On geometry influence on the behavior of a quantum mechanical scalar particle with intrinsic structure in external magnetic and electric fields

# On geometry influence on the behavior of a quantum mechanical scalar particle with intrinsic structure in external magnetic and electric fields

###### Abstract

Relativistic theory of the Cox’s scalar not point-like particle with intrinsic structure is developed on the background of arbitrary curved space-time. It is shown that in the most general form, the extended Proca-like tensor first order system of equations contains non minimal interaction terms through electromagnetic tensor and Ricci tensor .

In relativistic Cox’s theory, the limiting procedure to non-relativistic approximation is performed in a special class of curved space-time models. This theory is specified in simple geometrical backgrounds: Euclid’s, Lobachevsky’s, and Riemann’s. Wave equation for the Cox’s particle is solved exactly in presence of external uniform magnetic and electric fields in the case of Minkowski space. Non-trivial additional structure of the particle modifies the frequency of a quantum oscillator arising effectively in presence if external magnetic field. Extension of these problems to the case of the hyperbolic Lobachevsky space is examined. In presence of the magnetic field, the quantum problem in radial variable has been solved exactly; the quantum motion in z-direction is described by 1-dimensional Schrödinger-like equation in an effective potential which turns out to be too difficult for analytical treatment. In the presence of electric field, the situation is similar. The same analysis has been performed for spherical Riemann space model.

O.V. Veko111Kalinkovichi Gymnasium, Belarus,vekoolga@mail.ru, K.V. Kazmerchuk222Mosyr State Pedagogical University, Belarus, kristinash2@mail.ru, E.M. Ovsiyuk333Mosyr State Pedagogical University, Belarus, e.ovsiyuk@mail.ru, V.V. Kisel444Belarusian State University of Informatics and Radioelectronics, V.M. Red’kov555B.I. Stepanov Institute of Physics, NAS of Belarus, redkov@dragon.bas-net.by

PACS numbers: 02.30.Gp, 02.40.Ky, 03.65Ge, 04.62.+v

MSC 2010: 33E30, 34B30

KeywordsIntrinsic structure, scalar particle, curved space-time, generalized Schrödinger equation, magnetic field, electric field, Minkowski, Lobachevsky, Riemann space models

## 1 Scalar Cox’s particle with intrinsic structure

In 1982 W. Cox [1] proposed a special wave equation for a scalar particle with a larger set of tensor components than the usual Proca’s approach includes: he used the set of a scalar, 4-vector, antisymmetric and (irreducible) symmetric tensor, thus starting with the 20-component wave function (see Section 13).

First, let us consider the system of Cox’s equations [1] in the Minkowski space. We will use a Proca-like generalized system obtained after elimination from the initial system of Cox’s equations two second-rank tensors (see Section 13):

 (i√−g∂∂xα√−g−ecℏAα)gαβΦβ=McℏΦ,Kαρ(i∂α−ecℏAα)Φ=McℏΦρ. (1.1)

is a tensor inverse to ( stands for additional parameter responsible for non-trivia intrinsic structure of a scalar particle in Cox’s approach):

 Kαρ=λ1δβα+λ2Fβα+λ3FραFβρ+λ4FραFσρFβσ, (1.2)

are expressed through electromagnetic invariants (for more technical details see Section 13).

In geometrical models with metrics of special type one cap perform non-relativistic approximation and derive extended Schrödinger type equation (see Section 13):

 DtΨ=12M\lx@stackrel∘Dk(−gkj)(KljDl+McK0j)Ψ−12((K00−1)Mc2+Kj0cDj)Ψ, (1.3)

where the notation is used

 iℏ∂t−eA0=Dt,icℏ∂k−eAk=cDk,icℏ√−g∂∂xk√−g−eAk=c\lx@stackrel∘Dk,

It is a generalized Schrödinger equation for the particle with intrinsic structure.

In presence of a pure magnetic field, the above equation (1.3) takes a more simple form

 DtΨ=−12M\lx@stackrel∘Dkgkj(x)\lx@stackrel∗DjΨ, (1.4)

where the notation is used (let )

 \lx@stackrel∗D1=Kl1Dl=11+Γ2BiBi[D1+Γ(B2D3−B3D2)+Γ2B1(BiDi)], \lx@stackrel∗D2=Kl2Dl=11+Γ2BiBi[D2+Γ(B3D1−B1D3)+Γ2B2(BiDi)], \lx@stackrel∗D3=Kl3Dl=11+Γ2BiBi[D3+Γ(B1D2−B2D1)+Γ2B3(BiDi)], g22g33B1=B1,g33g11B2=B2,g11g22B3=B3,Fij=ϵijkBk.

