Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
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Second Quantization 6.4 Complementarity 119
For a Cartesian component Q
u
of the quasispin Q,
CQ
u
C
−1
=−Q
u
. (6.55)
Thus, C is the analog of the time-reversal operator T,
for which
TL
u
T
−1
=−L
u
, (6.56)
TS
u
T
−1
=−S
u
. (6.57)
Both C and T are antiunitary; thus,
CiC
−1
=−i . (6.58)
6.3.5 Dependence on Electron Number
Application of the Wigner–Eckart theorem to matrix
elements whose component parts have well-defined
quasispin ranks yields the dependence of the matrix el-
ements on the electron number N [6.18, 19]. For κ + k
even and nonzero, the quasispin rank of W
(κk)
is 1, and
N
ψ||W
(κk)
||
N
ψ
=
(6.59)
(2 + 1 − N)
(2 + 1 − v)
v
ψ || W
(κk)
||
v
ψ
.
For κ + k odd, W
(κk)
is necessarily a quasispin scalar,
and the matrix elements are diagonal with respect to the
seniority and independent of N. These properties were
first stated in Eqs. (69) and (70) of [6.8].
Application of these ideas to single-electron cfp
yields, for states ψ and
¯
ψ with seniorities v and v + 1,
respectively,
N
ψ{|
N−1
¯
ψ
=
[(N − v)(v + 2)/2N]
1
2
v+2
ψ
v+1
¯
ψ
.
(6.60)
6.3.6 The Half-filled Shell
Selection rules for operators of good quasispin rank K,
taken between states of the half-filled shell (for which
M
Q
= 0), can be found by inspecting the 3– j symbol
QKQ
00 0
,
which appears when the Wigner-Eckart theorem is ap-
plied in quasispin space. This 3– j symbol vanishes
unless Q + K + Q
is even. An equivalent result can be
obtained for W
(κk)
by referring to (6.53) and insisting
that y be even.
6.4 Complementarity
6.4.1 Spin–Quasispin Interchange
The operator R formally interchanges spin and qua-
sispin. The result for the creation and annihilation
operators for electrons can be expressed in terms of
triple tensors:
Ra
(qs)
ξ
R
−1
= a
(qs)
η
, (6.61)
where ξ ≡ (m
q
m
s
m
) and η ≡ (m
s
m
q
m
).Fortheten-
sors X
(Kκk)
defined in (6.44), we get
RX
(Kκk)
λ
R
−1
= X
(κKk)
µ
, (6.62)
where λ ≡ (M
K
M
κ
M
k
) and µ ≡ (M
κ
M
K
M
k
). For states
of the shell,
R|γQM
Q
SM
S
=(−1)
t
|γSM
S
QM
Q
, (6.63)
where the quasispin of the ket on the right is S and the
spin is Q. The phase factor t depends on S and Q and
on phase choices made for the coefficients of fractional
parentage. The symbol γ denotes the additional labels
necessary to completely define the state in question,
including L and M
L
.
For every γ , Racah [6.20] observed that there are
two possible pairs (v
1
, S
1
) and (v
2
, S
2
) satisfying
v
1
+ 2S
2
= v
2
+ 2S
1
= 2 + 1 . (6.64)
From (6.37) it follows that
S
1
= Q
2
, S
2
= Q
1
. (6.65)
6.4.2 Matrix Elements
Application of the complementarity operator R to the
component parts of a matrix element leads to the equa-
tion
γQM
Q
SM
S
|X
(Kκk)
λ
|γ
Q
M
Q
S
M
S
=
(6.66)
(−1)
y
γSM
S
QM
Q
|X
(κKk)
µ
|γ
S
M
S
Q
M
Q
,
where λ and µ have the same significance as in (6.62),
and where y, like t of (6.63), depends on the spins and
quasispins but not on the associated magnetic quantum
numbers. Equation (6.66) leads to a useful special case
when M
K
= M
κ
= 0andthetensorsX are converted to
Part A 6.4
120 Part A Mathematical Methods
those of type W,definedin(6.18). The sum K + κ + k is
taken to be odd, with the scalars κ = k = 0andK = k = 0
excluded. Application of the Wigner-Eckart theorem to
the spin and orbital spaces yields
γQM
Q
S||W
(κk)
||γ
Q
M
Q
S
γSM
S
Q||W
(Kk)
||γ
S
M
S
Q
=
(−1)
z
QKQ
−M
Q
0 M
Q
S κ S
−M
S
0 M
S
,
(6.67)
where z = y+ Q − M
Q
− S+ M
S
. An equivalent form
is
N
γv
1
S
1
||W
(κk)
||
N
γ
v
1
S
1
N
γv
2
S
2
||W
(Kk)
||
N
γ
v
2
S
2
=
(6.68)
(−1)
z
1
2
(2 + 1 − v
1
) K
1
2
(2 + 1 − v
1
)
1
2
(2 + 1 − N) 0
1
2
(N − 2 − 1)
1
2
(2 + 1 − v
2
)κ
1
2
(2 + 1 − v
2
)
1
2
(2 + 1 − N
) 0
1
2
(N
− 2 − 1)
,
where (6.64) is satisfied both for the unprimed and
primed quantities.
