284
9
Quantum Mechanics Formalism
that is, the timeevolution operator
U
has to be
unitary
. Moreover if
U
(
t
,
t
0
)
maps

a
,
t
0
into

a
,
t
,
U
(
t
,
t
0
)
−
1
=
U
(
t
,
t
0
)
†
maps

a
,
t
into

a
,
t
0
,
so that
U
(
t
,
t
0
)
†
=
U
(
t
0
,
t
)
. We finally require
U
to satisfy the condition
U
(
t
0
,
t
0
)
=
ˆ
I
.
In order to determine the timeevolution of

a
,
t
we compute the change of

a
,
t
under an infinitesimal change in the parameter
t
.
We have
d

a
,
t
dt
t
=
t
0
=
lim
t
→
t
0

a
,
t
− 
a
,
t
0
t
−
t
0
=
lim
t
→
t
0
U
−
ˆ
I
t
−
t
0

a
,
t
0
.
(9.74)
Let us denote the limit of the operator inside the curly brackets by
i
ˆ
H
;
we can the
write, at a generic time
t
,
the differential equation
ˆ
H

a
,
t
=
i
d
dt

a
,
t
.
(9.75)
The operator
ˆ
H
is the infinitesimal generator of timeevolution and, in analogy with
classical mechanics, is identified with the
quantum Hamiltonian
. If we substitute
(
9.73
) in (
9.74
) we obtain an equation for the evolution operator:
i
dU
(
t
,
t
0
)
dt
=
ˆ
HU
(
t
,
t
0
)
⇔
i
dU
(
t
,
t
0
)
†
dt
= −
U
(
t
,
t
0
)
†
ˆ
H
,
(9.76)
where we have used the hermiticity property of
ˆ
H
:
ˆ
H
†
=
ˆ
H
. If the Hamiltonian is
timeindependent, as it is the case for a free particle, we can easily write the formal
solution to the above equation with the initial condition
U
(
t
0
,
t
0
)
=
ˆ
I
:
U
(
t
,
t
0
)
=
U
(
t
−
t
0
)
=
e
−
i
ˆ
H
(
t
−
t
0
)
.
(9.77)
The equation for the wave function
ψ(
x
,
t
)
=
x

a
,
t
,
is obtained by scalar multiplication of both sides of (
9.75
) by the bra
x

. Taking into
account (
9.19
) we obtain
ˆ
H
ψ(
x
,
t
)
=
i
∂
∂
t
ψ(
x
,
t
).
(9.78)
that is the
Schrödinger equation
, where
ˆ
H
is now the Hamiltonian operator realized
as a differential operator on wave functions. For a free particle
ˆ
H
=
ˆ
p

2
2
m
= −
2
2
m
∇
2
,
and (
9.78
) reads: