Approximate Selection Rule for Orbital Angular Momentum
in Atomic Radiative Transitions
I.B. Khriplovich and D.V. Matvienko
Budker Institute of Nuclear Physics,
630090 Novosibirsk, Russia,
and Novosibirsk University
Abstract
We demonstrate that radiative transitions with ∆l = −1 are strongly dominating
for all values of n and l, except small region where l ≪ n.
It is well-known that the selection rule for the orbital angular momentum l in electro-
magnetic dipole transitions, dominating in atoms, is ∆l = ±1, i. e. in these transitions
the angular momentum can both increase and decrease by unity. Meanwhile, the classical
radiation of a charge in the Coulomb field is always accompanied by the loss of angular
momentum. Thus, at least in the semiclassical limit, the probability of dipole transitions
with ∆ l = − 1 is higher. Here we discuss the question how strongly and under what
exactly conditions the transitions with ∆l = −1 dominate in atoms. (To simplify the
presentation, we mean always, here and below, the radiation of a photon, i. e. transitions
with ∆n < 0. Obviously, in the case of photon absorption, i. e. for ∆n > 0, the angular
momentum predominantly increases.)
The analysis of numerical values for the transition probabilities in hydrogen presented
in [1] has demonstrated that even for n and l, comparable with unity, i. e. in a nonclassical
situation, radiation with ∆l = −1 can be much more probable than that with ∆l = 1.
Later, the relation between the probabilities of transitions with ∆l = −1 and ∆l = 1
was investigated in [2] by analyzing the corresponding matrix elements in the semiclassical
approximation. The conclusion made therein is also that the transitions with ∆l = −1
dominate, and the dominance is especially strong when l > n2/3.
Here we present a simple solution of the problem using the classical electrodynamics
and, of course, the correspondence principle. Our results describe the situation not only
in the semiclassical situation. Remarkably enough, they agree, at least qualitatively, with
the results of [1], although the latter refer to transitions with |∆n| ∼ n ∼ 1 and l ∼ 1,
which are not classical at all.
We start our analysis with a purely classical problem. Let a particle with a mass
m and charge − e moves in an attractive Coulomb field, created by a charge e, along
an ellipse with large semi-axis a and eccentricity ε. It is known [3] that the radiation
intensity at a given harmonic ν is here
4e2ω4
ξ2ν + η
; (1)
J ′ν(νε), ην =
1− ε2
Jν(νε). (2)
In expressions (2), Jν(νε) is the Bessel function, and J
ν(νε) is its derivative. We use the
Fourier transformation in the following form:
x(t) = a
iνω0t = 2a
ξν cos νω0t,
http://arxiv.org/abs/0704.0856v1
y(t) = a
iνω0t = 2a
ην sin νω0t,
where all dimensionless Fourier components ξν and ην are real, and ξ−ν = ξν , η−ν = − ην .
We note that the Cartesian coordinates x and y are related here to the polar coordinates r
and φ as follows: x = r cos φ, y = r sin φ, where φ increases with time. Thus, the angular
momentum is directed along the z axis (but not in the opposite direction).
We note also that, since 0 ≤ ε ≤ 1, both Jν(νε) and J ′ν(νε) are reasonably well
approximated by the first term of their series expansion in the argument. Therefore, all
the Fourier components ξν and ην are positive.
In the quantum problem (where ν = |∆n|), the probability of transition in the unit
time is
h̄ω0ν
4e2ω3
3c3h̄
ξ2ν + η
, ω0 =
h̄2n3
. (3)
Now, the loss of angular momentum with radiation is [3]
r× r... .
Going over here to the Fourier components, we obtain
Ṁν = −
4e2ω2
rν × ṙν ,
or (with our choice of the direction of coordinate axes, and with the angular momentum
measured in the units of h̄)
Ṁν = −
4e2ω3
3c3h̄
2ξνην . (4)
Obviously, the last expression is nothing but the difference between the probabilities
of transitions with ∆l = 1 and ∆l = −1 in the unit time:
Ṁν = W
ν −W−ν . (5)
Of course, the total probability (3) can be written as
Wν = W
ν . (6)
From explicit expressions (3) and (4) it is clear that inequality W+ν ≪ W−ν holds if
2ξνην ≈ ξ2ν + η2ν , or ην ≈ ξν . The last relation is valid for ε ≪ 1, i. e. for orbits close
to circular ones. (The simplest way to check it, is to use in formulae (2) the explicit
expression for the Bessel function at small argument: Jν(νε) = (νε)
ν/(2νν !).)