6.5 Quasiparticles
Sets of linear combinations of the creation and an-
nihilation operators for electrons in the shell can be
constructed such that every member of one set anticom-
mutes with a member of a different set. To preserve the
tensorial character of these quasiparticle operators with
respect to L, it is convenient to define [6.21]
λ
†
q
= 2
−
1
2
a
†
1
2
,q
+ (−1)
−q
a
1
2
,−q
,
(6.69)
µ
†
q
= 2
−
1
2
a
†
1
2
,q
− (−1)
−q
a
1
2
,−q
,
(6.70)
ν
†
q
= 2
−
1
2
a
†
−
1
2
,q
+ (−1)
−q
a
−
1
2
,−q
,
(6.71)
ξ
†
q
= 2
−
1
2
a
†
−
1
2
,q
− (−1)
−q
a
−
1
2
,−q
.
(6.72)
The four tensors θ
†
(≡ λ
†
, µ
†
, ν
†
, or ξ
†
) anticommute
with each other; the first two act in the spin-up space,
the second two in the spin-down space. The tensors θ,
whose components
˜
θ
q
are defined as in (6.17) with t =
and m
t
= q, are related to their adjoints by the equations
λ
†
= λ , µ
†
=−µ , (6.73)
ν
†
= ν , ξ
†
=−ξ . (6.74)
Under the action of the complementarity operator R
(see (6.61)) [6.22],
RλR
−1
= λ , R µR
−1
= µ , (6.75)
Rν R
−1
= ν , R ξ R
−1
=−ξ . (6.76)
The tensors λ, µ,andν, for a given component q,form
a vector with respect to S + Q. Every component of ξ is
scalar with respect to S + Q [6.23].
The compound quasiparticle operators defined by
[6.21]
Θ
†
q
= 2
−
1
2
θ
†
q
,θ
†
0
,
(6.77)
where q > 0andθ ≡ λ, µ, ν,orξ satisfy the anticom-
mutation relations
Θ
†
q
,Θ
†
q
+
= 0 , (6.78)
Θ
q
,Θ
q
+
= 0 , (6.79)
Θ
†
q
,Θ
q
+
= δ(q, q
), (6.80)
for q, q
> 0. The Θ
†
q
with q > 0 can thus be regarded as
the creation operators for a fermion quasiparticle with
components.
The connection between the creation and annihi-
lation operators for quasiparticles and for quarks
(appearing in the last two rows of Table 3.1)is
θ → 2
(−1)/2
θ
γ
θ
q
†
θ
q
θ
(10...0)
, (6.81)
where the γ
θ
are Dirac matrices satisfying
γ
θ
γ
φ
+ γ
φ
γ
θ
= 2δ(θ, φ) , (6.82)
and the
θ
are phases, to some extent dependent on the
definitions (6.69–6.72)[6.24]. The superscript (10 ...0)
indicates that q
†
θ
and q
θ
each of which belongs to the
elementary spinor (
1
2
1
2
...
1
2
) of SO
θ
(2 + 1),aretobe
coupled to the resultant (10 ...0), which matches the
group label for θ. In the quark model, the 2
4+2
states
of the atomic shell are given by
q
†
λ
q
†
µ
q
†
ν
q
†
ξ
|0
pp
, (6.83)
Part A 6.5
Second Quantization References 121
where p and p
are parity labels that distinguish the four
reference states |0 corresponding to the evenness and
oddness of the number of spin-up and spin-down elec-
trons. The scalar nature of ξ (and hence of q
ξ
) with
respect to S+ Q can be used to derive relations be-
tween spin-orbit matrix elements that go beyond those
expected from an application of the Wigner–Eckart the-
orem [6.25].
