Research on the relativistic theory of the spectra of hydrogen-like atoms
Research on the relativistic theory of the spectra of hydrogen-like atoms
Research on the relativistic theory of the spectra of hydrogen-like atoms Tyukhtyaev Yury Nikolaevich
I. Methods for studying the problem of two bodies in relativistic quantum theory.
The problem of two bodies in the nonrelativistic theory is reduced to two simpler ones: the uniform motion of the center of mass and the motion of a particle with a reduced mass in a potential field. In the relativistic case, the explicit separation of the motion of the center of mass, and at the same time the direct introduction of the concept of potential, is impossible.
The relativistic two-body problem covers the problems of scattering and bound states of particles. At present, the quantum theory of scattering is the most developed: the collision of particles can be described using free field operators, which makes it possible to use the apparatus S- matrices.
At the same time, the creation and development of quantum theory is closely connected with the study of the energy spectrum of the hydrogen atom and hydrogen-like (HL) atoms. As the theory, technique, and technique of the experiment improved, positronium, muonium, deuterium, and muonic and pionic hydrogen were included in the number of systems studied. Recently, quark structures of the charm-monium type have also been included here. In general, III atoms are the simplest bound states of particles, the most accessible for theoretical and experimental studies, and therefore an ideal object for testing the basic provisions of quantum theory.
Since in the relativistic case the problem of bound states of a particle in an external field and the problem of bound states of a system of two particles are essentially different, the direct use of the Dirac equation, and even more so the Schrödinger equation for studying the WP of an atom
mov is not enough. There are a number of specific methods in quantum relativistic theory for these purposes. Such are the Fock-Tamm-Dankov-Dyson approach [1-4] or the use of the dispersion relations strictly proven by N.N. Bogolyubov [5,6] in combination with the unitarity condition for the scattering amplitude. However, the practical application of these methods is associated with certain difficulties.
The Schwinger-Bethe-Salpeter formalism played a special role in the development of the theory of bound states [7, 8]. It is based on the use of operators of interacting fields and, consequently, of the complete two-particle Green's functions [9, 10].
The complete four-time Green's function G^X^X^X^X^) for two fermions with masses m4 and m2 in the coordinate representation satisfies the Bethe-Salpeter equation
+ b(chl-XxE in HbyY^NbYD*LA)
where Q.o is the Green's function of non-interacting particles, K, is the kernel of the interaction of two fermions, which is the sum of two-particle irreducible Feynman diagrams.)
K (X.X') pge^K(x, x'OS-T), *
u In the center-of-mass system (c.m.) the non-zero component of the total four-momentum*R
coincides with the total energy of two interacting particles.
In contrast to the nonrelativistic eigenfunction corresponding to a state with a certain value of energy E, the function ^(X) is non-stationary. The relative time parameter X0 has no direct physical interpretation. At the same time, it is clear that the interaction of charged particles in the nonrelativistic limit is Coulomb and the expansion of the function *P_ CX) in perturbation theory rad must be related to the Coulomb wave function, which does not depend on time. The construction of such a decomposition is a far from trivial task. Causes mathematical difficulties
also normalization and formulation of boundary conditions for a wave function depending on relative time >The quasipotential method in quantum theory, proposed by A.A. Logunov and A.N. Tavkhelidze, turns out to be very effective.
, in which the relative time parameter is excluded from the beginning. With this approach, the quantum field equations for a system of two particles are reduced to an equation of the Schrödinger type with a quasipotential determined in terms of the scattering amplitude. Despite the absence of explicit relativistic covariance, the Logunov-Tavkhelidze quasipotential method contains all the information about the properties of the scattering amplitude that can be obtained from the general principles of quantum field theory. Therefore, using the quasi-potential equation, it is possible to study both the properties of analyticity and the asymptotic behavior of the scattering amplitude and some regularities of potential scattering, in particular, at high energies [16–28]. The renormalization of the quasipotential equation reduces, as in the usual theory
matrix, to the renormalization of mass and charge
In some cases, for example, in the problem of finding the matrix elements of local operators between bound states [30–32], the explicit relativistic covariance of the equation turns out to be important. lang=en style=height: 28px;Over repeated variables in (I.I.I) integration is implied.
The Vethe-Salpeter equation can also be rewritten as:where the inverse Green's function of noninteracting fermions is G0 =57-,^1, Si – distribution function of the t-th particle, I – unit operator.
