PHYSICAL REVIEW A, VOLUME 65, 032112
Larmor diamagnetism and Van Vleck paramagnetism in relativistic quantum theory:
The Gordon decomposition approach
Radosław Szmytkowski*
Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics,
Technical University of Gdańsk, Narutowicza 11/12, PL 80-952 Gdańsk, Poland
共Received 18 June 2001; published 20 February 2002兲
We consider a charged Dirac particle bound in a scalar potential perturbed by a classical magnetic field
derivable from a vector potential A(r). Using a procedure based on the Gordon decomposition of a fieldinduced current, we identify diamagnetic and paramagnetic contributions to the second-order perturbationtheory correction to the particle’s energy. In contradiction to earlier findings, based on the sum-over-states
2
(0)
approach, it is found that the resulting diamagnetic term is E(2)
兩 ␤ A2 ⌿ (0) 典 , where ⌿ (0) (r) is an
d ⫽(q /2m) 具 ⌿
unperturbed eigenstate and ␤ is the matrix associated with the rest-energy term in the Dirac Hamiltonian.
DOI: 10.1103/PhysRevA.65.032112
PACS number共s兲: 03.65.Pm, 75.20.⫺g, 31.30.Jv
I. INTRODUCTION
A nonrelativistic quantum-mechanical problem of a spinparticle of electric charge q bound in a potential V(r)
perturbed by a classical static magnetic field derivable from a
vector potential A(r) is frequently encountered in physics. It
is well known 关1–3兴 that if the perturbing magnetic field is
weak and the potential A(r) has no singularities, the firstorder perturbation-theory correction to an unperturbed energy level E (0) associated with an unperturbed wave function
␺ (0) (r) may be expressed in the form
1
2
E (1) ⫽ 具 ␺ (0) 兩 Ĥ (1) ␺ (0) 典 ,
共1.1兲
and the second-order one in the form
(2)
E (2) ⫽E (2)
d ⫹E p ,
共1.2兲
with the contributions given by
(0)
E (2)
兩 Ĥ (2) ␺ (0) 典
d ⫽具␺
共1.3兲
(0)
兩 Ĥ (1) Ĝ (0) Ĥ (1) ␺ (0) 典 .
E (2)
p ⫽⫺ 具 ␺
共1.4兲
The contribution E (2)
d is clearly positive and is called a diamagnetic 共or Larmor兲 component of E (2) . If E (0) is the
is negaground-state energy of Ĥ (0) , the contribution E (2)
p
tive; it is called a paramagnetic component of E (2) and is
usually associated with the name of Van Vleck 关1,3兴. It is to
be mentioned that since the vector potential A(r) is gauge
and E (2)
is not
dependent, the splitting of E (2) into E (2)
d
p
(2)
(1)
unique. Still, E and E are gauge invariant.
Feneuille 关4兴 and, more recently, Aucar et al. 关5兴 共cf. also
Ref. 关6兴兲 attempted to explain the origin of diamagnetism
within the framework of the Dirac relativistic quantum mechanics. Considerations based on the sum-over-states approach led these authors to the conclusion that diamagnetism
had to be attributed to the ‘‘redressing’’ of a relativistic particle by a perturbing magnetic field. It was claimed that in
the relativistic theory the diamagnetic contribution to the
second-order energy expression was due to the negativeenergy Dirac sea and was given by the following formula:
E (2)
d ⫽
and
Here Ĝ (0) is the generalized Green operator of the unperturbed energy eigenproblem, associated with the energy level
E (0) , and the particle-field interaction operators Ĥ (1) and
Ĥ (2) are
Ĥ (1) ⫽
iqប
qប
␴•B共 r兲
关 “•A共 r兲 ⫹A共 r兲 •“ 兴 ⫺
2m
2m
共1.5兲
关with B(r)⫽curl A(r) and ␴ denoting the vector composed
of the Pauli matrices兴 and
Ĥ (2) ⫽
q2 2
A 共 r兲 .
2m
*Electronic address: radek@mif.pg.gda.pl
1050-2947/2002/65共3兲/032112共8兲/$20.00
共1.6兲
q2
具 ⌿ (0) 兩 A2 ⌿ (0) 典 ,
2m
共1.7兲
formally identical with its nonrelativistic counterpart 共1.3兲
关7兴. Here ⌿ (0) (r) is a four-component eigenfunction of the
zeroth-order 共i.e., with the magnetic field switched off兲 Dirac
Hamiltonian.
In the present paper, we reconsider the problem of determining diamagnetic and paramagnetic contributions to the
second-order energy correction within the framework of the
single-particle Dirac theory. Our approach is entirely different from that proposed in Refs. 关4,5兴. Instead of utilizing the
sum-over-states scheme, we find a field-induced electric current in a linear-response approximation 关8 –12兴 and identify
its diamagnetic and paramagnetic components. This is
achieved using an ingenious 共but, regrettably, not appreciated
enough兲 procedure devised by Gordon 关13–15兴 in the very
early days of relativistic quantum mechanics. Subsequent use
of the decomposed current in a formula linking the secondorder energy correction E (2) with a vector potential and the
induced current allows us to identify diamagnetic and paramagnetic contributions to the former. Somewhat surprisingly,
65 032112-1
©2002 The American Physical Society
RADOSL” AW SZMYTKOWSKI
PHYSICAL REVIEW A 65 032112
our procedure leads us to the conclusion that E (2)
is not
d
given by Eq. 共1.7兲 but rather has the form
q2
⫽
E (2)
具 ⌿ (0) 兩 ␤ A2 ⌿ (0) 典 ,
d
2m
共1.8兲
where ␤ is the matrix associated with the rest-energy term in
the Dirac Hamiltonian 关notice that both expressions 共1.7兲 and
共1.8兲 have the same nonrelativistic limit兴.