References
6.1 E. K. U. Gross, E. Runge, O. Heinonen: Many-Particle
Theory (Hilger, New York 1991)
6.2 G. Racah: Phys. Rev. 62,438(1942)
6.3 E. U. Condon, G. H. Shortley: The Theory of Atomic
Spectra (Cambridge Univ. Press, New York 1935)
6.4 A. R. Edmonds: Angular Momentum in Quantum Me-
chanics (Princeton Univ. Press, Princeton 1957)
6.5 C. W. Nielson, G. F. Koster: Spectroscopic Coefficients
for the p
n
,d
n
,andf
n
Configurations (MIT Press,
Cambridge 1963)
6.6 R. Karazija, J. Vizbarait
˙
e, Z. Rudzikas, A. P. Jucys: Ta-
bles for the Calculation of Matrix Elements of Atomic
Operators (Academy Sci. Computing Center, Moscow
1967)
6.7 B. R. Judd: Second Quantization and Atomic Spec-
troscopy (Johns Hopkins, Baltimore 1967)
6.8 G. Racah: Phys. Rev. 63,367(1943)
6.9 V. L. Donlan: Air Force Materials Laboratory Report
No. AFML-TR-70-249 (Wright–Patterson Air Force
Base, Ohio 1970)
6.10 D. D. Velkov: Multi-Electron Coefficients of Fractional
Parentage for the p, d, and f Shells. Ph.D. The-
sis (The Johns Hopkins University, Baltimore 2000)
http://www.pha.jhu.edu/groups/cfp/
6.11 P. J. Redmond: Proc. R. Soc. London A222,84
(1954)
6.12 B. H. Flowers, S. Szpikowski: Proc. Phys. Soc. London
84,673(1964)
6.13 Z. Rudzikas: Theoretical Atomic Spectroscopy (Cam-
bridge Univ. Press, New York 1997)
6.14 G. Gaigalas, Z. Rudzikas, C. Froese Fischer: At. Data
Nucl. Data Tables 70, 1 (1998)
6.15 B. R. Judd, E. Lo, D. Velkov: Mol. Phys. 98, 1151 (2000),
Table 4
6.16 B. R. Judd: J. Phys. C 14, 375 (1981)
6.17 J. S. Bell: Nucl. Phys. 12, 117 (1959)
6.18 H. Watanabe: Prog. Theor. Phys. 32,106(1964)
6.19 R. D. Lawson, M. H. Macfarlane: Nucl. Phys. 66,80
(1965)
6.20 G. Racah: Phys. Rev. 76, 1352 (1949), Table I
6.21 L. Armstrong, B. R. Judd: Proc. R. Soc. London A 315,
27, 39 (1970)
6.22 B. R. Judd, S. Li: J. Phys. B 22, 2851 (1989)
6.23 B. R. Judd, G. M. S. Lister, M. A. Suskin: J. Phys. B 19,
1107 (1986)
6.24 B. R. Judd: Phys. Rep. 285, 1 (1997)
6.25 B. R. Judd, E. Lo: Phys. Rev. Lett. 85,948
(2000)
Part A 6
123
Density Matric
7. Density Matrices
The density operator was first introduced by J. von
Neumann [7.1] in 1927 and has since been widely
used in quantum statistics. Over the past decades,
however, the application of density matrices
has spread to many other fields of physics.
Density matrices have been used to describe, for
example, coherence and correlation phenomena,
alignment and orientation and their effect on the
polarization of emitted radiation, quantum beat
spectroscopy, optical pumping, and scattering
processes, particularly when spin-polarized
projectiles and/or targets are involved. A thorough
introduction to the theory of density matrices
and their applications with emphasis on atomic
physics can be found in the book by Blum [7.2]
from which many equations have been extracted
for use in this chapter.
7.1 Basic Formulae .................................... 123
7.1.1 Pure States ............................... 123
7.1.2 Mixed States ............................. 124
7.1.3 Expectation Values .................... 124
7.1.4 The Liouville Equation ............... 124
7.1.5 Systems in Thermal Equilibrium .. 125
7.1.6 Relaxation Processes ................. 125
7.2 Spin and Light Polarizations ................. 125
7.2.1 Spin-Polarized Electrons ............ 125
7.2.2 Light Polarization ...................... 125
7.3 Atomic Collisions ................................. 126
7.3.1 Scattering Amplitudes................ 126
7.3.2 Reduced Density Matrices........... 126
7.4 Irreducible Tensor Operators................. 127
7.4.1 Definition ................................ 127
7.4.2 Transformation Properties .......... 127
7.4.3 Symmetry Properties of State
Multipoles ................................ 128
7.4.4 Orientation and Alignment......... 128
7.4.5 Coupled Systems ....................... 129
7.5 Time Evolution of State Multipoles ........ 129
7.5.1 Perturbation Coefficients............ 129
7.5.2 Quantum Beats ......................... 129
7.5.3 Time Integration over Quantum
Beats ....................................... 130
7.6 Examples ............................................ 130
7.6.1 Generalized STU-parameters ...... 130
7.6.2 Radiation from Excited States:
Stokes Parameters ..................... 131
7.7 Summary ............................................ 133
References .................................................. 133
The main advantage of the density matrix formalism is
its ability to deal with pure and mixed states in the same
consistent manner. The preparation of the initial state as
well as the details regarding the observation of the final
state can be treated in a systematic way. In particular,
averages over quantum numbers of unpolarized beams in
the initial state and incoherent sums over non-observed
quantum numbers in the final state can be accounted
for via the reduced density matrix. Furthermore, expan-
sion of the density matrix in terms of irreducible tensor
operators and the corresponding state multipoles allows
for the use of advanced angular momentum techniques,
as outlined in Chapts. 2, 3 and 12. More details can be
found in two recent textbooks [7.3, 4].
