182
7 Spherical Symmetry and Spins
It can easily be demonstrated that the Hamiltonian in this new basis is real:
φ|H|χ=
ϑφ|ϑ(Hχ)
=
ϑφ|ϑHϑ|ϑχ
= φ|H|χ
(7.59)
Hence, in this case, it is always possible to rewrite the basis in such a way that the
Hamiltonian matrix is completely real, and the states behave in all respects as a
real twofold-degenerate irrep, which can be split by symmetry-lowering electrostatic fields. In particular, such states will be subject to Jahn–Teller distortions.
• ϑ 2 =−1. In the case of a negative sign, it is impossible to obtain states that are
time invariant. This can be shown as follows. We start again with two states that
are each other’s time inverse (|Ψ and ϑ|Ψ ) and first show that these states must
be linearly independent:
Ψ |ϑΨ=
ϑΨ|ϑ 2 Ψ
=−ϑΨ|Ψ =−−Ψ |ϑΨ=0
(7.60)
In contrast to the previous case, all attempts to construct a linear combination of
these basis states that is invariant under time reversal, fail. Indeed, suppose that
|X is a linear combination with coefficients a and b, such that ϑ|X=|X. Then
we have:
|X=a|Ψ +bϑ|Ψ
ϑ|X= ¯
aϑ|Ψ − ¯
b|Ψ ≡a|Ψ +bϑ|Ψ
(7.61)
Since the two kets are linearly independent, their respective coefficients must
coincide, and this is possible only for a = b = 0. Hence, it is not possible to
remove the degeneracy by time-even external fields. In particular, these states
will not be subject to the JT effect.
Time-Reversal Selection Rules
The argument used in Eq. (7.59) can be generalized to describe selection rules that
depend on time reversal [12]. We first introduce two parities, τ and η, which describe the time-dependence of the state and of the Hamiltonian:
ϑ
2 = (−1)
τ ˆ
E
ϑHϑ
−1 = (−1)
η H
(7.62)
The first label, τ , indicates the parity of the state functions, as we have just introduced in this section. The second label, η, indicates whether the Hamiltonian is
time-even or time-odd. Time-even interactions are typically interactions associated
with the electrostatic potential, such as the Jahn–Teller and Stark effects. Time-odd
interactions are electrodynamic in nature, the most common one being the Zeeman
interaction. We shall now study a function space that is invariant under time reversal
7 Spherical Symmetry and Spins
It can easily be demonstrated that the Hamiltonian in this new basis is real:
φ|H|χ=
ϑφ|ϑ(Hχ)
=
ϑφ|ϑHϑ|ϑχ
= φ|H|χ
(7.59)
Hence, in this case, it is always possible to rewrite the basis in such a way that the
Hamiltonian matrix is completely real, and the states behave in all respects as a
real twofold-degenerate irrep, which can be split by symmetry-lowering electrostatic fields. In particular, such states will be subject to Jahn–Teller distortions.
• ϑ 2 =−1. In the case of a negative sign, it is impossible to obtain states that are
time invariant. This can be shown as follows. We start again with two states that
are each other’s time inverse (|Ψ and ϑ|Ψ ) and first show that these states must
be linearly independent:
Ψ |ϑΨ=
ϑΨ|ϑ 2 Ψ
=−ϑΨ|Ψ =−−Ψ |ϑΨ=0
(7.60)
In contrast to the previous case, all attempts to construct a linear combination of
these basis states that is invariant under time reversal, fail. Indeed, suppose that
|X is a linear combination with coefficients a and b, such that ϑ|X=|X. Then
we have:
|X=a|Ψ +bϑ|Ψ
ϑ|X= ¯
aϑ|Ψ − ¯
b|Ψ ≡a|Ψ +bϑ|Ψ
(7.61)
Since the two kets are linearly independent, their respective coefficients must
coincide, and this is possible only for a = b = 0. Hence, it is not possible to
remove the degeneracy by time-even external fields. In particular, these states
will not be subject to the JT effect.
Time-Reversal Selection Rules
The argument used in Eq. (7.59) can be generalized to describe selection rules that
depend on time reversal [12]. We first introduce two parities, τ and η, which describe the time-dependence of the state and of the Hamiltonian:
ϑ
2 = (−1)
τ ˆ
E
ϑHϑ
−1 = (−1)
η H
(7.62)
The first label, τ , indicates the parity of the state functions, as we have just introduced in this section. The second label, η, indicates whether the Hamiltonian is
time-even or time-odd. Time-even interactions are typically interactions associated
with the electrostatic potential, such as the Jahn–Teller and Stark effects. Time-odd
interactions are electrodynamic in nature, the most common one being the Zeeman
interaction. We shall now study a function space that is invariant under time reversal