Searching through the literature of the subject, we found
that our idea of applying the Gordon decomposition to determine diamagnetic and paramagnetic contributions to magnetic properties was anticipated by a similar idea of Pyper
who exploited it in the context of the theories of nuclear
shielding 关16 –19兴 and hyperfine interaction 关20兴. There are,
however, two main differences between Pyper’s and our
work. First, somewhat different physical problems are considered. Second, different procedures are used: while Pyper
used the sum-over-states approach performing decompositions of virtual transition currents between unperturbed
eigenstates of the zeroth-order Dirac Hamiltonian, we apply
the linear-response approach decomposing the field-induced
current.
关 Ĥ(0) ⫺E (0) 兴 ⌿ (0) 共 r兲 ⫽0
共for the sake of brevity, henceforth we shall assume that the
eigenvalue E (0) is nondegenerate兲, we may seek corresponding eigensolutions to the problem 共2.1兲 in the form
共2.7兲
E⫽E (0) ⫹E (1) ⫹E (2) ⫹•••.
共2.8兲
关 Ĥ(0) ⫺E (0) 兴 ⌿ (1) 共 r兲 ⫽⫺ 关 Ĥ(1) ⫺E (1) 兴 ⌿ (0) 共 r兲 ,
共2.9兲
关 Ĥ(0) ⫺E (0) 兴 ⌿ (2) 共 r兲 ⫽⫺ 关 Ĥ(1) ⫺E (1) 兴 ⌿ (1) 共 r兲
⫹E (2) ⌿ (0) 共 r兲 ,
共2.10兲
which are to be supplemented by pertinent boundary conditions. Imposing the constraints
具 ⌿ (0) 兩 ⌿ (0) 典 ⫽1,
具 ⌿ (0) 兩 ⌿ (1) 典 ⫽0,
共2.11兲
from Eqs. 共2.9兲 and 共2.10兲 we infer
E (1) ⫽ 具 ⌿ (0) 兩 Ĥ(1) ⌿ (0) 典 ,
共2.12兲
E (2) ⫽ 具 ⌿ (0) 兩 Ĥ(1) ⌿ (1) 典 .
共2.13兲
Formally, a solution to Eq. 共2.9兲 is
Consider a relativistic Dirac particle of electric charge q
共for an electron q⫽⫺e) bound in a field of force derivable
from a potential V(r) 共not necessarily of an electromagnetic
origin兲 and perturbed by a classical static magnetic field
characterized by a real nonsingular vector potential A(r).
The time-independent wave equation for the particle has the
form
关 Ĥ⫺E兴 ⌿ 共 r兲 ⫽0
⌿ 共 r兲 ⫽⌿ (0) 共 r兲 ⫹⌿ (1) 共 r兲 ⫹⌿ (2) 共 r兲 ⫹•••,
From this, in the standard way we obtain equations
II. RELATIVISTIC THEORY OF FIRST- AND SECONDORDER MAGNETIC CORRECTIONS TO ENERGY
A. Generalities
共2.6兲
共2.1兲
⌿ (1) 共 r兲 ⫽⫺Ĝ (0) Ĥ(1) ⌿ (0) 共 r兲 .
共2.14兲
where Ĝ (0) 关possessing the property Ĝ (0) ⌿ (0) (r)⫽0兴 is the
generalized Green operator for the zeroth-order Dirac Hamiltonian Ĥ(0) , associated with the eigenvalue E (0) of the latter.
The operator Ĝ (0) has an integral kernel G (0) (r,r⬘ ) which is
a solution to the equation
关 Ĥ(0) ⫺E (0) 兴 G (0) 共 r,r⬘ 兲 ⫽I ␦ 共 r⫺r⬘ 兲 ⫺⌿ (0) 共 r兲 ⌿ (0)† 共 r⬘ 兲
共2.15兲
with the Hamiltonian
Ĥ⫽c ␣• 关 ⫺iប“⫺qA共 r兲兴 ⫹ ␤ mc 2 ⫹V 共 r兲 .
共2.2兲
Here ␣ and ␤ are standard Dirac matrices. The Hamiltonian
Ĥ may be conveniently written as
Ĥ⫽Ĥ ⫹Ĥ ,
共2.3兲
Ĥ(0) ⫽⫺icប ␣•“⫹ ␤ mc 2 ⫹V 共 r兲 ,
共2.4兲
Ĥ(1) ⫽⫺qc ␣•A共 r兲 .
共2.5兲
(0)
(1)
where
Henceforth, we shall assume that the magnetic field is weak,
which implies that the operator Ĥ(1) may be treated as a
small perturbation of Ĥ(0) . Then, if ⌿ (0) (r) and E (0) are
particular eigensolutions to the zeroth-order Dirac eigenproblem
共here I is the 4⫻4 unit matrix兲 supplemented by the same
boundary conditions that have been imposed on ⌿ (0) (r). On
substituting the solution 共2.14兲 into Eq. 共2.13兲, we find
E (2) ⫽⫺ 具 ⌿ (0) 兩 Ĥ(1) Ĝ (0) Ĥ(1) ⌿ (0) 典 .
