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Journal of Modern Physics

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

Kunnat J. Sebastian, Wangyao Li

Published Jan 1, 2020• 0 citations

Abstract

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 . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , 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, j2, l2, s2 and jz, 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 Z2 a2 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 (ECM ) of the CM circular motion of the He+ ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He+ ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He+ ions in magnetized astrophysical objects.

DOI: 10.4236/jmp.2020.113029

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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 . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , 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, j2, l2, s2 and jz, 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 Z2 a2 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 (ECM ) of the CM circular motion of the He+ ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He+ ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He+ ions in magnetized astrophysical objects.

Abstract In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z -axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a + and a − . We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency ω c = ∣ q ∣ B m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a + in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a − . The phase of the circular motion is the same as the phase of the complex eigen value of a + . So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z -axis. We also show that as the magnitude of the eigen value of a + becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.

The Land\'e

g

factor describes the response of an atomic energy level to an external perturbation by a uniform and constant magnetic field. In the case of many-electron systems, the leading term is given by the interaction ${\ensuremath{\mu}}_{B}\phantom{\rule{0.16em}{0ex}}(\stackrel{P\vec}{L}+2\phantom{\rule{0.16em}{0ex}}\stackrel{P\vec}{S})\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{B}$, where $\stackrel{P\vec}{L}$ and $\stackrel{P\vec}{S}$ are the orbital and spin angular momentum operators, respectively, summed over all electrons. For helium, a long-standing experimental-theoretical discrepancy for

P

states motivates a re-evaluation of the higher order terms which follow from relativistic quantum theory and quantum electrodynamics (QED). The tensor structure of relativistic corrections involves scalar, vector, and symmetric and antisymmetric tensor components. We perform a tensorial reduction of these operators in a Cartesian basis, using an approach which allows us to separate the internal atomic from the external degrees of freedom (magnetic field) right from the start of the calculation. The evaluation proceeds in a Cartesian basis of helium eigenstates, using a weighted sum over the magnetic projections. For the relativistic corrections, this leads to a verification of previous results obtained using the Wigner-Eckhart theorem. The same method, applied to the radiative correction (Bethe logarithm term) leads to a spin-dependent correction, which is different for singlet versus triplet

P

states. Theoretical predictions are given for singlet and triplet

2P

and triplet

3P

states and compared to experimental results where available.

For neutral particle systems in a homogeneous magnetic field, there exists a constant of motion, the pseudomomentum, which allows a complete pseudoseparation of the center-of-mass motion. For ions such a separation is not possible, and the Landau orbit of the center-of-mass and internal (electronic) motion are intimately coupled. We investigate different physical situations for which this interaction becomes strong at laboratory magnetic-field strengths. In particular, we find that the Landau orbit itself can change substantially upon this interaction.

In the relativistic description of atomic systems in external fields, the total momentum and the external electric field couple to the angular momentum of the individual particles. Therefore, the motional state of an ion in a particle trap influences measurements of internal observables such as energy levels or the

g

factor. We calculate the resulting relativistic shift of the Larmor frequency and the corresponding

g

-factor correction for a bound electron in a hydrogenlike ion in the

1 S

state due to the ion moving in a Penning trap and show that it is negligible at the current precision of measurements. We also show that the analogous energy shift for measurements with an ion in the ground state of a Paul trap vanishes in leading order.

The approximate constant of motion introduced previously for the description of a charged system in a homogeneous magnetic field is interpreted physically as the kinetic momentum of the collective motion. The algebra satisfied by this operator and by the exact constants of motion describes the behavior in a magnetic field of a single particle possessing the total charge of the system. With this algebra, we generalize the approximate constant of motion and obtain a family of operators depending on arbitrary parameters. Canonical transformations based on these new operators separate the Hamiltonian into collective, internal, and coupling terms. These terms take the same form for charged and neutral systems although the collective energy represents different physical behaviors. The coupling between the internal and collective motions is small for some choices of the parameters if at least one particle is much heavier than the other ones.

Introduction The claim that it is impossible to construct a realistic physical model of quantum matter is often presented as definitively and finally proved in the case of systems with spin ½, or internal angular momentum of magnitude ħ/2, which exhibit a characteristic ‘nonclassical two-valuedness’. Nevertheless, the extension of the causal interpretation to embrace nonrelativistic spin ½ systems was successfully carried through in the 1950s (Bohm, Schiller & Tiomno, 1955; Bohm & Schiller, 1955; Takabayasi, 1955; Takabayasi & Vigier, 1957) and developed more recently (Dewdney, Holland & Kyprianidis, 1986, 1987a, b; Dewdney, Holland, Kyprianidis & Vigier, 1988). The idea is to assume that the spinor wave represents a new type of physical field propagating in spacetime that exerts an influence on a corpuscle moving within it. The additional information conveyed by the spinor beyond that contained in the spinless Schrödinger wave implies that the state of the particle is not exhausted by specifying its position and momentum, but that it is endowed with further properties such as an internal angular momentum. The equation governing the dynamics of the spinor wavefunction is the Pauli equation: where the Hamiltonian operator is given by Here e, m and μ are respectively the charge, mass and magnetic moment of the particle, A 0 and A are the electromagnetic potentials, B = ∇ × A is the magnetic field, V is any other external scalar potential, and σ is a vector whose components are the 2 × 2 Pauli matrices. The Hamiltonian (9.1.2) istherefore a 2 × 2 matrix operator and the wavefunction ψ( x , t ) has two complex components.