The state of a two-particle system is determined by the two-time wave function Ch* – the solution of the corresponding (I.I.2) homogeneous equation The problem for the eigenvalues of the total energy of a system of two particles can be posed as a result of the transition to the coordinate of the center of mass X and the relative coordinate x. [P,12] : X ^FromChChi+PGIgChgMti+tg) x \u003d X1 -Xg (I.I.4) Then the state with a total four-momentum (p corresponds to the wave functionV (Xi, X,) e lWt -f9 (x) (I.I.5)
The relative coordinate function Hls) satisfies the equation [13]: C G^frX)-Xy(x,x')]-f0y
(x') Oh
(i.i.b) [4] .
. (x, x') = [14,15]$tz $-m [29].
'\fct+M*-*))S$X-U**) [38].
K (X.X') pge^K(x, x'OS-T), *
u 1 In the center-of-mass system (c.m.) the non-zero component of the total four-momentum *R
coincides with the total energy of two interacting particles.
In contrast to the nonrelativistic eigenfunction corresponding to a state with a certain value of energy E, the function ^(X) is non-stationary. The relative time parameter X0 has no direct physical interpretation. At the same time, it is clear that the interaction of charged particles in the nonrelativistic limit is Coulomb and the expansion of the function *P_ CX) in perturbation theory rad must be related to the Coulomb wave function, which does not depend on time. The construction of such a decomposition is a far from trivial task. Causes mathematical difficulties
also normalization and formulation of boundary conditions for a wave function depending on relative time
The quasipotential method in quantum theory, proposed by A.A. Logunov and A.N. Tavkhelidze, turns out to be very effective. , in which the relative time parameter is excluded from the beginning. With this approach, the quantum field equations for a system of two particles are reduced to an equation of the Schrödinger type with a quasipotential determined in terms of the scattering amplitude. Despite the absence of explicit relativistic covariance, the Logunov-Tavkhelidze quasipotential method contains all the information about the properties of the scattering amplitude that can be obtained from the general principles of quantum field theory. Therefore, using the quasi-potential equation, it is possible to study both the properties of analyticity and the asymptotic behavior of the scattering amplitude and some regularities of potential scattering, in particular, at high energies [16–28]. The renormalization of the quasipotential equation reduces, as in the usual theorymatrix, to the renormalization of mass and charge
In some cases, for example, in the problem of finding the matrix elements of local operators between bound states [30–32], the explicit relativistic covariance of the equation turns out to be important.)
K (H.Kh ') Rch^k (x, x'os-g), *
u
In the system of the Center for the masses (S.Ts.M.) different from zero component of complete fourchimpulse
*R
coincides with the complete energy of two interacting particles. In contrast to the non -nobility of its own function, corresponding to the state with a certain value of energy E, the function ^(x) is non -stationary. The parameter of the relative time x0 does not have a direct physical interpretation. At the same time, it is clear that the interaction of charged particles in a non -nobtic limit is Kulonovsky and the decomposition of the function *p_ sca) in the radiation theory should be associated with the Kulon’s wave function, which independent of time. The construction of such decomposition is far from non -trivial. Mathematical difficulties causealso normalizing and formulating the boundary conditions for the wave function, depending on the relative time A quasi -potential method in quantum theory proposed by A.A. Logunov and A.N. Tavhelidze turns out to be very effective
in which the parameter of relative time is excluded from the very beginning. With this approach, quant field equations for the system of two particles are brought to the equation of the type of shredinger with a quasi -focus determined through the scattering amplitude. lang=en style=height: 28px;By repeating variables in (I.I.I), integration is meant.