共2.16兲
In principle, at this stage the problem of finding the firstand the second-order corrections to E (0) may be considered
as solved since Eq. 共2.12兲 and, depending on the particular
computational technique used, either Eq. 共2.13兲 or Eq. 共2.16兲
are suitable for computational purposes. Still, on a purely
aesthetic ground, one may be slightly dissatisfied with the
above results. First, Eqs. 共1.1兲 and 共2.12兲 are only seemingly
similar since the operators Ĥ (1) and Ĥ(1) have completely
different structures 关cf. Eqs. 共1.5兲 and 共2.5兲兴. Second, while
in the nonrelativistic theory the second-order correction is
the sum of the diamagnetic and paramagnetic parts 关cf. Eqs.
共1.2兲–共1.4兲兴, in the relativistic case the second-order correction has the compact form 共2.16兲 which is, but only superfi-
032112-2
LARMOR DIAMAGNETISM AND VAN VLECK . . .
PHYSICAL REVIEW A 65 032112
cially 关cf. again Eqs. 共1.5兲 and 共2.5兲兴, similar to the nonrelativistic paramagnetic term 共1.4兲. Thus, it is natural to ask the
question: is it possible to transform the relativistic corrections 共2.12兲 and 共2.16兲 into expressions closely resembling
their nonrelativistic counterparts? Below we shall show that
the answer to this question is affirmative.
Comparison of Eqs. 共1.1兲, 共1.5兲, 共2.23兲, and 共2.24兲 shows
their remarkable similarity. Thus, in the case of the firstorder correction we have reached our goal.
Encouraged by the success achieved in the first-order
case, we shall attempt to apply a similar procedure in the
second-order case. On substituting the perturbation expansion 共2.7兲 into the expression
B. The Gordon decomposition approach
J共 r兲 ⫽qc⌿ † 共 r兲 ␣⌿ 共 r兲 ,
At first we focus on the first-order correction. The starting
point is to rewrite Eq. 共2.12兲 using the notion of the particle’s
current. According to the Dirac’s theory, the current in the
state ⌿ (0) (r) is
J(0) 共 r兲 ⫽qc⌿ (0)† 共 r兲 ␣⌿ (0) 共 r兲 .
E
⫽⫺
冕
d rA共 r兲 •J 共 r兲 .
3
R3
(0)
defining the particle’s current in the state ⌿(r), and collecting terms of the same order in the perturbation, we obtain
共2.17兲
共2.18兲
In the next step, following Gordon 关13–15兴, we rewrite Eq.
共2.17兲 in the form
the unperturbed Dirac equation 共2.6兲 in the form
iប
E (0) ⫺V 共 r兲
␤␣•“⌿ (0) 共 r兲 ⫹
␤ ⌿ (0) 共 r兲 ,
mc
mc 2
共2.20兲
and substitute Eq. 共2.20兲 into the first term on the right-hand
side of Eq. 共2.19兲 and its Hermitian conjugate into the second term. After some elementary movements exploiting
properties of the Dirac matrices ␣ and ␤ , we arrive at
J(0) 共 r兲 ⫽
qប
Im关 ⌿ (0)† 共 r兲 ␤ “⌿ (0) 共 r兲兴
m
⫹
qប
“⫻ 关 ⌿ (0)† 共 r兲 ␤ ⌺⌿ (0) 共 r兲兴 ,
2m
⌺⫽
冉 冊
0
0
␴
共2.21兲
共2.27兲
冕
R3
d 3 rA共 r兲 •J(1) 共 r兲 .
共2.28兲
The decomposition of J(1) (r) analogous to that in Eq.
共2.21兲 is achieved by performing the Gordon decomposition
of the current J(r) and making use of Eq. 共2.26兲. After steps
completely analogous to those which have led us to Eq.
共2.21兲, we arrive at
J共 r兲 ⫽
qប
qប
Im关 ⌿ † 共 r兲 ␤ “⌿ 共 r兲兴 ⫹
“⫻ 关 ⌿ † 共 r兲 ␤ ⌺⌿ 共 r兲兴
m
2m
⫺
q2
A共 r兲 ⌿ † 共 r兲 ␤ ⌿ 共 r兲 .
m
共2.29兲
Then the desired form of J(1) (r) is obtained by substituting
the expansion 共2.7兲 into Eq. 共2.29兲, subtracting Eq. 共2.21兲
from the result, and retaining only first-order terms. This
yields
(1)
J(1) 共 r兲 ⫽J(1)
d 共 r 兲 ⫹Jp 共 r 兲
共2.30兲
with
.
共2.22兲
Now, substitution of the decomposed current 共2.21兲 into Eq.
共2.18兲 leads, after elementary integration by parts, to the expression
E (1) ⫽ 具 ⌿ (0) 兩 Ĥ␤(1) ⌿ (0) 典 ,
共2.23兲
J(1)
d 共 r 兲 ⫽⫺
J(1)
p 共 r兲 ⫽
共2.31兲
qប
Im关 ⌿ (1)† 共 r兲 ␤ “⌿ (0) 共 r兲 ⫹⌿ (0)† 共 r兲 ␤ “⌿ (1) 共 r兲兴
m
⫹
iqប
qប
␤ 关 “•A共 r兲 ⫹A共 r兲 •“ 兴 ⫺
␤ ⌺•B共 r兲 .