7.1 Basic Formulae
7.1.1 Pure States
Consider a system in a quantum state that is represented
by a single wave function |Ψ . The density operator for
this situation is defined as
ρ =|Ψ Ψ | .
(7.1)
If |Ψ is normalized to unity, i.e., if
Ψ |Ψ =1 ,
(7.2)
then
ρ
2
= ρ. (7.3)
Part A 7
124 Part A Mathematical Methods
Equation (7.3) is the basic equation for identifying pure
quantum mechanical states represented by a density
operator.
Next, consider the expansion of |Ψ in terms of
a complete orthonormal set of basis functions {|Φ
n
},
i.e.,
|Ψ =
n
c
n
|Φ
n
. (7.4)
The density operator then becomes
ρ =
n,m
c
n
c
∗
m
|Φ
n
Φ
m
|=ρ
nm
|Φ
n
Φ
m
| , (7.5)
where the star denotes the complex conjugate
quantity. Note that the density matrix elements
ρ
nm
=Φ
n
|ρ |Φ
m
depend on the choice of the
basis and that the density matrix is Hermitian,
i.e.,
ρ
∗
mn
= ρ
nm
. (7.6)
Finally, if |Ψ =|Φ
i
is one of the basis functions,
then
ρ
mn
= δ
ni
δ
mi
, (7.7)
where δ
ni
is the Kronecker δ. Hence, the density mat-
rix is diagonal in this representation with only one
nonvanishing element.
7.1.2 Mixed States
The above concepts can be extended to treat statistic-
al ensembles of pure quantum states. In the simplest
case, such mixed states can be represented by a diagonal
density matrix of the form
ρ =
n
w
n
|Ψ
n
Ψ
n
| , (7.8)
where the weight w
n
is the fraction of systems in the
pure quantum state |Ψ
n
. The standard normalization
for the trace of ρ is
Tr{ρ}=
n
w
n
= 1 . (7.9)
Since the trace is invariant under unitary transformations
of the basis functions, (7.9) also holds if the |Ψ
n
states
themselves are expanded in terms of basis functions as
in (7.4). For a pure state and the normalization (7.9), one
finds in an arbitrary basis
Tr{ρ}=Tr
ρ
2
= 1 .
(7.10)
7.1.3 Expectation Values
The density operator contains the maximum available in-
formation about a physical system. Consequently, it can
be used to calculate expectation values for any operator
A that represents a physical observable. In general,
A=Tr{Aρ}/Tr{ρ} ,
(7.11)
where Tr{ρ} in the denominator of (7.11) ensures the
correct result even for a normalization that is different
from (7.9). The invariance of the trace operation ensures
the same result – independent of the particular choice of
the basis representation.
7.1.4 The Liouville Equation
Suppose (7.8) is valid for a time t = 0. If the functions
|Ψ
n
(r, t) obey the Schrödinger equation, i.e.
i
∂
∂t
|Ψ
n
(r, t)=H(t) |Ψ
n
(r, t) , (7.12)
the density operator at the time t can be written as
ρ(t) =U(t)ρ(0) U
†
(t). (7.13)
In (7.13), U(t) is the time evolution operator which re-
lates the wave functions at times t = 0andt according
to
|Ψ
n
(r, t)=U(t) |Ψ
n
(r, 0) , (7.14)
and U
†
(t) denotes its adjoint. Note that
U(t) = e
−iHt
, (7.15)
if the Hamiltonian H is time-independent.
Differentiation of (7.13) with respect to time and in-
serting (7.14) into the Schrödinger equation (7.12) yields
the equation of motion
i
∂
∂t
ρ(t) =[H(t), ρ(t)] ,
(7.16)
where [A, B ] denotes a commutator.