The equation of a vete-soldier can also be rewritten in the form:
where is the reverse function of the grind of non-current-providing enzyme G0 = 57-,^1,
Si – The distribution function of the T -th particle, I is a single operator.The condition of the two -idist system determines the two -time wave function h* – the solution of the corresponding (I.I.2) homogeneous equationThe task for its own values of the full energy of the system of two particles can be set as a result of the transition to the coordinate of the center of mass X and the relative coordinate X. [P, 12]:
X ^Father+PGGGHMTI+TG)
x = x1 -kg (I.I.4)
Then the state with full fourchimpulse (R answers the wave function
V (XI, X,) E lwt -f9 (x) (I.I.5)
The function depending on the relative coordinates, the function NL C) satisfies the equation [13]:
With g^frx) -xy (x, x ')] -f y
(x ') About ^ (I.I.B) . (x, x ') =$ t z
m
'\ FCT+M*-*)) S $ x -U **)
&K (H.Kh ') Rch^k (x, x'os-g), * u ++ In the system of the Center for the masses (S.Ts.M.) different from zero component of complete fourchimpulse *R 0 coincides with the complete energy of two interacting particles. 2 In contrast to the non -nobility of its own function, corresponding to the state with a certain value of energy E, the function ^(x) is non -stationary. The parameter of the relative time x0 does not have a direct physical interpretation. At the same time, it is clear that the interaction of charged particles in a non -nobtic limit is Kulonovsky and the decomposition of the function *p_ sca) in the radiation theory should be associated with the Kulon’s wave function, which independent of time. The construction of such decomposition is far from non -trivial. Mathematical difficulties cause
also normalizing and formulating the boundary conditions for the wave function, depending on the relative timeA quasi -potential method in quantum theory proposed by A.A. Logunov and A.N. Tavhelidze turns out to be very effective in which the parameter of relative time is excluded from the very beginning. With this approach, quant field equations for the system of two particles are brought to the equation of the type of shredinger with a quasi -focus determined through the scattering amplitude.)»K (X.X') pge^K(x, x'OS-T), * u In the center-of-mass system (c.m.) the non-zero component of the total four-momentum *Rcoincides with the total energy of two interacting particles. In contrast to the nonrelativistic eigenfunction corresponding to a state with a certain value of energy E, the function ^(X) is non-stationary. The relative time parameter X0 has no direct physical interpretation. At the same time, it is clear that the interaction of charged particles in the nonrelativistic limit is Coulomb and the expansion of the function *P_ CX) in perturbation theory rad must be related to the Coulomb wave function, which does not depend on time. lang=en style=height: 28px;Over repeated variables in (I.I.I) integration is implied.
The Vethe-Salpeter equation can also be rewritten as:
where the inverse Green's function of noninteracting fermions is G0 =57-,^1,
Si + – distribution function of the t-th particle, I – unit operator. (1.2.9)
The state of a two-particle system is determined by the two-time wave function Ch* – the solution of the corresponding (I.I.2) homogeneous equation
& + The problem for the eigenvalues of the total energy of a system of two particles can be posed as a result of the transition to the coordinate of the center of mass X and the relative coordinate x. [P,12] :1 X ^FromChChi+PGIgChgMti+tg)2 x \u003d X1 -Xg (I.I.4)Then the state with a total four-momentum (p corresponds to the wave function V (Xi, X,) e lWt -f9 (x) (I.I.5)1 The relative coordinate function Hls) satisfies the equation [13]: (1 – C G^frX)-Xy(x,x')]-f
y
(x') Oh
(i.i.b) . (x, x') = $tz
m
'\fct+M*-*))S$X-U**)
K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.) the non-zero component of the total four-momentum
*R 1 coincides with the total energy of two interacting particles. 1 In contrast to the nonrelativistic eigenfunction corresponding to a state with a certain value of energy E, the function ^(X) is non-stationary. The relative time parameter X0 has no direct physical interpretation. At the same time, it is clear that the interaction of charged particles in the nonrelativistic limit is Coulomb and the expansion of the function *P_ CX) in perturbation theory rad must be related to the Coulomb wave function, which does not depend on time.) 1 (1 – K (X.X') pge^K(x, x'OS-T), *
u $0 :
In the center-of-mass system (c.m.) the non-zero component of the total four-momentum 1 (1 – *R
coincides with the total energy of two interacting particles. lang=en style=height: 28px;Over repeated variables in (I.I.I) integration is implied.
The Vethe-Salpeter equation can also be rewritten as:+ where the inverse Green's function of noninteracting fermions is G0 =57-,^1, Si1 – distribution function of the t-th particle, I – unit operator.1 The state of a two-particle system is determined by the two-time wave function Ch* – the solution of the corresponding (I.I.2) homogeneous equation »- 2 – The problem for the eigenvalues of the total energy of a system of two particles can be posed as a result of the transition to the coordinate of the center of mass X and the relative coordinate x. [P,12] : >
X ^FromChChi+PGIgChgMti+tg)
x \u003d X1 -Xg (I.I.4)
Then the state with a total four-momentum (p corresponds to the wave function+ V (Xi, X,) e lWt -f9 (x) (I.I.5) The relative coordinate function Hls) satisfies the equation [13]: C G^frX)-Xy(x,x')]-f – 2 – y
(x') Oh
(i.i.b)
. (x, x') = $tz m +.
