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APAS 5110. Internal Processes in Gases. Fall 1999.
Transition Probabilities and Selection Rules
- Correspondence between Classical and Quantum Mechanical Transition Rates
According to the correspondence principle between classical and quantum mechanics (e.g. Landau and Lifshitz, Quantum Mechanics, §48), if A is a classically time-varying quantity, then there is a correspondence between its Fourier components Aω and the matrix elements of the quantum mechanical operator between two energy eigenstates ψL = φL(x)e−iωLt^ and ψU = φU (x)e−iωU^ t^ differing in energy by ω = ωU − ωL
Aω → 〈φL|Aω|φU 〉. (1.1)
Equation (1.1) is used to obtain the quantum mechanical transition probabilities (2.11), (3.5), and (4.3) given below.
- Electric Dipole Start from the classical nonrelativistic formula for electric dipole radiation (e.g. Ry- bicki & Lightman (1979) Radiative Processes in Astrophysics, eq. (3.23b); Weinberg (1995) Quantum Theory of Fields Vol. 1, eq. (1.2.19); note Weinberg uses Heaviside rather than Gaussian units for charge e) dE dt
3 c^3
d¨^2 , (2.1)
where d ≡
charges q
qr (2.2)
is the electric dipole moment. The Fourier expansion of the dipole moment d is
d =
ω
dωeiωt^ , (2.3)
so the Fourier expansion of d¨ is
d¨ =
ω
−ω^2 dωeiωt^ , (2.4)
and the Fourier expansion of d¨^2 is
d¨^2 =
ω
−ω^2 dωeiωt^ ·
ω′
−ω′^2 dω′ eiω
′t
. (2.5)
The thing which is really of interest is the dipole radiation rate averaged over time, or averaged over a period if the motion is periodic. Averaged over time, the periodic terms in equation (2.5) disappear, leaving only the constant terms, which are those satisfying ω′^ = −ω: d¨^2 =
ω
ω^4 dω · d−ω = 2
ω> 0
ω^4 dω · d−ω. (2.6)
Since the dipole moment d is real, it satisfies
d−ω = d∗ ω. (2.7) 1
Hence equation (2.6) reduces to
d¨^2 = 2
ω> 0
ω^4 |dω|^2. (2.8)
Thus equation (2.1) for the classical dipole radiation rate, averaged over time, becomes
dE dt
ω> 0
4 ω^4 3 c^3
|dω|^2. (2.9)
To make the transition to quantum mechanics, according to the general prescription (1.1), the Fourier coefficient dω must be replaced by the matrix element 〈φL|d|φU 〉, where φU and φL represent initial (Upper) and final (Lower) spatial wave functions differing in energy by ωU − ωL = ω. The quantum mechanical equivalent of equation (2.9) is therefore
dE dt
L
4 ω^4 3 c^3
|〈φL|d|φU 〉|^2. (2.10)
Equation (2.10) gives the mean energy loss per unit time by electric dipole transitions out of an initial state φU to a set of final states φL. The spontaneous transition probability, or Einstein A coefficient, or simply A-value, for electric dipole transitions is obtained by dividing the energy loss rate (2.10) by the transition energy ℏω (this is Rybicki & Lightman, eq. (10.28a)),
AU L(electric dipole) =
L
4 ω^3 3 c^3 ℏ
|〈φL|d|φU 〉|^2. (2.11)
Note that the spontaneous transition probability is of order α^3 (the fine-structure constant 1/137 cubed) in atomic units e = me = ℏ. The probability depends on nuclear charge Z approximately as AU L ∼ Z^4 , since ω ∼ Z^2 and d ∼ r ∼ Z−^1. For a single electron, the electric dipole moment is d = −er. (2.12)
The radial vector operator r can be written as the product r = rrˆ of the radial operator r and the dipole operator ˆr, which is the unit vector in the r direction. The matrix elements of the dipole operator ˆr are given in the notes on Angular Momentum. From these matrix elements follow the electric dipole selection rules for a single electron:
(1) ∆L = ±1, ∆M = 0, ±1; (2) ∆S = 0, ∆MS = 0.
