One way to motivate the quantum theory of a free particle is that, whatever
it is, it should have analogous behavior to that of the classical case under trans-
lations in space and time. In chapter 14 we will see that in the Hamiltonian form
of classical mechanics, the components of the momentum vector give a basis of
the Lie algebra of the spatial translation groupR^3 , the energy a basis of the
Lie algebra of the time translation groupR. Invoking the classical relationship
between energy and momentum
E=
|p|^2
2 m
used in non-relativistic mechanics relates the Hamiltonian and momentum op-
erators by
H=
|P|^2
2 m
On wavefunctions, for this choice ofH the abstract Schr ̈odinger equation 1.1
becomes the partial differential equation
i~
∂
∂t
ψ(q,t) =
−~^2
2 m
∇^2 ψ(q,t)
for the wavefunction of a free particle.
10.1 The groupRand its representations
Some of the most fundamental symmetries of nature are translational symme-
tries, and the basic example of these involves the Lie groupR, with the group
law given by addition. Note thatRcan be treated as a matrix group with a
multiplicative group law by identifying it with the group of matrices of the form
(
1 a
0 1
)
fora∈R. Since (
1 a
0 1
)(
1 b
0 1
)
=
(
1 a+b
0 1
)
multiplication of matrices corresponds to addition of elements ofR. Using the
matrix exponential one finds that
e
^0 a
0 0
=
(
1 a
0 1
)
so the Lie algebra of the matrix groupRis matrices of the form
(
0 a
0 0