The Liouville equation (7.16) can be used to deter-
mine the density matrix and to treat transitions from
nonequilibrium to equilibrium states in quantum mech-
anical systems. Especially for approximate solutions in
the presence of small time-dependent perturbation terms
in an otherwise time-independent Hamiltonian, i.e., for
H(t) = H
0
+V(t), (7.17)
the interaction picture is preferably used. The Liouville
equation then becomes
i
∂
∂t
ρ
I
(t) =
V
I
(t), ρ
I
(t)
, (7.18)
Part A 7.1
Density Matrices 7.2 Spin and Light Polarizations 125
where the subscript I denotes the operator in the inter-
action picture. In first-order perturbation theory, (7.18)
can be integrated to yield
ρ
I
(t) = ρ
I
(0) −i
t
0
[V
I
(τ), ρ
I
(0)] dτ, (7.19)
and higher-order terms can be obtained through subse-
quent iterations.
7.1.5 Systems in Thermal Equilibrium
According to quantum statistics, the density operator
for a system which is in thermal equilibrium with a
surrounding reservoir R at a temperature T (canonical
ensemble), can be expressed as
ρ =
exp(−βH)
Z
,
(7.20)
where H is the Hamiltonian, and β = 1/k
B
T with k
B
being the Boltzmann constant. The partition sum
Z = Tr
exp(−βH)
,
(7.21)
ensures the normalization condition (7.9). Expectation
values are calculated according to (7.11), and extensions
to other types of ensembles are straightforward.
7.1.6 Relaxation Processes
Transitions from nonequilibrium to equilibrium states
can also be described within the density matrix for-
malism. One of the basic problems is to account for
irreversibility in the energy (and sometimes particle)
exchange between the system of interest, S,andthe
reservoir, R . This is usually achieved by assuming
that the interaction of the system with the reservoir is
negligible and, therefore, the density matrix represen-
tation for the reservoir at any time t is the same as the
representation for t = 0.
Another important assumption that is frequently
made is the Markov approximation. In this approxi-
mation, one assumes that the system “forgets” all
knowledge of the past, so that the density matrix elem-
ents at the time t +∆t depend only on the values of
these elements, and their first derivatives, at the time t.
When (7.19) is put back into (7.18), the result in the
Markov approximation can be rewritten as
∂
∂t
ρ
SI
(t) =−iTr
R
V
I
(t), ρ
SI
(0)ρ
R
(0)
−
t
0
dτTr
R
V
I
(t),[V
I
(τ),ρ
SI
(t)ρ
R
(0)]
,
(7.22)
where Tr
R
denotes the trace with regard to all variables
of the reservoir. Note that the integral over dτ contains
the system density matrix in the interaction picture, ρ
SI
,
at the time t, rather than at all times τ which are inte-
grated over (the Markov approximation), and that the
density matrix for the reservoir is taken as ρ
R
(0) at all
times. For more details, see Chapter 7 of Blum [7.2]and
references therein.
Equations such as (7.22) are the basis for the master
or rate equation approach used, for example, in quantum
optics for the theory of lasers and the coupling of atoms
to cavity modes. For more details, see Chapts. 68, 69,
70 and 78.
7.2 Spin and Light Polarizations
Density matrices are frequently used to describe the
polarization state of spin-polarized particle beams as
well as light. The latter can either be emitted from ex-
cited atomic or molecular ensembles or can be used, for
example, for laser pumping purposes.
7.2.1 Spin-Polarized Electrons
The spin polarization of an electron beam with respect
to a given quantization axis
ˆ
n is defined as [7.5]
P
ˆ
n
=
N
↑
−N
↓
N
↑
+N
↓
, (7.23)
where N
↑
(N
↓
) is the number of electrons with spin up
(down) with regard to this axis. An arbitrary polarization
state is described by the density matrix
ρ =
1
2
1 + P
z
P
x
−iP
y
P
x
+iP
y
1 − P
z
,
(7.24)
where P
x,y,x
are the cartesian components of the spin
polarization vector. The individual components can be
obtained from the density matrix as
P
i
= Tr{σ
i
ρ} , (7.25)
where the σ
i
(i = x, y, z) are the standard Pauli spin
matrices.
Part A 7.2
126 Part A Mathematical Methods
7.2.2 Light Polarization
Another important use of the density matrix formalism
is the description of light polarization in terms of the so-
called Stokes parameters [7.6]. For a given direction of
observation, the general polarization state of light can be
fully determined by the measurement of one circular and
two independent linear polarizations. Using the notation
of Born and Wolf [7.7], the density matrix is given by
ρ =
I
tot
2
1 − P
3
P
1
−iP
2
P
1
+iP
2
1 + P
3
,
(7.26)
where P
1
and P
2
are linear light polarizations while P
3
is the circular polarization (see also Sect. 7.6). In (7.26),
the density matrix is normalized in such as way that
Tr{ρ}=I
tot
, (7.27)
where I
tot
is the total light intensity. Other frequently
used names for the various Stokes parameters are
P
1
= η
3
= M , (7.28)
P
2
= η
1
= C , (7.29)
P
3
=−η
2
= S . (7.30)
The Stokes parameters of electric dipole radiation can be
related directly to the charge distribution of the emitting
atomic ensemble. As discussed in detail in Chapt. 46,
one finds, for example,
L
⊥
=−P
3
(7.31)
for the angular momentum transfer perpendicular to the
scattering plane in collisional (de-)excitation, and
γ =
1
2
arg{P
1
+iP
2
} (7.32)
for the alignment angle.