This conclusion looks quite natural from the quantum point of view. Indeed, it is
the state with the orbital quantum number l equal to n − 1 (i. e. with the maximum
possible value for given n) which corresponds to the circular orbit. In result of radiation
n decreases, and therefore l should decrease as well.
The surprising fact is, however, that in fact the probabilities W−ν of transitions with
∆l = −1 dominate numerically everywhere, except small vicinity of the maximum possible
eccentricity ε = 1. For instance, if ε ≃ 0.9 (which is much more close to 1 than to 0 !),
then at ν = 1 the discussed probability ratio is very large, it constitutes
≃ 12 .
The change with ε of the ratio of W+ν to W
ν for two values of ν is illustrated in Fig. 1.
The curves therein demonstrate in particular that with the increase of ν, the region
0.2 0.4 0.6 0.8 1.0 Ε
�������� ����
Fig. 1
where W−ν and W
ν are comparable, gets more and more narrow, i. e. when ν grows, the
corresponding curves tend more and more to a right angle.
Let us go over now to the quantum problem. In the semiclassical limit, the classical
expression for the eccentricity
is rewritten with usual relations E = −me4/(2h̄2n2) and M = h̄l as
. (8)
In fact, the exact expression for ε, valid for arbitrary l and n, is [3]:
l(l + 1) + 1
. (9)
Clearly, in the semiclassical approximation the eccentricity is close to unity only under
condition l ≪ n. If this condition does not hold, one may expect that in the semiclas-
sical limit the transitions with ∆l = −1 dominate. In other words, as long as l ≪ n,
the probabilities of transitions with decrease and increase of the angular momentum are
comparable. But if the angular momentum is not small, it is being lost predominantly in
radiation. This situation looks quite natural.
The next point is that with the increase of |∆n| = ν, the region where W−ν and W+ν
are comparable, gets more and more narrow in agreement with the observation made in
However, we do not see any hint at some special role (advocated in [2]) of the condition
l > n2/3 for the dominance of transitions with ∆l = −1.
As mentioned already, the analysis of the numerical values of transition probabilities [1]
demonstrates that even for n and l comparable with unity and |∆n| ≃ n, i. e. in the
absolutely nonclassical regime, the transitions with ∆l = −1 are still much more probable
than those with ∆l = 1. The results of this analysis for the ratio W−/W+ in some
transitions are presented in Table 3.1 (first line). Then we indicate in Table 3.1 (last line)
W4p→3s
W4p→3d
W5p→4s
W5p→4d
W5d→4p
W5d→4f
W6f→5d
W6f→5g
W5p→3s
W5p→3d
W6p→3s
W6p→3d
exact
value 10 3.75 28 72 10.67 13.7
ε̄ 0.87 0.92 0.81 0.75 0.90 0.92
ν = |∆n| 1 1 1 1 2 3
semiclassical
value 17.6 8.7 34 58 17.2 15.7
Table 3.1
the values of these ratios obtained in the näıve (semi)classical approximation. Here for
the eccentricity ε̄ we use the value of expression (9), calculated with l corresponding to
the initial state; as to n, we take its value average for the initial and final states.
The table starts with the smallest possible quantum numbers where the transitions,
which differ by the sign of ∆l, occur, i. e. with the ratio W4p→3s/W4p→3d. This table
demonstrates that the ratio of the classical results to the exact quantum-mechanical ones
remains everywhere within a factor of about two. In fact, if one uses as ε̄ expression (8),
calculated in the analogous way, the numbers in the last line change considerably. It is
clear, however, that the classical approximation describes here, at least qualitatively, the
real situation.
References
[1] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,
Springer, 1957; §63.
[2] N.B. Delone and V.P. Krainov, FIAN Preprint No. 18, 1979; J. Phys. B 27, (1994)
4403.
[3] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Nauka, 1973; §70,
problem 2 to §72.
[4] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Nauka, 1974; §36.

  We demonstrate that radiative transitions with \Delta l = - 1 are strongly
dominating for all values of n and l, except small region where l << n.

