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3 Quantum Mechanics – II
where P
m
l (cos θ) are the associated Legendre functions
Now, P
m
l (cos θ) = (1 − cos
2
θ)
m/2 d
m P l (cos θ )/d cos
m
θ
∴ P
m
l (cos(π − θ)) = (1 − cos
2
θ)
m/2 d
m P l (− cos θ )/d(− cos θ )
m
= (1 − cos
2
θ)
m/2 (−1)
l d
m P l (cos θ)/d(− cos θ )
m
= (1 − cos
2
θ)
m/2 (−1)
l+m d
m P l (cos θ)/d cos
m
θ = (−1)
l+m P l
m (cos θ )
Thus P l
m (cos(π − θ)) → (−1)
l+m P l
m (cos θ )
Further e
im(ϕ+π)
= e
imϕ
.e
imπ
= e
imϕ
.(cos mπ + i sin mπ )
= (−1)
m e
imϕ
where m is an integer, positive or negative. So, under parity ( p) operation,
the function overall F(r, θ, ϕ), goes as
P F(r, θ, ϕ) = P f (r )P l
m (cos θ)e
imϕ
= f (r )P l
m (cos θ )e
imϕ (−1)
l+m (−1)
m
= (−1)
l+2m F(r, θ, ϕ)
= (−1)
l F(r, θ, ϕ)
All the atomic functions with even values of l have even parity while
those with odd values of l have 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 is l initial − l final = Δl = 0, 2, 4 . . .
This condition for the dipole radiation emission is known as Laportes’s
rule. Actually a more restrictive rule applies
Δ l = ±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 2s state of hydrogen can not decay to the 1s state via dipole radiation
because that would imply Δl = 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.
3 Quantum Mechanics – II
where P
m
l (cos θ) are the associated Legendre functions
Now, P
m
l (cos θ) = (1 − cos
2
θ)
m/2 d
m P l (cos θ )/d cos
m
θ
∴ P
m
l (cos(π − θ)) = (1 − cos
2
θ)
m/2 d
m P l (− cos θ )/d(− cos θ )
m
= (1 − cos
2
θ)
m/2 (−1)
l d
m P l (cos θ)/d(− cos θ )
m
= (1 − cos
2
θ)
m/2 (−1)
l+m d
m P l (cos θ)/d cos
m
θ = (−1)
l+m P l
m (cos θ )
Thus P l
m (cos(π − θ)) → (−1)
l+m P l
m (cos θ )
Further e
im(ϕ+π)
= e
imϕ
.e
imπ
= e
imϕ
.(cos mπ + i sin mπ )
= (−1)
m e
imϕ
where m is an integer, positive or negative. So, under parity ( p) operation,
the function overall F(r, θ, ϕ), goes as
P F(r, θ, ϕ) = P f (r )P l
m (cos θ)e
imϕ
= f (r )P l
m (cos θ )e
imϕ (−1)
l+m (−1)
m
= (−1)
l+2m F(r, θ, ϕ)
= (−1)
l F(r, θ, ϕ)
All the atomic functions with even values of l have even parity while
those with odd values of l have 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 is l initial − l final = Δl = 0, 2, 4 . . .
This condition for the dipole radiation emission is known as Laportes’s
rule. Actually a more restrictive rule applies
Δ l = ±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 2s state of hydrogen can not decay to the 1s state via dipole radiation
because that would imply Δl = 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.