2m
2m
共2.24兲
q2
A共 r兲 ⌿ (0)† 共 r兲 ␤ ⌿ (0) 共 r兲
m
and
where we have defined
Ĥ␤(1) ⫽
J(1) 共 r兲 ⫽2qc Re关 ⌿ (0)† 共 r兲 ␣⌿ (1) 共 r兲兴
E (2) ⫽⫺ 21
where ⌺ is the 4⫻4 matrix defined as
␴
共2.26兲
is the induced current linear in the perturbation. Then from
Eqs. 共2.5兲, 共2.13兲, and 共2.27兲 and from the reality of E (2)
关implied by Eq. 共2.16兲兴, it follows that
J(0) 共 r兲 ⫽ 21 qc⌿ (0)† 共 r兲 ␣⌿ (0) 共 r兲 ⫹ 21 qc⌿ (0)† 共 r兲 ␣⌿ (0) 共 r兲 ,
共2.19兲
⌿ (0) 共 r兲 ⫽
J共 r兲 ⫽J(0) 共 r兲 ⫹J(1) 共 r兲 ⫹•••,
where
It is then evident from Eqs. 共2.5兲, 共2.12兲, and 共2.17兲 that
(1)
共2.25兲
qប
“⫻Re关 ⌿ (0)† 共 r兲 ␤ ⌺⌿ (1) 共 r兲兴 .
m
共2.32兲
The above expressions for the induced currents J(1)
d (r) and
J(1)
(r)
are
strikingly
similar
to
the
following
expressions:
p
032112-3
RADOSL” AW SZMYTKOWSKI
j(1)
d 共 r 兲 ⫽⫺
j(1)
p 共 r兲 ⫽
PHYSICAL REVIEW A 65 032112
q2
A共 r兲 ␺ (0)† 共 r兲 ␺ (0) 共 r兲 ,
m
具 ⌿ (0) 兩 ⌿ ␤(1) 典 ⫽0,
共2.33兲
is a solution to the inhomogeneous equation
qប
Im关 ␺ (1)† 共 r兲 “ ␺ (0) 共 r兲 ⫹ ␺ (0)† 共 r兲 “ ␺ (1) 共 r兲兴
m
qប
“⫻Re关 ␺ (0)† 共 r兲 ␴␺ (1) 共 r兲兴
⫹
m
关 Ĥ(0) ⫺E (0) 兴 ⌿ ␤(1) 共 r兲 ⫽⫺ 关 Ĥ␤(1) ⫺E (1) 兴 ⌿ (0) 共 r兲 共2.45兲
共2.34兲
(1)
for induced diamagnetic 关 j(1)
d (r) 兴 and paramagnetic 关 jp (r) 兴
currents derivable within the framework of the nonrelativistic Pauli theory 关15兴. Therefore, it seems judicious to identify
(1)
J(1)
d (r) and Jp (r) as induced relativistic diamagnetic and
paramagnetic currents, respectively. Accordingly, from Eqs.
共2.28兲 and 共2.30兲 we have
(2)
E (2) ⫽E (2)
d ⫹E p ,
共2.35兲
where
1
E (2)
d ⫽⫺ 2
冕
d 3 rA共 r兲 •J(1)
d 共 r兲
共2.36兲
冕
d 3 rA共 r兲 •J(1)
p 共 r兲
共2.37兲
R3
and
1
E (2)
p ⫽⫺ 2
R3
共2.44兲
are the diamagnetic and paramagnetic contributions to E (2) ,
respectively. If we define
subject to the same boundary conditions that have been imposed on ⌿ (1) (r).
The expression 共2.39兲 differs from Eq. 共1.7兲 obtained in
earlier works 关4,5兴 based on the sum-over-states procedure.
However, we believe that the physical character of our argumentation, based on the natural notion of the induced current, testifies in favor of our identification of the expression
共2.39兲 as the relativistic analog of the nonrelativistic Larmor
diamagnetic term 共1.3兲.
With such an interpretation of E (2)
d , it seems judicious to
identify the paramagnetic correction 共2.41兲 as the relativistic
analog of the nonrelativistic Van Vleck term 共1.4兲. This point
of view is supported by the fact that since in the nonrelativ(2)
and E (2)
istic limit E (2) and E (2)
d tend to E
d , respectively,
(2)
(2)
E p tends to E p . On the other hand, one might raise an
objection pointing out the evident asymmetry between the
right-hand sides of Eqs. 共2.41兲 and 共1.4兲. Continuing, one
might argue that E (2)
p should be still decomposed in the following way:
(2)
(2)
E (2)
p ⫽E p1 ⫹E p2 ,
共2.46兲
(2)
E p1
⫽⫺ 具 ⌿ (0) 兩 Ĥ␤(1) Ĝ (0) Ĥ␤(1) ⌿ (0) 典 ,
共2.47兲
where
2
Ĥ␤(2) ⫽
q
␤ A2 共 r兲 ,
2m
Eqs. 共2.36兲 and 共2.38兲 imply that
E (2)
d
共2.38兲
may be rewritten as
(0)
兩 Ĥ␤(2) ⌿ (0) 典
E (2)
d ⫽具⌿
共2.39兲
(2)
E p2
⫽⫺Re具 ⌿ (0) 兩 Ĥ␤(1) Ĝ (0) 关 Ĥ(1) ⫺Ĥ␤(1) 兴 ⌿ (0) 典 ,
(2)
rather than E (2)
is a right counterpart of the
and that E p1
p
nonrelativistic Van Vleck term E (2)
p . With such an interpre(2)
, which may be also rewritten in the form
tation of E p1
关cf. Eq. 共1.8兲兴. Similarly, after elementary integrations by
parts, Eqs. 共2.37兲 and 共2.32兲 yield
(0)
兩 Ĥ␤(1) ⌿ (1) 典 ,
E (2)
p ⫽Re具 ⌿
共2.40兲
with Ĥ␤(1) defined in Eq. 共2.24兲, or equivalently, after making
use of Eq. 共2.14兲,
(0)
兩 Ĥ␤(1) Ĝ (0) Ĥ(1) ⌿ (0) 典 .