7.3 Atomic Collisions
7.3.1 Scattering Amplitudes
Transitions from an initial state |J
0
M
0
;k
0
m
0
to a final
state |J
1
M
1
;k
1
m
1
are described by scattering ampli-
tudes
f(M
1
m
1
; M
0
m
0
) =J
1
M
1
;k
1
m
1
|T |J
0
M
0
;k
0
m
0
,
(7.33)
where T is the transition operator. Furthermore, J
0
(J
1
)
is the total electronic angular momentum in the initial
(final) state of the target and M
0
(M
1
) its corresponding
z-component, while k
0
(k
1
) is the initial (final) momen-
tum of the projectile and m
0
(m
1
) its spin component.
7.3.2 Reduced Density Matrices
While the scattering amplitudes are the central elements
in a theoretical description, some restrictions usually
need to be taken into account in a practical experiment.
The most important ones are: (i) there is no “pure” ini-
tial state, and (ii) not all possible quantum numbers are
simultaneously determined in the final state. The solu-
tion to this problem can be found by using the density
matrix formalism. First, the complete density operator
after the collision process is given by [7.2]
ρ
out
= T ρ
in
T
†
, (7.34)
where ρ
in
is the density operator before the collision.
The corresponding matrix elements are given by
(
ρ
out
)
k
1
,M
1
M
1
m
1
m
1
=
m
0
m
0
M
0
M
0
ρ
m
0
m
0
ρ
M
0
M
0
× f
M
1
m
1
; M
0
m
0
× f
∗
M
1
m
1
; M
0
m
0
,
(7.35)
where the term ρ
m
0
m
0
ρ
M
0
M
0
describes the preparation of
the initial state (i). Secondly, “reduced” density matrices
account for (ii). For example, if only the scattered pro-
jectiles are observed, the corresponding elements of the
reduced density matrix are obtained by summing over
the atomic quantum numbers as follows:
(
ρ
out
)
k
1
m
1
m
1
=
M
1
(
ρ
out
)
k
1
,M
1
M
1
m
1
m
1
. (7.36)
The differential cross section for unpolarized projectile
and target beams is given by
dσ
dΩ
= C
m
1
(
ρ
out
)
k
1
m
1
m
1
, (7.37)
where C is a constant that depends on the normalization
of the continuum waves in a numerical calculation.
On the other hand, if only the atoms are observed
(for example, by analyzing the light emitted in optical
transitions), the elements
(
ρ
out
)
M
1
M
1
=
d
3
k
1
m
1
(
ρ
out
)
k
1
,M
1
M
1
m
1
m
1
(7.38)
Part A 7.3
Density Matrices 7.4 Irreducible Tensor Operators 127
determine the integrated Stokes parameters [7.8,9], i.e.,
the polarization of the emitted light. They contain in-
formation about the angular momentum distribution in
the excited target ensemble.
Finally, for electron–photon coincidence experi-
ments without spin analysis in the final state, the
elements
(
ρ
out
)
k
1
M
1
M
1
=
m
1
(
ρ
out
)
k
1
,M
1
M
1
m
1
m
1
(7.39)
simultaneously contain information about the project-
iles and the target. This information can be extracted
by measuring the angle-differential Stokes parameters.
In particular, for unpolarized electrons and atoms, the
“natural coordinate system”, where the quantization
axis coincides with the normal to the scattering plane,
allows for a simple physical interpretation of the various
parameters [7.10] (see Chapt. 46).
The density matrix formalism outlined above
is very useful for obtaining a qualitative descrip-
tion of the geometrical and sometimes also of
the dynamical symmetries of the collision pro-
cess [7.11]. Two explicit examples are discussed
in Sect. 7.6.
7.4 Irreducible Tensor Operators
The general density matrix theory can be formulated
in a very elegant fashion by decomposing the density
operator in terms of irreducible components whose ma-
trix elements then become the state multipoles. In such
a formulation, full advantage can be taken of the most
sophisticated techniques developed in angular momen-
tum algebra (see Chapt. 2). Many explicit examples can
be found in [7.3, 4].
