1000 Solved Problems in Modern Physics

160 3 Quantum Mechanics – II

wherePlm(cosθ) are the associated Legendre functions

Now, Plm(cosθ)=(1−cos^2 θ)m/^2 dmPl(cosθ)/dcosmθ

∴ Plm(cos(π−θ))=(1−cos^2 θ)m/^2 dmPl(−cosθ)/d(−cosθ)m

=(1−cos^2 θ)m/^2 (−1)ldmPl(cosθ)/d(−cosθ)m

=(1−cos^2 θ)m/^2 (−1)l+mdmPl(cosθ)/dcosmθ=(−1)l+mPlm(cosθ)

ThusPlm(cos(π−θ))→(−1)l+mPlm(cosθ)

Furthereim(φ+π)=eimφ.eimπ=eimφ.(cosmπ+isinmπ)

=(−1)meimφ

wheremis an integer, positive or negative. So, under parity (p) operation,

the function overallF(r,θ,φ), goes as

PF(r,θ,φ)=Pf(r)Plm(cosθ)eimφ=f(r)Plm(cosθ)eimφ(−1)l+m(−1)m

=(−1)l+^2 mF(r,θ,φ)

=(−1)lF(r,θ,φ)

All the atomic functions with even values oflhave even parity while

those with odd values oflhave odd parity. Considering that an integral

vanishes between symmetrical limits if the integrand has odd parity, and

that the operator of the electric dipole moment has odd parity, the following

selection rule may be stated:- The expectation value of the electric dipole

moment, as well as the transition probability vanishes unless initial and

final state have different parity, that islinitial−lfinal=Δl

= 0 , 2 , 4 ...

This condition for the dipole radiation emission is known as Laportes’s

rule. Actually a more restrictive rule applies

Δl=± 1

Note that even if the matrix element of electric dipole moment vanishes,

an atom will eventually go to the ground state by an alternative mechanism

such as magnetic dipole or electric quadrupole etc for which the transition

probability is much smaller than the dipole radiation.

(b) The 2sstate of hydrogen can not decay to the 1sstate via dipole radiation

because that would implyΔl=0. Furthermore, there are no other electric

or magnetic moments to facilitate the transition. However, de-excitation

may occur in collision processes with other atoms. Even in perfect vacuum

transition may take place via two-photon emission, probability for which

is again very small compared to one-photon emission. The result is that

such a state is allowed to live for considerable time. Such states are known

as metastable states.