The Coherent State of the Landau Hamiltonian and the Relativistic Corrections to the Zeeman Effect in He + Ions

We calculate the energy levels of He + ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order 2 2 v c . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order 2 2 v c , of a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He + ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He + ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h, 2 j , 2 l , 2 s and z j , where j , l and s are the internal total, orbital and spin angular moments of the He + ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He + ion. In addition to the 2 2 Z α corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections lines of He + ions in magnetized astrophysical objects.

a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He + ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He + ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h, 2 j , 2 l , 2 s and z j , where j , l and s are the internal total, orbital and spin angular moments of the He + ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He + ion. In addition to the 2 2 Z α corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections which depend on the kinetic energy ( CM E ) of the CM circular motion of the He + ion, in a nontrivial way. Even though these corrections are proportional to 1. Introduction When a composite system with a nonzero net electric charge is placed in a uniform magnetic field, neither the total canonical or mechanical momenta are conserved. So there is no inertial frame where the total momentum is zero at all times. So the problem of separating the c.m. motion from the internal motion for such a system will be different from that of an isolated system where the total or the c.m. momentum is conserved. In the past, several authors [1]- [8] have studied the problem of calculating the corrections to the internal energy levels of composite ions in a magnetic field. Some of them [1] [2] [3] [4] are based on a constant of motion introduced initially by Baye. And some of the works treated N-body problem [6] [8]. There are also aothors introducing new momentas using the coordinates transformation [5], however the coupling term is not small when one of the particles is much heavier than the other. In the paper [7], the the center of mass motion in an electromagnetic radiation is researched. In this paper, we take a different approach to this problem which is more physical. First of all we note that the overall or the c.m. motion of the ion in a uniform magnetic field under ordinary circumstances, is the familiar classical circular motion, if the uniform magnetic field is directed perpendicular to the plane of motion of the ion. In a recent paper [9] we have shown that the quantum state which most closely resembles the classical state of a charged particle moving in a uniform magnetic field, is the coherent state of the Landau quantum Hamiltonian of such a particle. In this paper, we will calculate the corrections to the energy levels of a H like ion (specifically He + ion) in a uniform magnetic field, by treating the state of the c.m. motion of the ion as a coherent state of the Landau Hamiltonian. One of the interesting features of our results is that the shifts in energy levels of the ion, including their first order relativistic corrections, depend on the energy of the c.m. motion, in a significant way. Even though these corrections are of the order of times the usual results where M is the mass of the He + ion, and hence small for nonrelativistic c.m. motion, which we assume, it is important to include them, since in the future, more precise spectroscopic measurements may be able to detect such small corrections.
The format of the rest of the paper is as follows: In Section 2, we describe the approximately relativistic (correct to the order 2 2 v c ) Hamiltonian of the isolated and the c.m. variables. We then express these Hamiltonians in terms of these variables. In Section 3, we give the details of our perturbative calculation of the corrections to the energy levels due to the external uniform magnetic field. We also give particular attention to the basis states used in the first order perturbation calculations. The basis states we used, are the direct product of an internal state and the c.m. state which is the coherent state of the Landau Hamiltonian of a particle of charge ( )