7.4.1 Definition
The density operator for an ensemble of particles in
quantum states labeled as |JM where J and M are the
total angular momentum and its magnetic component,
respectively, can be written as
ρ =
J
JM
M
ρ
J
J
M
M
|J
M
JM| , (7.40)
where
ρ
J
J
M
M
=J
M
|ρ|JM (7.41)
are the matrix elements. (For simplicity, interactions out-
side the single manifold of momentum states |JM are
neglected). Alternatively, one may write
ρ =
J
JKQ
T
J
J
†
KQ
T
J
J
KQ
, (7.42)
where the irreducible tensor operators are defined in
terms of 3–j symbols as
T
J
J
KQ
=
M
M
(−1)
J
−M
√
2K +1
×
J
JK
M
−M −Q
|J
M
JM| ,
(7.43)
and the state multipoles or statistical tensors are given
by
T
J
J
†
KQ
=
M
M
(−1)
J
−M
√
2K +1
×
J
JK
M
−M −Q
J
M
|ρ|JM .
(7.44)
Hence, the selection rules for the 3–j symbols imply
that
|J − J
|≤K ≤ J + J
, (7.45)
M
−M = Q . (7.46)
Equation (7.44) can be inverted through the orthogonal-
ity condition of the 3–j symbols to give
J
M
|ρ|JM=
KQ
(−1)
J
−M
√
2K +1
×
J
JK
M
−M −Q
T
J
J
†
KQ
.
(7.47)
7.4.2 Transformation Properties
Suppose a coordinate system (X
2
, Y
2
, Z
2
) is obtained
from another coordinate system (X
1
, Y
1
, Z
1
) through
a rotation by a set of three Euler angles (γ,β,α) as
defined in Edmonds [7.12]. The irreducible tensor oper-
ators (7.43) defined in the (X
1
, Y
1
, Z
1
) system are then
related to the operators T(J
J)
†
KQ
in the (X
2
, Y
2
, Z
2
)
Part A 7.4
128 Part A Mathematical Methods
system by
T
J
J
KQ
=
q
T
J
J
Kq
D(γ,β,α)
K
qQ
, (7.48)
where
D(γ,β,α)
J
M
M
= e
iM
γ
d(β)
J
M
M
e
iMα
(7.49)
is a rotation matrix (see Chapt. 2). Note that the rank K
of the tensor operator is invariant under such rotations.
Similarly,
T
J
J
†
KQ
=
q
T
J
J
†
Kq
D(γ,β,α)
K
qQ
∗
(7.50)
holds for the state multipoles.
The irreducible tensor operators fulfill the orthogon-
ality condition
Tr
T
J
J
KQ
T
J
J
†
K
Q
= δ
K
K
δ
Q
Q
, (7.51)
with
T
J
J
00
=
1
√
2J +1
δ
J
J
1 (7.52)
being proportional to the unit operator 1, it follows that
all tensor operators have vanishing trace, except for the
monopole T(J
J)
00
.
Reduced tensor operators fulfill the Wigner–Eckart
theorem (see Sect. 2.8.4)
J
M
|T
J
J
KQ
|JM
= (−1)
J
−M
J
KJ
−M
QM
×
J
T
K
J
, (7.53)
where the reduced matrix element is simply given by
J
T
K
J
=
1
√
2K +1
.
(7.54)
7.4.3 Symmetry Properties
of State Multipoles
The Hermiticity condition for the density matrix implies
T
J
J
†
KQ
∗
= (−1)
J
−J+Q
T( JJ
)
†
K−Q
,
(7.55)
which, for sharp angular momentum J
= J, yields
T( J)
†
KQ
∗
= (−1)
Q
T( J)
†
K−Q
.
(7.56)
Hence, the state multipoles
T( J)
†
K0
are real numbers.
Furthermore, the transformation property (7.50)of
the state multipoles imposes restrictions on nonvan-
ishing state multipoles to describe systems with given
symmetry properties. In detail, one finds:
1. For spherically symmetric systems,
T
J
J
†
KQ
=
T
J
J
†
KQ
rot
(7.57)
for all sets of Euler angles. This implies that only the
monopole term
T( J)
†
00
can be different from zero.
2. For axially symmetric systems,
T
J
J
†
KQ
=
T
J
J
†
KQ
rot
(7.58)
for all Euler angles φ that describe a rotation around the
z-axis. Since this angle enters via a factor exp(−iQφ)
into the general transformation formula (7.50), it follows
that only state multipoles with Q = 0, i.e.,
T
J
J
†
K0
,
can be different from zero in such a situation.
3. For planar symmetric systems with fixed J
= J,
T( J)
†
KQ
= (−1)
K
T( J)
†
KQ
∗
(7.59)
if the system properties are invariant under reflection in
the xz-plane. Hence, state multipoles with even rank K
are real numbers, while those with odd rank are purely
imaginary in this case.
