C.2 Structure of the Hamiltonian Matrix
425
If the Hamiltonian contains no explicit dependence on time, then
d ˆ
H
dt = 0 and ˆ
H is a
constant of the motion. If
d ˆ
H
dt = 0, then the eigenvalues of ˆ
H (the allowed energies
of the system) are constant in time.
C.2.1.2 Space Translation Invariance
If the Hamiltonian depends only on the relative displacements of the particles,
ˆ
q α − ˆ
q β = ˆ
q α,β , then the Hamiltonian will be invariant under a translation of the
entire system in space. When this occurs, the total momentum (the center-of-mass
momentum) ˆ
P =
α ˆ
p α commutes with Hamiltonian [ ˆ
H , ˆ
P] = 0, and therefore ˆ
P
is a constant of the motion. In such a case, we can write the Hamiltonian in the form
ˆ
H =
ˆ
P 2
2M
+ ˆ
H rel ({ ˆ
p α }, { ˆ
q α,β }, {ˆ s α }),
(C.29)
where M =
N
α=1 m α is the total mass, m α is the mass of the αth particle, and ˆ
H rel
is the Hamiltonian describing relative motion of the particles.
C.2.1.3 Rotational Invariance
If the Hamiltonian is invariant under a rotation of the entire system in space, then
the Hamiltonian will commute with the total angular momentum operator, ˆ
J = ˆ
L +
ˆ
S, where ˆ
L is the total orbital angular momentum and ˆ
S is the total spin. Thus
[ ˆ
H , ˆ
J] = 0, and ˆ
J is a constant of the motion. Because [ ˆ
H , ˆ
J] = 0, one can find
simultaneous eigenstates of J 2 = ˆ
J· ˆ
J, ˆ
J z , and ˆ
H (components of ˆ
J do not commute
with one another). There will be no coupling between states with different values
of the angular momentum quantum number, J , and the Hamiltonian matrix can be
written in block diagonal form, with each block containing matrix elements that all
depend on the same value of J .
C.2.2 Time Reversal Invariance
In order to discuss the effect of time reversal invariance on the Hamiltonian matrix,
we must consider separately the cases of integer spin and half-integer spin.
C.2.2.1 Integer Spin
For time reversal invariant systems with integer total spin, the time reversal operator,
ˆ
K = e
i
¯
h π ˆ
S y ˆ
K o , has the property ˆ
K 2 = +1, where ˆ
K o is the complex conjugation
425
If the Hamiltonian contains no explicit dependence on time, then
d ˆ
H
dt = 0 and ˆ
H is a
constant of the motion. If
d ˆ
H
dt = 0, then the eigenvalues of ˆ
H (the allowed energies
of the system) are constant in time.
C.2.1.2 Space Translation Invariance
If the Hamiltonian depends only on the relative displacements of the particles,
ˆ
q α − ˆ
q β = ˆ
q α,β , then the Hamiltonian will be invariant under a translation of the
entire system in space. When this occurs, the total momentum (the center-of-mass
momentum) ˆ
P =
α ˆ
p α commutes with Hamiltonian [ ˆ
H , ˆ
P] = 0, and therefore ˆ
P
is a constant of the motion. In such a case, we can write the Hamiltonian in the form
ˆ
H =
ˆ
P 2
2M
+ ˆ
H rel ({ ˆ
p α }, { ˆ
q α,β }, {ˆ s α }),
(C.29)
where M =
N
α=1 m α is the total mass, m α is the mass of the αth particle, and ˆ
H rel
is the Hamiltonian describing relative motion of the particles.
C.2.1.3 Rotational Invariance
If the Hamiltonian is invariant under a rotation of the entire system in space, then
the Hamiltonian will commute with the total angular momentum operator, ˆ
J = ˆ
L +
ˆ
S, where ˆ
L is the total orbital angular momentum and ˆ
S is the total spin. Thus
[ ˆ
H , ˆ
J] = 0, and ˆ
J is a constant of the motion. Because [ ˆ
H , ˆ
J] = 0, one can find
simultaneous eigenstates of J 2 = ˆ
J· ˆ
J, ˆ
J z , and ˆ
H (components of ˆ
J do not commute
with one another). There will be no coupling between states with different values
of the angular momentum quantum number, J , and the Hamiltonian matrix can be
written in block diagonal form, with each block containing matrix elements that all
depend on the same value of J .
C.2.2 Time Reversal Invariance
In order to discuss the effect of time reversal invariance on the Hamiltonian matrix,
we must consider separately the cases of integer spin and half-integer spin.
C.2.2.1 Integer Spin
For time reversal invariant systems with integer total spin, the time reversal operator,
ˆ
K = e
i
¯
h π ˆ
S y ˆ
K o , has the property ˆ
K 2 = +1, where ˆ
K o is the complex conjugation