'\fct+M*-*))S$X-U**)
K (X.X') pge^K(x, x'OS-T), * , u 1 In the center-of-mass system (c.m.) the non-zero component of the total four-momentum 4 *R coincides with the total energy of two interacting particles.Over repeated variables in (I.I.I) integration is implied. ) The Vethe-Salpeter equation can also be rewritten as: 4 where the inverse Green's function of noninteracting fermions is G0 =57-,^1, Si
– distribution function of the t-th particle, I – unit operator. The state of a two-particle system is determined by the two-time wave function Ch* – the solution of the corresponding (I.I.2) homogeneous equation The problem for the eigenvalues of the total energy of a system of two particles can be posed as a result of the transition to the coordinate of the center of mass X and the relative coordinate x. [P,12] : (1.2.20)
X ^FromChChi+PGIgChgMti+tg)
x \u003d X1 -Xg (I.I.4)
Then the state with a total four-momentum (p corresponds to the wave function
)
K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.) the non-zero component of the total four-momentum
*R
coincides with the total energy of two interacting particles.
In contrast to the nonrelativistic eigenfunction corresponding to a state with a certain value of energy E, the function ^(X) is non-stationary. The relative time parameter X0 has no direct physical interpretation. At the same time, it is clear that the interaction of charged particles in the nonrelativistic limit is Coulomb and the expansion of the function *P_ CX) in perturbation theory rad should be related to the Coulomb wave function, lang=en style=height: 28px;V (Xi, X,) e lWt -f9 (x) (I.I.5)
The relative coordinate function Hls) satisfies the equation [13]:
C G^frX)-Xy(x,x')]-f
y 2 ^* (x') Oh 1 (i.i.b) – 2 – 24)
. (x, x') =
$tz
m
'\fct+M*-*))S$X-U**)
K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.) the non-zero component of the total four-momentum
*R
^ = coincides with the total energy of two interacting particles. In contrast to the nonrelativistic eigenfunction corresponding to a state with a certain value of energy E, the function ^(X) is non-stationary. The relative time parameter X0 has no direct physical interpretation. At the same time, it is clear that the interaction of charged particles in the nonrelativistic limit is Coulomb and the expansion of the function *P_ CX) in perturbation theory rad should be related to the Coulomb wave function,) *
K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.), the non-zero component of the total four-momentum *Rcoincides with the total energy of two interacting particles. In contrast to the nonrelativistic eigenfunction corresponding to a state with a certain value of energy E, the function ^(X) is non-stationary. Parameter lang=en style=height: 28px;V (Xi, X,) e lWt -f9 (x) (I.I.5)The relative coordinate function Hls) satisfies the equation [13]:
C G^frX)-Xy(x,x')]-f
y (x') Oh .
(i.i.b)
. (x, x') = =4 ( $tz + &+%) ^- 3 – 2 )
m '\fct+M*-*))S$X-U**)K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.), the non-zero component of the total four-momentum *Rcoincides with the total energy of two interacting particles.
In contrast to the nonrelativistic eigenfunction corresponding to a state with a certain value of energy E, the function ^(X) is non-stationary. Parameter)
K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.), the non-zero component of the total four-momentum
*R
coincides with the total energy of two interacting particles.
Unlike the nonrelativistic eigenfunction, lang=en style=height: 28px;V (Xi, X,) e lWt -f9 (x) (I.I.5)
The relative coordinate function Hls) satisfies the equation [13]:
C G^frX)-Xy(x,x')]-f
y
(x') Oh
(i.i.b)
. (x, x') =
$tz
m
'\fct+M*-*))S$X-U**)
K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.), the non-zero component of the total four-momentum
*R
coincides with the total energy of two interacting particles.
Unlike the nonrelativistic eigenfunction,)
K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.), the non-zero component of the total four-momentum lang=en style=height: 28px;V (Xi, X,) e lWt -f9 (x) (I.I.5)
The relative coordinate function Hls) satisfies the equation [13]:
C G^frX)-Xy(x,x')]-f
y
(x') Oh
(i.i.b)
. (x, x') =
$tz
m
'\fct+M*-*))S$X-U**)
K (X.X') pge^K(x, x'OS-T), *
u
In the center-of-mass system (c.m.), the non-zero component of the total four-momentum)
K (H.Kh ') Rch^k (x, x'os-g), *
u lang=en style=height: 28px;V (XI, X,) E lwt -f9 (x) (I.I.5)
The function depending on the relative coordinates, the function NL C) satisfies the equation [13]:
With g^frx) -xy (x, x ')] -f
y
(x ') About
(I.I.B)
. (x, x ') =
$ t z
m
'\ FCT+M*-*)) S $ x -U **)
K (H.Kh ') Rch^k (x, x'os-g), *