Subatomic Physics

9.6. The Two-State Problem 261
Hamiltonian is invariant under reflections through the origin, and H and the parity
operator P commute,
[H, P] =[H0 + Hint,P]=0. (9.61)
With the choice of coordinates shown in Fig. 9.9, the parity operator gives
P |L〉, = |R〉 P |R〉 = |L〉. (9.62)
The simultaneous eigenfunctions of H0 and P are easy to find; they are the symmetric
and antisymmetric combinations of the unperturbed states |L〉 and |R〉:
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1
1
|s〉 = 2 {|L〉 + |R〉}, |a〉 = 2 {|L〉−|R〉}. (9.63)
These combinations indeed are eigenstates of P ,
P |s〉 =+|s〉, P|a〉 = −|a〉. (9.64)
Eqs. (9.61) and (9.64) together prove that H does not connect |a〉 and |s〉:
or
〈a|H|s〉 = 〈a|HP|s〉 = 〈a|PH|s〉 = 〈a|P † H|s〉 = −〈a|H|s〉,
〈a|H|s〉 =0. (9.65)
Ordinary perturbation theory can consequently be applied to the states |a〉 and |s〉.
The energy shift caused by the perturbation, Hint, is given by the expectation value
of Hint, or
where
〈s|Hint|s〉 = E ′ +∆E
〈a|Hint|a〉 = E ′ − ∆E, (9.66)
〈L|Hint|L〉 = 〈R|Hint|R〉 = E ′
〈L|Hint|R〉 = 〈R|Hint|L〉 =∆E. (9.67)
The interaction lowers the center of the energy levels by E ′ and splits the degenerate
levels by an amount 2∆E, as indicated in Fig. 9.9(b). The splitting shows up in
the hydrogen molecule ion and particularly clearly in the inversion spectrum of
ammonia. (24)
24Two-state systems and the ammonia MASER are beautifully treated in R. P. Feynman, R. B.
Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. III, Addison-Wesley, Reading,
Mass., 1965, Chapters 8–11.