2.2 Transitions 33selection rules. The parity transformation is an inversion through the
origin given byr→−r. This is equivalent to the following transforma-
tion of the polar coordinates:
θ−→π−θ: a reflection,
φ−→φ+π: a rotation.The reflection produces a mirror image of the original system and parity
is also referred to as mirror symmetry. The mirror image of a hydrogen
atom has the same energy levels as those in the original atom since the
Coulomb potential is the same after reflection. It turns out that all the
electric and magnetic interactions ‘look the same’ after reflection and
all atoms have parity symmetry.^36 To find the eigenvalues for parity we^36 This can be proved formally in
quantum mechanics by showing that
the Hamiltonians for these interactions
commute with the parity operator. The
weak interaction in nuclear physics does
not have mirror symmetry and violates
parity conservation. The extremely
small effect of the weak interaction on
atoms has been measured in exceed-
ingly careful and precise experiments.
use the full quantum mechanical notation, with hats to distinguish the
operatorP̂from its eigenvaluePin the equation
Pψ̂ =Pψ, (2.40)from which it follows thatP̂^2 ψ=P^2 ψ. Two successive parity operations
correspond to there being no change (the identity operator), i.e.r→
−r→r.ThusP^2 = 1. Therefore the parity operator has eigenvalues
P=1and−1 that correspond to even and odd parity wavefunctions,
respectively:
Pψ̂ =ψ or Pψ̂ =−ψ.
Both eigenvalues occur for the spherical harmonic functions,
PŶ l,m=(−1)lYl,m. (2.41)The value of the angular integral does not change in a parity trans-
formation^37 so^37 See, for example, Mandl (1992).
Iang=(−1)l^2 +l^1 +1Iang. (2.42)
Thus the integral is zero unless the initial and final states have opposite
parity (see Exercise 2.12). In particular, electric dipole transitions re-
quire an odd change in the orbital angular momentum quantum number
(∆l=0).^3838 The radial integral is not changed by
The treatment above of the parity operator acting on a wavefunction the parity transformation.
is quite general and even in complex atoms the wavefunctions have a
definite parity. The selection rules we have discussed in this section and
others are tabulated in Appendix C. If the electric dipole matrix element
is zero between two states then other types of transition may occur but
at a rate many orders of magnitude slower than allowed transitions.
The allowed transitions between then=1,2 and 3 shells of atomic
hydrogen are shown in Fig. 2.2, as an example of the selection rules.
The 2s configuration has no allowed transitions downwards; this makes
it metastable, i.e. it has a very long lifetime of about 0.125 s.^3939 This special feature is used in the ex-
Finally, a comment on the spectroscopic notation. It can be seen periment described in Section 2.3.4.
in Fig. 2.2 that the allowed transitions give rise to several series of