E (2)
p ⫽⫺Re具 ⌿
共2.41兲
For the sake of completeness, we observe that it is evident
from Eq. 共2.41兲 and from the reality of E (2)
p that the latter
quantity may be rewritten in the form
(0)
兩 Ĥ(1) ⌿ ␤(1) 典 ,
E (2)
p ⫽Re具 ⌿
共2.42兲
⌿ ␤(1) 共 r兲 ⫽⫺Ĝ (0) Ĥ␤(1) ⌿ (0) 共 r兲 ,
共2.43兲
共2.48兲
(2)
E p1
⫽ 具 ⌿ (0) 兩 Ĥ␤(1) ⌿ ␤(1) 典 ,
共2.49兲
(2)
the term E p2
is a separate correction to E (2) of a purely
relativistic origin, vanishing in the nonrelativistic limit.
(2)
It seems to us that the discussion whether E (2)
p or E p1 is
the relativistic counterpart of the Van Vleck term is of an
academic character. Still, leaving aside interpretative questions, we observe that the decomposition 共2.46兲 may be useful in actual calculations. Therefore, below we shall show
(2)
may be written in a simpler form, which does not
that E p2
involve the generalized Green operator Ĝ(0) . To this end, we
observe that Eq. 共2.15兲 may be rewritten as
G (0) 共 r,r⬘ 兲 ⫽
where
iប
E (0) ⫺V 共 r兲
␤␣•“G (0) 共 r,r⬘ 兲 ⫹
␤ G (0) 共 r,r⬘ 兲
mc
mc 2
⫹
1
mc
2
␤ ␦ 共 r⫺r⬘ 兲 ⫺
1
mc 2
␤ ⌿ (0) 共 r兲 ⌿ (0)† 共 r⬘ 兲 .
共2.50兲
obeying
032112-4
LARMOR DIAMAGNETISM AND VAN VLECK . . .
PHYSICAL REVIEW A 65 032112
Interchanging the variables r↔r⬘ and making use of the
property
G (0)† 共 r⬘ ,r兲 ⫽G (0) 共 r,r⬘ 兲 ,
共2.51兲
Equation 共2.56兲 may be still simplified. Indeed, since the
matrix element 具 ⌿ (0) 兩 Ĥ␤(1) ⌿ (0) 典 is evidently real while, because of the easily proved relation
␤ Ĥ(1) ⫹ 共 ␤ Ĥ(1) 兲 † ⫽0,
we transform Eq. 共2.50兲 into
G (0) 共 r,r⬘ 兲 ⫽⫺
⫹
⫺
the matrix element 具 ⌿ (0) 兩 ␤ Ĥ(1) ⌿ (0) 典 is purely imaginary,
Eq. 共2.56兲 becomes
iប
“ ⬘ G (0) 共 r,r⬘ 兲 • ␣␤
mc
E (0) ⫺V 共 r⬘ 兲
mc
1
mc
2
2
G (0) 共 r,r⬘ 兲 ␤ ⫹
1
mc
2
␤ ␦ 共 r⫺r⬘ 兲
⌿ (0) 共 r兲 ⌿ (0)† 共 r⬘ 兲 ␤ .
共2.52兲
After these preparatory steps, we rewrite Eq. 共2.14兲 as
⌿ (1) 共 r兲 ⫽ 21 qc
冕
R3
⫹ 12 qc
d 3 r⬘ G (0) 共 r,r⬘ 兲 ␣•A共 r⬘ 兲 ⌿ (0) 共 r⬘ 兲
冕
R3
d 3 r⬘ G (0) 共 r,r⬘ 兲 ␣•A共 r⬘ 兲 ⌿ (0) 共 r⬘ 兲
共2.53兲
and substitute Eqs. 共2.52兲 and 共2.20兲 into the first and second
integrals, respectively, on the right-hand side of Eq. 共2.53兲.
Further integration by parts of the term containing
“ ⬘ G (0) (r,r⬘ ) yields
⌿ (1) 共 r兲 ⫽⫺Ĝ (0) Ĥ␤(1) ⌿ (0) 共 r兲 ⫺
⫹
1
2mc
2
1
2mc 2
(2)
⫽⫺
E p2
1
2mc 2
具 ⌿ (0) 兩 ␤ Ĥ(1) ⌿ (0) 典 ⌿ (0) 共 r兲 .
⫹
2mc
2
1
2mc 2
As an example illustrating the above general discussion,
we shall consider a Dirac particle bound in a central field of
potential V(r) perturbed by a static uniform magnetic field of
induction B directed along the z axis of a coordinate system.