Isolated He + Ion
First let us consider the isolated He + ion, which is a weakly bound composite system of two particles where the first particle is the electron of mass m and electric charge -e, and spin 1 2 s = and the second particle is the He 4 nucleus of electric charge Ze and mass N m with zero spin and zero magnetic moment. We will put 2 Z = only when we do numerical calculations. The Hamiltonian of such an electromagnetically bound system can be written, correct to order 2 2 v c using the methods of Close and Osborn [12]. 2  4  4  2  2  1  2  1  2  1  2  3 2  3 2  2  1  2  1  2   2  1  1  2  2  1  1  2  1  2  2  3  3  2  1  2  1  2   2  2  1  1  2  1  1  2  3  2 2  2 2  1  2   1  2  2  8  8 The first line in the above equation represents the kinetic energy terms, including their first order corrections and the Coulomb potential energy of the two charges Ze and -e. The second and third lines are the straight forward Breit interaction, resulting from the unretarded transverse one photon exchange. The fourth line represents the interaction between the spin magnetic moment of the electron and the magnetic field produced by the motion of the charged nucleus. The fifth line represents the effect of the zitterbewegung of the electron's motion, coming from the Darwin term, in the nonrelativistic reduction of the Dirac Hamiltonian of the electron. The sixth line represents the conventional spin-orbit interaction, including the so called Thomas precession. The above expression for the Hamiltonian is also the same as the expression for the Hamiltonian of a two particle system interacting electromagnetically, given by Krajcik and Foldy [10], specialized to our case, where one of the particles He 4 nucleus has zero spin and zero magnetic moment. We have also put the electronic spin magnetic moment e µ in their expressions [10] [11] as 2 e e mc µ = − which neglects the radiative corrections. This is consistent with the spirit of their approach which neglected the radiative corrections to the one photon exchange between two charged particles. It should also be noted we have used the Gaussian system of units, where as Close and Osborn [12] and Krajcik and Foldy [10] used rationalized Gaussian units. Also in Equation (1)  c.m. and internal variables are defined by the requirement that when the ten generators are expressed in terms of them, they take the single particle form [10] [11] [12] to the same order 2 1 c .
Using this requirement, the relativistic relations between the constituent variables ( µ r , µ p and µ s , 1,2 µ = ) and the c.m. and the internal variables ( R , P , q , p and µ σ , ( 1,2 µ = )) for a two particle composite system made up of two particles, particle 1 being the electron of mass m and charge -e and particle 2 being the nucleus of mass N m and charge Ze + , are given by [ We also notice, since we put Also R and P commute with the internal variables q and p as well as e σ and N σ . In our case, for the He 4 nucleus, N s and N σ both vanish. N σ is the spin operator of the electron in the 0 = P frame. The reader can easily verify that when expressed in terms of the c.m. and the internal variables, the Hamiltonian of Equation (1) takes the single particle form, to order In Equation (16) µ is the reduced mass It should be stressed that Equation (1) would not have taken the single particle form of Equation (12) if we had used the nonrelativistic c.m. variables, which means neglecting the 2 1 c terms in Equation (2) and Equation (3). Also we will find with the nonrelativistic c.m. variables, the center of mass itself does not move uniformly and that the internal angular momentum, = × l q p is not conserved. We should emphasize that e σ in Equation (14) and elsewhere in the paper is not the Pauli matrix, but the spin 1 2 operator of the electron, e s , in the 0 = P frame, which is really one half the usual Pauli matrix.
In order to derive Equations (12)- (14) we made use of the following convenient relations between the constituent and linearly independent c.m. and internal variables R , P , q and p , which can be derived from Equations (2)- (4) using Equation (6) and Equation (7).
The nonrelativistic internal Hamiltonian ( ) 0 h of Equation (13) gives the Bohr energy levels of a particle of reduced mass µ given by Equation (15). The first order relativistic correction to this Hamiltonian, ( )

The Hamiltonian of the He + Ion in a Uniform Magnetic Field
The main goal of this paper is to calculate the corrections to the eigenvalues or energy levels of ( ) where χ is an arbitrary function of x and t , and A µ is the four vector potential representing the external e.m. field = ∇ × B A. In our case where there is only an external magnetic field, this requirement reduce to, 2) The resulting Hamiltonian should reduce to the sum of the Foldy-Wouthuysen reduced Hamiltonians (to order The last two terms on the right hand side of Equation (25) represents the terms obtained from the second requirement mentioned above. In Equation (25) we have to keep terms up to order 3 1 A where q is the electric charge of the particle.
We have reasons to believe we are on the right track with the Hamiltonian of Equation (21). In previous works, [14] we have shown that the Hamiltonian in the presence of an external radiation field obtained by means requirements (1) and (2) where we have chosen the symmetric gauge to define the vector potential. If B is along the Z-axis, the vector potential A will only have x and y components, so that Substituting Equation (25) where the c.m. orbital angular momentum operator, the operator e σ is the spin operator of the electron in the