The above results can be applied immediately to the
description of atomic collisions where the incident beam
axis is the quantization axis (the so-called “collision sys-
tem”). For example, impact excitation of unpolarized
targets by unpolarized projectiles without observation
of the scattered projectiles is symmetric both with re-
gard to rotation around the incident beam axis and with
regard to reflection in any plane containing this axis.
Consequently, the state multipoles
T( J)
†
00
,
T( J)
†
20
,
T( J)
†
40
,... fully characterize the atomic ensemble of
interest. Using (7.50), similar relationships can be de-
rived for state multipoles defined with regard to other
coordinate systems, such as the “natural system” where
the quantization axis coincides with the normal vector
to the scattering plane (see Chapt. 46).
7.4.4 Orientation and Alignment
From the above discussion, it is apparent that the
description of systems that do not exhibit spherical
symmetry requires the knowledge of state multipoles
with rank K = 0. Frequently, the multipoles with K =1
and K = 2 are determined via the angular correla-
tion and the polarization of radiation emitted from
Part A 7.4
Density Matrices 7.5 Time Evolution of State Multipoles 129
an ensemble of collisionally excited targets. The state
multipoles with K = 1 are proportional to the spheri-
cal components of the angular momentum expectation
value and, therefore, give rise to a nonvanishing
circular light polarization (see also Sect. 7.6). This
corresponds to a sense of rotation or an orientation
in the ensemble which is therefore called oriented
(see Sect. 46.1).
On the other hand, nonvanishing multipoles with
rank K = 2 describe the alignment of the system.
Some authors, however, use the terms “alignment”
or “orientation” synonymously for all nonvanishing
state multipoles with ranks K = 0, thereby de-
scribing any system with anisotropic occupation of
magnetic sublevels as “aligned” or “oriented”. For
details on alignment and orientation, see Chapt. 46
and [7.3, 4].
7.4.5 Coupled Systems
Tensor operators and state multipoles for coupled sys-
tems are constructed as direct products (⊗)ofthe
operators for the individual systems. For example, the
density operator for two subsystems in basis states
|L, M
L
and |S, M
S
is constructed as [7.2]
ρ =
KQkq
T(L)
†
KQ
⊗T(S)
†
kq
T(L)
KQ
⊗T(S)
kq
.
(7.60)
If the two systems are uncorrelated, the state multipoles
factor as
T(L)
†
KQ
⊗T(S)
†
kq
=
T(L)
†
KQ
T(S)
†
kq
;
(7.61)
More generally, irreducible representations of coupled
operators can be defined in terms of a 9–j symbol as
T
J
, J
K
Q
=
KQkq
ˆ
K
ˆ
k
ˆ
J
ˆ
J
KQ, kq|K
Q
×
KkK
LSJ
LSJ
T(L)
KQ
⊗T(S)
kq
,
(7.62)
where
ˆ
x ≡
√
2x +1, and ( j
1
m
1
, j
2
m
2
|j
3
m
3
) is a stand-
ard Clebsch-Gordan coefficient.
7.5 Time Evolution of State Multipoles
7.5.1 Perturbation Coefficients
From the general expansions
ρ(t) =
j
jkq
T
j
j;t
†
kq
T
j
j
kq
(7.63)
in terms of irreducible components, together with (7.42)
for time t = 0and(7.13) for the time development of
the density operator, it follows that
T
j
j;t
†
kq
=
J
JKQ
T
J
J;0
†
KQ
× G
J
J, j
j;t
Qq
Kk
, (7.64)
where the perturbation coefficients are defined as
G
J
J, j
j;t
Qq
Kk
= Tr
U(t)T
J
J
KQ
U(t)
†
T( j
j)
†
kq
.
(7.65)
Hence, these coefficients relate the state multipoles at
time t to those at t = 0.
7.5.2 Quantum Beats
An important application of the perturbation coefficients
is the coherent excitation of several quantum states
which subsequently decay by optical transitions. Such
an excitation may be performed, for example, in beam-
foil experiments or electron–atom collisions where the
energy width of the electron beam is too large to resolve
the fine structure (or hyperfine structure) of the target
states.
Suppose, for instance, that explicitly relativistic ef-
fects, such as the spin–orbit interaction between the
projectile and the target, can be neglected during acol-
lision process between an incident electron and a target
atom. In that case, the orbital angular momentum (L)
system of the collisionally excited target states may be
oriented, depending on the scattering angle of the pro-
jectile. On the other hand, the spin (S) system remains
unaffected (unpolarized), provided that both the target
and the projectile beams are unpolarized. During the
lifetime of the excited target states, however, the spin–
orbit interaction within the target produces an exchange
of orientation between the L and the S systems, which
results in a net loss of orientation in the L system.
Part A 7.5