
H
and
F
. Thus the expectation value of
F
is
conserved
if and
only if the commutator [
H
,
F
]=0. Because an operator commutes with itself, the Kinetic energy opera-
tor and the “speed operator” commute: I can measure the kinetic energy and the speed simultaneously.
A particle in a box, which has only elastic collisions with the wall, will conserve energy, and the speed
is a conserved quantity. This is not so for the position measurement: the measurement of “
x
” is not