Perturbative Calculation of Corrections to the Energy Levels of He + Ion Due to the External Uniform Magnetic Field
For the first order perturbation calculation, we first write down our explicit expressions for 0 H and V.
We can write 0 H as, where ( ) There are three types of terms in the expression for V given by Equation (31): 1) terms which depend only on the CM variables, contained as the first two lines of the right hand side of Equation (31) 2) the terms which depend only on the internal variables which are the remaining terms in Equation (31), except for the third, the fourth and the last four lines and 3) coupled terms which depend on the internal as well as the c.m. variables, namely, the third, the fourth and the last four lines of Equation (31). The fact that there are coupled terms is not surprising. When the composite system with a net electric charge is in an external magnetic field, the total momentum P is not conserved and we can not go to a frame where 0 = P for all times and so we do not expect the Hamiltonian to be uncoupled between internal and c.m. variables, as in the case of an isolated composite system.
For future reference, we will write the perturbation V as h h + , 2 j , z j , 2 l and 2 s , and the coherent states of the Landau Hamiltonian, described above. The coherent state of the Landau Hamiltonian is the simultaneous eigenstate of the annihilation operators a + and a − of the two dimensional simple harmonic oscillator [9], as defined in reference [9]. In this reference, we show that the coherent state is the best approximation to the classical state where the charged particle moves in a circle in the xy plane with the cyclotron angular frequency which for all practical purposes [9] is the kinetic energy of circular motion of the He + ion in the uniform magnetic field. The eigenvalues of the internal Hamiltonian ( ) are themselves calculated in the first order perturbation theory, treating ( ) 1 h as a perturbation to ( ) 0 h . So in our calculation we implicitly assume that the corrections to the energy levels due to the external magnetic field, the so called Zeeman splittings, are much smaller than the fine structure splittings induced by ( ) 0 h . This assumption is justified for any external magnetic field whose strength is such that, ( ) 4 2 . Numerically this condition is satisfied for any B whose magnitude is less than 10 Tesla or 10 5 gauss. For 10 Tesla, where 0c X and 0c Y are the x and y coordinates of the center of the circular

Results of the Perturbative Calculations in the First Order
We Several comments are in order about Equation (60) and Equation (61). In these equations CM E is the energy of the circular motion of the C.M. of the He + ion, corresponding to the coherent state of the Landau Hamiltonian of Equation (30). Its explicit expression is given by Equation (46). Since the CM motion represented by the coherent state [9] is practically classical, the eigenvalue of the annihilation operator a + satisfies the condition, which means the 1 2 in Equation (46) can be neglected. So, where c ω is given by Equation (33). In obtaining Equation (61) In deriving Equation (31)

Summary and Concluding Remarks
We have calculated the order 2 2 v c relativistic correction to the Zeeman Effect in He + ion, when its CM moves in a circular orbit under the action of a uniform magnetic field perpendicular to its plane of motion. We have assumed the weak field approximation in the sense that the splittings of energy levels due to the magnetic field are much smaller than the fine structure splittings. This is justified so long as the strength of the magnetic field is less than two Tesla or 20,000 Gauss. We also assumed the CM circular motion is approximately nonrelativistic.
Even if the kinetic energy of the CM circular motion is 1 MeV, it is much less than the rest mass energy of the He + ion which is about 9 3.76 10 eV × .
What is novel in our approach is that we have chosen the basis states, for the first order degenerate perturbation theory, to be the direct product of the cohe- The study of Zeeman effect in He + ion is important in astrophysical situation [17] [18], where there are He + ions in magnetic fields. We have only considered He 4 ion, where the He 4 nucleus has zero spin and zero magnetic moment. But our treatment can be easily extended to He 3 ion, where He 3 nucleus has spin 1 2 and a spin magnetic moment. In this case there will be extra terms in the Hamiltonian of Equation (1) and hence in Equation (22) which will give extra terms in the expressions of Equation (60) and Equation (61). Equation (60) will then include hyperfine splittings in the energy levels, due to the interaction of the electronic and the nuclear spin magnetic moments. The study of composite systems with net electric charge in a magnetic field has also received considerable attention [19] [20] in atomic and solid state physics.