The second rule follows because the radial operator acts only on the spatial part of the wave function, not the spin. The first rule implies that parity must change, which is also evident from the fact that r has odd parity (it changes sign under coordinate inversion). These selection rules reflect the fact that electric dipole photons have unit angular momentum and odd parity. For a system of electrons, which all have the same charge −e, the electric dipole moment is
d = −e
electrons
r. (2.13)
is the magnetic dipole moment. The magnetic dipole transition probability is then
AU L(magnetic dipole) =
L
4 ω^3 3 c^3 ℏ
|〈φL|μ|φU 〉|^2. (3.3)
The magnetic dipole transition probability is of order Z^6 α^5 in atomic units, down by a factor Z^2 α^2 from the electric dipole probability. For a system of nonrelativistic electrons, the magnetic moment μ is
μ =
−e 2 mec
(L + 2S). (3.4)
The minus sign in equation (3.4) is because electrons have a negative charge −e, and I’ve added in the spin contribution S to the magnetic moment. Classically the spin S is absent, but it has to be included quantum mechanically. The factor of 2 in front of S comes from the nonrelativistic limit of the Dirac equation, which is the relativistic equivalent of Schr¨odinger’s equation for spin 12 particles (see the notes on Spin in Atoms). Bunging (3.4) into (3.3) gives the magnetic dipole transition probability
AU L(magnetic dipole) =
L
e^2 ω^3 3 m^2 ec^5 ℏ
|〈φL|L + 2S|φU 〉|^2 (3.5)
(don’t confuse L for Lower with L for angular momentum). The selection rules for magnetic dipole transitions follow from equation (3.5): only transi- tions between states in which the matrix elements of either L or S are nonzero are magnetic- dipole allowed. The matrix elements of L, which are also those of S, are given in the notes on Angular Momentum. Since L and S both have even parity (they remain unchanged under coordinate inversion), the wave functions φL and φU must have the same parity. Since L and S act only on the angular part of the wave function, not the radial part, the n quantum numbers are unchanged. Since L^2 commutes with L and S, the magnitude of the total orbital angular momentum is unchanged. Likewise since S^2 commutes with L and S, the magnitude of the total spin angular momentum is unchanged. The remaining selection rules follow most straightforwardly from the requirement that the departing photon has unit angular momentum. The magnetic dipole selection rules are, then:
(1) Parity is unchanged; (2) ∆J = 0, ±1; (3) ∆MJ = 0, ±1; (4) ∆J = 0 together with ∆MJ = 0 is not allowed; in particular, J = 0 ↔ 0 is not allowed; (5) No change in electronic configuration; (6) ∆L = 0; (7) ∆S = 0.
Rules 1-4 follow from the magnetic dipole nature of the photon, and are always valid. Rules 5-8 are exact for H-like ions, but can break down in heavier ions because of mixing of LS terms.
- Electric Quadrupole The classical electric quadrupole radiation rate is dE dt
180 c^5
D
2 , (4.1)
where
D
2 means
Dij
Dij (implicit summation over^ i, j^ = 1,^2 ,^ 3), and
Dij ≡
charges q
qr^2 (3ˆri ˆrj − δij ) (4.2)
is the (3-dimensional) electric quadrupole moment tensor. The electric quadrupole transi- tion probability is then
AU L(electric quadrupole) =
L
ω^5 90 c^5 ℏ
|〈φL|Dij |φU 〉|^2. (4.3)
which is of order Z^6 α^5 in atomic units, the same as the magnetic dipole transition proba- bility. You can check that, in a single electron atom, the matrix elements are nonzero only for ∆L = 2, ∆M = 0, ± 1 , ±2. Parity is conserved, since Dij has even parity, and spin is conserved, because Dij acts only on spatial coordinates. These rules reflect the fact that the emitted electric quadrupole photon has angular momentum 2 and even parity. The electric quadrupole selection rules are:
(1) Parity is unchanged; (2) ∆J = 0, ± 1 , ±2; (3) ∆MJ = 0, ± 1 , ±2; (4) J = 0 ↔ 0, 0 ↔ 1, 1 / 2 ↔ 1 /2 are not allowed; (5) Either zero or one electron changes its nl state; ∆n = any, ∆l = 0, ±2; (6) ∆L = 0, ± 1 , ±2; (7) L = 0 ↔ 0, 0 ↔ 1 are not allowed; (8) ∆S = 0.
Rules 1-4 define what is meant by an electric quadrupole transition. Rules 5-8 are exact for H-like atoms, but can break down in heavier atoms because of mixing of LS terms.
- Two-Photon Transitions Since photons must have an angular momentum of at least one, the selection rule J = 0 ↔ 0 is not allowed (5.1)
is absolute. Furthermore, in H-like ions, the selection rule
L = 0 ↔ 0 is not allowed (5.2)
is also absolute since there is no violation of LS coupling for single electrons. Thus for example the 1s^2 − 1 s 2 s 1 S 0 transition in He-like ions, and the 1s − 2 s transition in H-like ions, are absolutely forbidden. However, the 2s levels of both He-like and H-like ions can decay by the emission of two photons, and in fact two-photon emission is the dominant mode of radiative decay from these levels.
with one photon in an angular frequency interval dω 1 :
dAU L(2-photon) (5.5)
L
4 ω 13 3 c^3 ℏ
4 ω 23 3 c^3 ℏ
I
ωIU + ω 1
〈φL|d 2 |φI 〉〈φI |d 1 |φU 〉 + (1 ↔ 2)
2 dω 1 2 π
where the summation is over all intermediate states I. If the upper state is 2s, the interme- diate states I must be p-states, by the dipole selection rules: 2p, 3p, .... The intermediate states include continuum as well as discrete states. Energy conservation requires that the sum of the two photon energies be equal to the transition energy ℏω,
ω 1 + ω 2 = ω. (5.6)