In the symmetric gauge, which we adopt here, the vector
potential of the magnetic field is
A共 r兲 ⫽ 21 B⫻r.
共2.54兲
⌿ (0)
M 共 r兲 ⫽
Re具 ⌿ (0) 兩 Ĥ␤(1) ␤ Ĥ(1) ⌿ (0) 典
⫹
2mc 2
1
2mc
2
冉
冊
1 P (0) 共 r 兲 ⍀ ⫺1M 共 nr 兲
.
r iQ (0) 共 r 兲 ⍀ ⫹1M 共 nr 兲
共3.2兲
Here M ⫽⫾1/2 is the magnetic quantum number, ⍀ ␬ M (nr ),
with nr ⫽r/r, are spherical spinors while the real radial functions P (0) (r) and Q (0) (r), normalized according to
Re关 具 ⌿ (0) 兩 Ĥ␤(1) ⌿ (0) 典具 ⌿ (0) 兩 ␤ Ĥ(1) ⌿ (0) 典 兴 .
The first term on the right-hand side of Eq. 共2.55兲 is seen to
(2)
given by Eq. 共2.47兲, hence, we infer
be identical with E p1
(2)
E p2
⫽⫺
共3.1兲
We shall be interested in those solutions to the zeroth-order
共i.e., magnetic-field-free兲 problem, associated with the
ground-state energy E (0) , which are eigenfunctions of the
projection of the total angular momentum on the field direction. Such solutions are of the form
冕
共2.55兲
1
共2.58兲
III. AN EXAMPLE: CHARGED PARTICLE IN A CENTRAL
FIELD PERTURBED BY A UNIFORM
MAGNETIC FIELD
(0)
兩 Ĥ␤(1) Ĝ (0) Ĥ␤(1) ⌿ (0) 典
E (2)
p ⫽⫺ 具 ⌿
1
Re具 ⌿ (0) 兩 Ĥ␤(1) ␤ Ĥ(1) ⌿ (0) 典 .
(2)
,
This is the sought suitable expression for the correction E p2
(0)
which does not contain the generalized Green operator Ĝ .
Concluding this section, we mention that while the corrections E (1) and E (2) are gauge invariant, the decompositions 共2.35兲 and 共2.46兲 are not 关cf. the remarks following Eq.
共1.6兲兴.
␤ Ĥ(1) ⌿ (0) 共 r兲
Inserting the above result into Eq. 共2.40兲, we arrive at
⫺
共2.57兲
0
Re关 具 ⌿ (0) 兩 Ĥ␤(1) ⌿ (0) 典具 ⌿ (0) 兩 ␤ Ĥ(1) ⌿ (0) 典 兴 .
共2.56兲
dr 关 P (0)2 共 r 兲 ⫹Q (0)2 共 r 兲兴 ⫽1,
共3.3兲
obey
冉
Re具 ⌿ (0) 兩 Ĥ␤(1) ␤ Ĥ(1) ⌿ (0) 典
⬁
mc 2 ⫹V 共 r 兲 ⫺E (0)
cប 共 ⫺d/dr⫺1/r 兲
cប 共 d/dr⫺1/r 兲
⫺mc 2 ⫹V 共 r 兲 ⫺E (0)
冊冉
P (0) 共 r 兲
Q (0) 共 r 兲
冊
⫽0
共3.4兲
subject to the vanishing boundary conditions for r→0 and
r→⬁.
032112-5
RADOSL” AW SZMYTKOWSKI
PHYSICAL REVIEW A 65 032112
Although the energy level E (0) is twofold degenerate with
respect to the magnetic quantum number M, we observe that
the perturbation 共2.5兲 does not mix states with different M.
Therefore we may directly apply the results of Sec. II to
determine the first and the second-order corrections to E (0) .
We begin with the first-order correction. In the particular
case considered in this section, the ‘‘non-Gordon’’ Eq. 共2.12兲
takes the form
1
⌿ (1)
M 共 r 兲 ⫽ 2 qcB
冉
冊
(1)
1 P ␬ 共 r 兲 ⍀ ␬ M 共 nr 兲
a␬M
,
r iQ ␬(1) 共 r 兲 ⍀ ⫺ ␬ M 共 nr 兲
共3.13兲
兺␬
where
a ␬ M ⫽i
冖
4␲
d 2 nr ⍀ ␬† M 共 nr 兲共 nr ⫻ ␴兲 z ⍀ ⫹1M 共 nr 兲 .
共3.5兲
共3.14兲
or, after utilizing the explicit form 共3.2兲 of the unperturbed
eigenfunction and after some angular momentum algebra,
The coefficients a ␬ M do not vanish only for ␬ ⫽⫺1 or ␬ ⫽
⫹2. In these two cases their values are
1
(0)
(0)
E (1)
M ⫽⫺ 2 qcB 具 ⌿ M 兩 共 r⫻ ␣ 兲 z ⌿ M 典
E (1)
M ⫽
4M
qcB
3
冕
⬁
0
dr r P (0) 共 r 兲 Q (0) 共 r 兲 .