Approximate Selection Rule for Orbital Angular Momentum
in Atomic Radiative Transitions
I.B. Khriplovich and D.V. Matvienko
Budker Institute of Nuclear Physics,
630090 Novosibirsk, Russia,
and Novosibirsk University
Abstract
We demonstrate that radiative transitions with ∆l = −1 are strongly dominating
for all values of n and l, except small region where l ≪ n.
It is well-known that the selection rule for the orbital angular momentum l in electro-
magnetic dipole transitions, dominating in atoms, is ∆l = ±1, i. e. in these transitions
the angular momentum can both increase and decrease by unity. Meanwhile, the classical
radiation of a charge in the Coulomb field is always accompanied by the loss of angular
momentum. Thus, at least in the semiclassical limit, the probability of dipole transitions
with ∆ l = − 1 is higher. Here we discuss the question how strongly and under what
exactly conditions the transitions with ∆l = −1 dominate in atoms. (To simplify the
presentation, we mean always, here and below, the radiation of a photon, i. e. transitions
with ∆n < 0. Obviously, in the case of photon absorption, i. e. for ∆n > 0, the angular
momentum predominantly increases.)
The analysis of numerical values for the transition probabilities in hydrogen presented
in [1] has demonstrated that even for n and l, comparable with unity, i. e. in a nonclassical
situation, radiation with ∆l = −1 can be much more probable than that with ∆l = 1.
Later, the relation between the probabilities of transitions with ∆l = −1 and ∆l = 1
was investigated in [2] by analyzing the corresponding matrix elements in the semiclassical
approximation. The conclusion made therein is also that the transitions with ∆l = −1
dominate, and the dominance is especially strong when l > n2/3.
Here we present a simple solution of the problem using the classical electrodynamics
and, of course, the correspondence principle. Our results describe the situation not only
in the semiclassical situation. Remarkably enough, they agree, at least qualitatively, with
the results of [1], although the latter refer to transitions with |∆n| ∼ n ∼ 1 and l ∼ 1,
which are not classical at all.
We start our analysis with a purely classical problem. Let a particle with a mass
m and charge − e moves in an attractive Coulomb field, created by a charge e, along
an ellipse with large semi-axis a and eccentricity ε. It is known [3] that the radiation
intensity at a given harmonic ν is here
4e2ω4
ξ2ν + η
; (1)
J ′ν(νε), ην =
1− ε2
Jν(νε). (2)
In expressions (2), Jν(νε) is the Bessel function, and J
ν(νε) is its derivative. We use the
Fourier transformation in the following form:
x(t) = a
iνω0t = 2a
ξν cos νω0t,
http://arxiv.org/abs/0704.0856v1
y(t) = a
iνω0t = 2a
ην sin νω0t,
where all dimensionless Fourier components ξν and ην are real, and ξ−ν = ξν , η−ν = − ην .
We note that the Cartesian coordinates x and y are related here to the polar coordinates r
and φ as follows: x = r cos φ, y = r sin φ, where φ increases with time. Thus, the angular
momentum is directed along the z axis (but not in the opposite direction).
We note also that, since 0 ≤ ε ≤ 1, both Jν(νε) and J ′ν(νε) are reasonably well
approximated by the first term of their series expansion in the argument. Therefore, all
the Fourier components ξν and ην are positive.
In the quantum problem (where ν = |∆n|), the probability of transition in the unit
time is
h̄ω0ν
4e2ω3
3c3h̄
ξ2ν + η
, ω0 =
h̄2n3
. (3)
Now, the loss of angular momentum with radiation is [3]
r× r... .
Going over here to the Fourier components, we obtain
Ṁν = −
4e2ω2
rν × ṙν ,
or (with our choice of the direction of coordinate axes, and with the angular momentum
measured in the units of h̄)
Ṁν = −
4e2ω3
3c3h̄
2ξνην . (4)
Obviously, the last expression is nothing but the difference between the probabilities
of transitions with ∆l = 1 and ∆l = −1 in the unit time:
Ṁν = W
ν −W−ν . (5)
Of course, the total probability (3) can be written as
Wν = W
ν . (6)
From explicit expressions (3) and (4) it is clear that inequality W+ν ≪ W−ν holds if
2ξνην ≈ ξ2ν + η2ν , or ην ≈ ξν . The last relation is valid for ε ≪ 1, i. e. for orbits close
to circular ones. (The simplest way to check it, is to use in formulae (2) the explicit
expression for the Bessel function at small argument: Jν(νε) = (νε)
ν/(2νν !).)
