the spectrum. We do not assume, however, thatE^0 n′=En^0 , so that we can applynon-degenerate
perturbation theory. First order perturbation theory will split the levels significantly and lift the
approximate degeneracy unless we haveEn^1 =E^1 n′, namely when〈En^0 |H 1 |En〉=〈E^0 n′|H 1 |En′〉. In
this case, we need to go to second order perturbation theory.
At second order perturbation theory, we may continue to usenon-degenerate perturbation
theoryto deduce those second order corrections, and they are givenby
En^2 = −∑m 6 =n|〈Em^0 |H 1 |En^0 〉|^2
E^0 m−En^0E^2 n′ = −∑m 6 =n′|〈E^0 m|H 1 |E^0 n′〉|^2
E^0 m−En^0 ′(10.82)
This is a complicated sum, in general with an infinite number of terms. The assumption that
|En^0 ′−En^0 |≪ |Em^0 −E^0 n|for allm 6 =n,n′, which we made above, however, allows us to retain the
essential contributions of these sums only. The smallness ofEn^0 ′−En^0 allows us to approximate
each sum by just a single term, and we get
En^2 = −
|〈En^0 ′|H 1 |En^0 〉|^2
E^0 n′−En^0+O(1)
E^2 n′ = −|〈En^0 |H 1 |E^0 n′〉|^2
E^0 n−E^0 n′+O(1) (10.83)
where we neglected terms that have finite limits asEn^0 ′−E^0 n→0. We now read off the second
order behavior,
E^0 n′> E^0 n ⇒ En^2 ′> 0 , E^2 n< 0
E^0 n′< E^0 n ⇒ En^2 ′< 0 , E^2 n> 0 (10.84)In either case, the levels repel one another. This effect is sometimes referred to as theno-level
crossing Theorem.
There is, however, one very important exception to this effect. We tacitly assumed that the
matrix element〈En^0 ′|H 1 |E^0 n〉does not vanish in deriving the non-level crossing theorem.Naturally,
if this matrix element vanishes, then the degeneracy is not lifted, and levels can actually cross.
Putting together the conditions from first and second order perturbation theory for levels to cross,
we have
〈En^0 |H 1 |En〉 = 〈En^0 ′|H 1 |En′〉
〈En^0 ′|H 1 |E^0 n〉 = 0 (10.85)which means thatH 1 restricted to the two-dimensional subspace of states|En〉and|En′〉is actually
proportional to the identity operator. The full Hamiltonian then necessarily has anSU(2) symmetry
which rotates these two states into one another.
Thus, we have discovered an important amendement to theno level crossing theorem, namely
that levels can cross if and only if the HamiltonianHhas a symmetry at the point where the levels
are to cross.