共3.6兲
再
a␬M⫽
E (1)
M
from the
Alternatively, we might compute the correction
Gordon formula 共2.23兲, which in the present case becomes
qបB
(0)
ˆ
E (1)
具 ⌿ (0)
M ⫽⫺
M 兩 ␤ 共 ⌳z ⫹⌺ z 兲 ⌿ M 典 ,
2m
共3.7兲
共3.8兲
After utilizing Eq. 共3.2兲, Eq. 共3.7兲 yields
E (1)
M ⫽⫺
2M qបB
3 m
冉冕
⬁
0
dr 关 P
(0)2
共 r 兲 ⫺Q
(0)2
冕
4M qបB
E (1)
M ⫽⫺
3 m
冉冕
⬁
0
冊
共3.10兲
Equations 共3.6兲 and 共3.9兲 关or Eq. 共3.10兲兴 provide two alternative methods for evaluating E (1)
M . In the nonrelativistic
limit either from Eq. 共3.9兲 or from Eq. 共3.10兲, one infers the
well-known result
冉
E (1)
M
共3.11兲
Having computed the first-order correction, we turn to the
problem of evaluating the second-order one. We wish to
avoid utilizing the generalized Green function. In the nonGordon approach this means we have to use
(1)
E (2) ⫽⫺ 21 qcB 具 ⌿ (0)
M 兩 共 r⫻ ␣ 兲 z ⌿ M 典 ,
共3.12兲
which is the specialized form of Eq. 共2.13兲 and requires solving directly the inhomogeneous equation 共2.9兲 for ⌿ (1)
M (r).
共Here and below we do not put the subscript M in secondorder energy corrections since, as we shall see, they are independent of M.兲 To this end, we decompose
3
共3.15兲
␬ ⫽⫹2,
for
(1)
(1)
dr 关 P (0) 共 r 兲 P ⫺1
共 r 兲 ⫹Q (0) 共 r 兲 Q ⫺1
共 r 兲兴 ⫽0. 共3.16兲
mc 2 ⫹V 共 r 兲 ⫺E (0)
cប 共 ⫺d/dr⫹ ␬ /r 兲
cប 共 d/dr⫹ ␬ /r 兲
⫺mc 2 ⫹V 共 r 兲 ⫺E (0)
⫽r
冉
Q (0) 共 r 兲
P (0) 共 r 兲
冊 冉
⫹ ⑀ (1)
P (0) 共 r 兲
Q (0) 共 r 兲
冊
冊冉
P ␬(1) 共 r 兲
Q ␬(1) 共 r 兲
␦ ␬ ,⫺1 ,
冊
共3.17兲
with ⑀ (1) ⫽⫺3E (1)
M /2M qcB, which is to be solved subject to
the vanishing boundary conditions for r→0 and r→⬁. With
the knowledge of the radial functions P ␬(1) (r) and Q ␬(1) (r),
Eq. 共3.12兲 becomes
E (2) ⫽⫺ 41 q 2 c 2 B 2
兺 a ␬2 M 冕0 dr r 关 P (0)共 r 兲 Q ␬(1)共 r 兲
␬ ⫽⫺1,⫹2
⬁
⫹Q (0) 共 r 兲 P ␬(1) 共 r 兲兴 .
c→⬁
qបB
→ ⫺M
.
m
冑2
␬ ⫽⫺1,
for
On substituting the expansion 共3.13兲 into Eq. 共2.9兲, after employing the angular momentum algebra, we arrive at the inhomogeneous radial system
冊
1
dr P (0)2 共 r 兲 ⫺ .
4
⬁
0
1
共 r 兲兴 ⫹ ,
2
共3.9兲
or equivalently, in virtue of the normalization condition
共3.3兲,
⫺
4M
3
共notice that a ␬2 M is independent of M ). To enforce the orthogonality constraint in Eq. 共2.11兲, we require
ˆ z is the zth component of the orbital angular momenwhere ⌳
tum operator
ˆ ⫽⫺ir⫻“.
⌳
⫺
共3.18兲
To find E (2) , we may use the Gordon approach as well. In
the particular case considered here, the diamagnetic contribution 共2.39兲 is
E (2)
d ⫽
q 2B 2
(0)
2
2
具 ⌿ (0)
M 兩 ␤ 共 r ⫺z 兲 ⌿ M 典
8m
共3.19兲
and is easily evaluated to be
E (2)
d ⫽
q 2B 2
12m
冕
⬁
0
dr r 2 关 P (0)2 共 r 兲 ⫺Q (0)2 共 r 兲兴 ,
共3.20兲
which in the nonrelativistic limit yields the well-known result
032112-6
LARMOR DIAMAGNETISM AND VAN VLECK . . .
c→⬁ q 2 B 2
E (2)
d →
12m
冕
⬁
0
PHYSICAL REVIEW A 65 032112
where for brevity we have denoted
共3.21兲
dr r 2 P (0)2 共 r 兲 .
I ␤(0) ⫽
In turn, the paramagnetic components, according to Eqs.