This conclusion looks quite natural from the quantum point of view. Indeed, it is
the state with the orbital quantum number l equal to n − 1 (i. e. with the maximum
possible value for given n) which corresponds to the circular orbit. In result of radiation
n decreases, and therefore l should decrease as well.
The surprising fact is, however, that in fact the probabilities W−ν of transitions with
∆l = −1 dominate numerically everywhere, except small vicinity of the maximum possible
eccentricity ε = 1. For instance, if ε ≃ 0.9 (which is much more close to 1 than to 0 !),
then at ν = 1 the discussed probability ratio is very large, it constitutes
≃ 12 .
The change with ε of the ratio of W+ν to W
ν for two values of ν is illustrated in Fig. 1.
The curves therein demonstrate in particular that with the increase of ν, the region
0.2 0.4 0.6 0.8 1.0 Ε
�������� ����
Fig. 1
where W−ν and W
ν are comparable, gets more and more narrow, i. e. when ν grows, the
corresponding curves tend more and more to a right angle.
Let us go over now to the quantum problem. In the semiclassical limit, the classical
expression for the eccentricity
is rewritten with usual relations E = −me4/(2h̄2n2) and M = h̄l as
. (8)
In fact, the exact expression for ε, valid for arbitrary l and n, is [3]:
l(l + 1) + 1
. (9)
Clearly, in the semiclassical approximation the eccentricity is close to unity only under
condition l ≪ n. If this condition does not hold, one may expect that in the semiclas-
sical limit the transitions with ∆l = −1 dominate. In other words, as long as l ≪ n,
the probabilities of transitions with decrease and increase of the angular momentum are
comparable. But if the angular momentum is not small, it is being lost predominantly in
radiation. This situation looks quite natural.
The next point is that with the increase of |∆n| = ν, the region where W−ν and W+ν
are comparable, gets more and more narrow in agreement with the observation made in
However, we do not see any hint at some special role (advocated in [2]) of the condition
l > n2/3 for the dominance of transitions with ∆l = −1.
As mentioned already, the analysis of the numerical values of transition probabilities [1]
demonstrates that even for n and l comparable with unity and |∆n| ≃ n, i. e. in the
absolutely nonclassical regime, the transitions with ∆l = −1 are still much more probable
than those with ∆l = 1. The results of this analysis for the ratio W−/W+ in some
transitions are presented in Table 3.1 (first line). Then we indicate in Table 3.1 (last line)
W4p→3s
W4p→3d
W5p→4s
W5p→4d
W5d→4p
W5d→4f
W6f→5d
W6f→5g
W5p→3s
W5p→3d
W6p→3s
W6p→3d
exact
value 10 3.75 28 72 10.67 13.7
ε̄ 0.87 0.92 0.81 0.75 0.90 0.92
ν = |∆n| 1 1 1 1 2 3
semiclassical
value 17.6 8.7 34 58 17.2 15.7
Table 3.1
the values of these ratios obtained in the näıve (semi)classical approximation. Here for
the eccentricity ε̄ we use the value of expression (9), calculated with l corresponding to
the initial state; as to n, we take its value average for the initial and final states.
The table starts with the smallest possible quantum numbers where the transitions,
which differ by the sign of ∆l, occur, i. e. with the ratio W4p→3s/W4p→3d. This table
demonstrates that the ratio of the classical results to the exact quantum-mechanical ones
remains everywhere within a factor of about two. In fact, if one uses as ε̄ expression (8),
calculated in the analogous way, the numbers in the last line change considerably. It is
clear, however, that the classical approximation describes here, at least qualitatively, the
real situation.
References
[1] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,
Springer, 1957; §63.
[2] N.B. Delone and V.P. Krainov, FIAN Preprint No. 18, 1979; J. Phys. B 27, (1994)
4403.
[3] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Nauka, 1973; §70,
problem 2 to §72.
[4] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Nauka, 1974; §36.
