^{3}.

Quantum effects on a Landau-type system associated with a moving atom with a magnetic quadrupole moment subject to confining potentials are analysed. It is shown that the spectrum of energy of the Landau-type system can be modified, where the degeneracy of the energy levels can be broken. In three particular cases, it is shown that the analogue of the cyclotron frequency is modified, and the possible values of this angular frequency of the system are determined by the quantum numbers associated with the radial modes and the angular momentum and by the parameters associated with confining potentials in order that bound states solutions can be achieved.

It is well-known in the literature that the Landau quantization [

The aim of this work is to analyse quantum effects on a Landau-type system associated with an atom with a magnetic quadrupole moment subject to some confining potentials. A great deal of work can be found in the literature with respect to studies of quadrupole moments of atoms and molecules, for instance, in single crystals [

The structure of this paper is as follows: in Section

In this section, we make a brief introduction of the Landau quantization associated with neutral particle (atom or molecule) with a magnetic quadrupole moment. First of all, by following [

An analogue of the Landau quantization for a moving atom that possesses a magnetic quadrupole moment was proposed in [

In this section, we analyse the effects of the confinement of the Landau-type system established in the previous section to a hard-wall confining potential. By substituting (

A particular solution to (

In condensed matter physics, a hard-wall confining potential is used with the purpose of describing a more realistic geometry of quantum dots and quantum rings as shown in [

Note that the parameter

Thereby, (

In this section, our focus is on the effects of a Coulomb-type interaction on the Landau-type system discussed in Section

From (

Observe that the asymptotic behaviour is determined for

By substituting function (

We proceed with using the Frobenius method [

Let us start with

By focusing on achieving bound states solutions, then, we need to impose the notion that the biconfluent Heun series becomes a polynomial of degree

Next, we analyse the condition

Hence, the general expression for the energy levels (

In contrast to [

Several works have dealt with a linear scalar potential in molecular and atomic physics through the perturbation theory [

By analysing the asymptotic behaviour as in the previous section, we can write the function

By substituting the function (

By imposing the notion that the biconfluent Heun series becomes a polynomial of degree

Further, let us analyse the second condition

Hence, the general form of the energy levels of the Landau-type system under the influence of a linear confining potential can be written as

By comparing the spectrum of energy (

Linear plus Coulomb-type potential has been studied in the context of the perturbation theory [

Again, we must impose the notion that the biconfluent Heun series becomes a polynomial of degree

From the second condition

Hence, we have obtained analytical solutions to the Landau-type system for an atom/molecule with a magnetic quadrupole moment to be subject to the Coulomb-type and linear confining potentials. Note that the possible values of the angular frequency

We have investigated the behaviour of a neutral particle (atom or molecule) with a magnetic quadrupole moment in a region with a uniform effective magnetic field subject to confining potentials. We have analysed the confinement of the Landau-type system to a hard-wall confining potential, a Coulomb-type potential, a linear confining potential, and a linear plus Coulomb-type potential. In the confinement to a hard-wall confining potential, we have seen that the spectrum of energy is modified in contrast to the Landau-type levels, where the energy levels are parabolic with respect to the quantum number associated with the radial modes.

On the other hand, with respect to the confinement to a Coulomb-type potential, a linear confining potential, and a linear plus Coulomb-type potential, we have obtained different spectrum of energies. In these three cases, the ground state of the system becomes determined by the quantum number

The authors declare that they have no competing interests.

The authors would like to thank the Brazilian agencies CNPq and CAPES for financial support.

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