Elements of Quantum Theory
71
This equation is valid if the system is nonrelativistic. It requires some
modifications if the particle has additional variables such as intrinsic angular
momentum called spin. If the system has a well-defined energy, its Schrödinger
equation reduces to its simpler time-independent form
2
2 ( , )
( ) ( , )
2
−
∇ ψ
+
ψ
t V r
r t
m
r
=
( , )
∂ψ
∂
t
i
t
r
= Eψ (r, t) (3.24)
Postulates 1 and 2 allow us to deduce average values of general dynamical
variables. Equation (3.24) on being multiplied by ψ*(r, t) and integrated over the
entire space and on rearrangement of terms leads to
∫ψ* (r, t) V (r) ψ (r, t) d
3
r = ∫ψ* (r, t) i t
∂
∂
ψ (r, t) d
3
r
– ∫ ψ* (r, t)
2
2
2m
−
∇
ψ (r, t) d
3
r
(3.25)
The term on the left hand side, as seen from Eq. (3.21), is the average
potential energy. It is therefore reasonable to identify the terms on the right
hand side as the average total energy and the average kinetic energy:
〈 E 〉 = ∫ ψ* (r, t) i t
∂
∂
ψ (r, t) d
3
r
(3.26)
〈
2
1
2
p
m
〉 = ∫ψ* (r, t)
2
2
2m
−
∇
ψ(r, t) d
3
r
(3.27)
Now, since ,
E
c
p
transforms as a 4-vector, the relationships in Eqs. (3.10)
and (3.11) suggest that, in addition to Eq. (3.26)
〈 p 〉 = ∫ ψ* (r, t) (
)
i
− ∇
ψ (r, t) d
3
r
(3.28)
These results are generalized in the following postulate.
Postulate 3: The average values of E and the dynamical variable F (p, r) are
given by
〈 E 〉 = ∫ ψ* (r, t) i t
∂
∂
ψ (r, t) d
3
r
(3.29)
and
〈 F (p, r) 〉 = ∫ ψ* (r, t) F (
, )
i
r
− ∇
ψ (r, t) d
3
r
(3.30)
In quantum mechanics, the choice of dynamical variables which are
dynamical observables is not obvious. A minimal requirement is imposed on the
71
This equation is valid if the system is nonrelativistic. It requires some
modifications if the particle has additional variables such as intrinsic angular
momentum called spin. If the system has a well-defined energy, its Schrödinger
equation reduces to its simpler time-independent form
2
2 ( , )
( ) ( , )
2
−
∇ ψ
+
ψ
t V r
r t
m
r
=
( , )
∂ψ
∂
t
i
t
r
= Eψ (r, t) (3.24)
Postulates 1 and 2 allow us to deduce average values of general dynamical
variables. Equation (3.24) on being multiplied by ψ*(r, t) and integrated over the
entire space and on rearrangement of terms leads to
∫ψ* (r, t) V (r) ψ (r, t) d
3
r = ∫ψ* (r, t) i t
∂
∂
ψ (r, t) d
3
r
– ∫ ψ* (r, t)
2
2
2m
−
∇
ψ (r, t) d
3
r
(3.25)
The term on the left hand side, as seen from Eq. (3.21), is the average
potential energy. It is therefore reasonable to identify the terms on the right
hand side as the average total energy and the average kinetic energy:
〈 E 〉 = ∫ ψ* (r, t) i t
∂
∂
ψ (r, t) d
3
r
(3.26)
〈
2
1
2
p
m
〉 = ∫ψ* (r, t)
2
2
2m
−
∇
ψ(r, t) d
3
r
(3.27)
Now, since ,
E
c
p
transforms as a 4-vector, the relationships in Eqs. (3.10)
and (3.11) suggest that, in addition to Eq. (3.26)
〈 p 〉 = ∫ ψ* (r, t) (
)
i
− ∇
ψ (r, t) d
3
r
(3.28)
These results are generalized in the following postulate.
Postulate 3: The average values of E and the dynamical variable F (p, r) are
given by
〈 E 〉 = ∫ ψ* (r, t) i t
∂
∂
ψ (r, t) d
3
r
(3.29)
and
〈 F (p, r) 〉 = ∫ ψ* (r, t) F (
, )
i
r
− ∇
ψ (r, t) d
3
r
(3.30)
In quantum mechanics, the choice of dynamical variables which are
dynamical observables is not obvious. A minimal requirement is imposed on the