共2.49兲 and 共2.58兲, are
(2)
⫽⫺
E p1
(2)
⫽⫺
E p2
qបB
(1)
ˆ
具 ⌿ (0)
M 兩 ␤ 共 ⌳z ⫹⌺ z 兲 ⌿ ␤ M 典 ,
2m
q 2 បB 2
8m 2 c
共3.22兲
(2)
⫽
E p1
共3.23兲
q បB
2
2
12m c
冕
⬁
0
dr r P (0) 共 r 兲 Q (0) 共 r 兲 ,
兺
␬ ⫽⫺1,⫹2
a␬M
共3.24兲
冉
冊
(1)
1 P ␤␬ 共 r 兲 ⍀ ␬ M 共 nr 兲
(1)
r iQ ␤␬
共 r 兲 ⍀ ⫺ ␬ M 共 nr 兲
共3.25兲
with the coefficients a ␬ M given by Eq. 共3.15兲 and with the
constraint
冕
⬁
0
dr 关 P ␤(1),⫺1 共 r 兲 P (0) 共 r 兲 ⫹Q ␤(1),⫺1 共 r 兲 Q (0) 共 r 兲兴 ⫽0
共3.26兲
enforcing the orthogonality condition 共2.44兲. Substitution of
this expansion into the inhomogeneous equation 共2.45兲 followed by application of Eq. 共3.9兲 gives the following differ(1)
(r) and
ential systems obeyed by the radial functions P ␤␬
(1)
Q ␤␬ (r):
冉
mc 2 ⫹V 共 r 兲 ⫺E (0)
cប 共 ⫺d/dr⫺1/r 兲
cប 共 d/dr⫺1/r 兲
⫺mc 2 ⫹V 共 r 兲 ⫺E (0)
⫽
冉
冉
共 I ␤(0) ⫺1 兲 P (0) 共 r 兲
共 I ␤(0) ⫹1 兲 Q (0) 共 r 兲
冊
cប 共 ⫺d/dr⫹2/r 兲
cប 共 d/dr⫹2/r 兲
⫺mc 2 ⫹V 共 r 兲 ⫺E (0)
⫽⫺
冉
0
Q
(0)
共r兲
冊
,
冊冉
P ␤(1),⫺1 共 r 兲
Q ␤(1),⫺1 共 r 兲
冊
共3.27兲
,
mc 2 ⫹V 共 r 兲 ⫺E (0)
冕
⬁
0
2q 2 ប 2 B 2
9m
2
冕
⬁
0
冊冉
P ␤(1),⫹2 共 r 兲
Q ␤(1),⫹2 共 r 兲
冊
dr P (0)2 共 r 兲 ⫺1.
共3.29兲
dr 关 P (0) 共 r 兲 P ␤(1),⫺1 共 r 兲
⫹ 14 Q (0) 共 r 兲 Q ␤(1),⫹2 共 r 兲兴 .
which obviously vanishes in the nonrelativistic limit. To find
(2)
, we decompose
E p1
qបB
2m
0
dr 关 P (0)2 共 r 兲 ⫺Q (0)2 共 r 兲兴 ⫽2
(0)
ˆ
Re具 ⌿ (0)
M 兩 共 ⌳z ⫹⌺ z 兲共 r⫻ ␣ 兲 z ⌿ M 典 .
2
⌿ ␤(1)M 共 r兲 ⫽
⬁
Equations 共3.27兲 and 共3.28兲 are to be solved subject to the
vanishing boundary conditions for r→0 and r→⬁. Once the
(1)
(1)
(r) and Q ␤␬
(r) have been found, from Eqs.
solutions P ␤␬
共3.22兲, 共3.25兲, 共3.15兲, and 共3.26兲 we obtain
(2)
The contribution E p2
is readily found to be
(2)
E p2
⫽
冕
共3.30兲
In the nonrelativistic limit the right-hand sides of Eqs. 共3.27兲
and 共3.28兲 vanish. In conjunction with the constraint 共3.26兲
this implies that in this limit both equations have only trivial
(2)
→ c→⬁ 0 共notice that
solutions. From this we infer that E p1
this result is specific for the example considered here兲.
IV. CONCLUSIONS
We have presented the method, based on the Gordon decomposition of the field-induced current, for determination
of diamagnetic and paramagnetic contributions to the
second-order energy correction within the framework of the
relativistic quantum theory based on the Dirac equation. Our
results contradict earlier findings that the diamagnetic 共Larmor兲 contribution is given by Eq. 共1.7兲. Instead, we have
found that this contribution is given by Eq. 共1.8兲. We have
also analyzed the paramagnetic 共Van Vleck兲 contribution and
found that it may be naturally split into two parts, one of
which has a purely relativistic origin and vanishes in the
nonrelativistic limit while the second one tends in the same
limit to the nonrelativistic Van Vleck term. These inferences
are in a qualitative agreement with those obtained by Pyper
关16 –19兴 in the context of the nuclear-shielding theory. The
results presented by the latter author in Ref. 关16兴 justify the
supposition that a procedure analogous to that described in
the present work should be applicable to many-electron systems within the framework of the Dirac-Hartree-Fock formalism, at least at the single-determinantal level.
ACKNOWLEDGMENTS
共3.28兲
关1兴 J. H. Van Vleck, Phys. Rev. 34, 587 共1929兲.
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关3兴 J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities 共Oxford University Press, Oxford, 1932兲.
I am grateful to Professor Cz. Szmytkowski for commenting on the manuscript. This work was supported in part by
the Polish State Committee for Scientific Research under
Grant No. 228/P03/99/17.
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032112-7
RADOSL” AW SZMYTKOWSKI
PHYSICAL REVIEW A 65 032112
关7兴 Equation 共1.6兲 does not appear in the explicit form in Ref. 关5兴.
It is, however, an immediate consequence of Eqs. 共4兲 and 共18兲
共together with the text following the latter兲 of that paper and
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032112-8