Quantum Mechanics for Mathematicians

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

)