In presence of a pure electric field, the above equation (1.3) takes the form

 (Dt−cΓ2EiEiMc+ΓEjDj2(1+Γ2EiEi))Ψ=12M\lx@stackrel∘Dk(−gkj)(Dj+Γ2Ej(EiDi)+McΓEj1+Γ2EiEi)Ψ (1.5)

where

 g11E1=E1,g22E2=E2,g33E3=E3,Ei=F0i.

## 2 Cox’s particle in the magnetic field, Minkowski space

Let the homogeneous magnetic field be directed along the axis :

 (Aj)=→A=12(x2B,−x1B,0).

Recalculating the potential to cylindrical coordinates by the formulas

 Aj′=∂xj∂xj′Aj,xj=(x,y,z),xj′=(r,ϕ,z),x1=rcosϕ,x2=rsinϕ,x3=x3′=z,

we obtain

 Ar=0,Aϕ=−Br22,Az=0,Frϕ=−Br. (2.1)

The metric tensor in these coordinates and field variables are determined by

 dS2=c2dt2−dr2−r2dϕ2−dz2,√−g=r,B3=−Br,B3=−Br−1,BiBi=B2. (2.2)

The Schrödinger equation for this case reads

 DtΨ=12M(\lx@stackrel∘D1\lx@stackrel∗D1+\lx@stackrel∘D2r−2\lx@stackrel∗D2+\lx@stackrel∘D3\lx@stackrel∗D3)Ψ,

where

 D1=iℏ∂r,D2=iℏ∂ϕ+ecBr22,D3=iℏ∂z,\lx@stackrel∘D1=iℏ(∂r+1r),\lx@stackrel∘D2=iℏ∂ϕ+ecBr22, \lx@stackrel∘D3=iℏ∂z,\lx@stackrel∗D1=11+Γ2B2(D1−ΓB3D2)=11+Γ2B2(iℏ∂r−ΓBr(iℏ∂ϕ+ecBr22)),
 \lx@stackrel∗D2=11+Γ2B2(D2+ΓB3D1)=11+Γ2B2((iℏ∂ϕ+ecBr22)+iℏΓBr∂r),
 \lx@stackrel∗D3=(D3+Γ2B3B3D3)1+Γ2B2=iℏ∂z;

below we will use the notation

 eB2ℏc=b,ΓB=γ.

We compute

 12M\lx@stackrel∘D1\lx@stackrel∗D1=−ℏ22M(1+γ2)(∂2r+1r∂r−γr∂r∂ϕ+iγbr∂r+2iγb), 12M\lx@stackrel∘D2\lx@stackrel∗D2=−ℏ22M(1+γ2)(1r2(∂ϕ−ibr2)2+γ(∂ϕ−ibr2)1r∂r), 12M\lx@stackrel∘D3\lx@stackrel∗D3=−ℏ22M(1+γ2)(1+γ2)∂2z. (2.3)

After using the substitution for the wave function

 Ψ=e−iEt/ℏeimϕeikzR(r),ϵ=2mEℏ2(1+γ2)

we get the radial Schrödinger equation

 [d2dr2+1rddr+2iγb+ϵ−(m−br2)2r2−(1+γ2)k2]R=0. (2.4)

By physical reasons parameter must be purely imaginary (see Section 13): ; so the radial equation reads

 [d2dr2+1rddr+2ηb+ϵ−(m−br2)2r2−(1−η2)k2]R=0. (2.5)

With the use of notation equation (2.5) can be written as

 (d2dr2+1rddr−(m−br2)2r2+ϵ′)=0, (2.6)

which coincides with the equation arising in the problem of the usual particle in the magnetic field. Its solutions are known. We present here only an expression for the energy spectrum

 ϵ=4b(n+m+∣m∣+12)+(1−η2)k2−2ηb; (2.7)

from this after translating to ordinary units we obtain

 E=p22M+11−η2eBMcℏ(n+m+∣m∣+12)−η211−η2eBMcℏ. (2.8)

With the use of notation the formula for the energy levels can be written as

 E=p22M+ωℏ1−(ΓB)2(n+m+∣m∣+12)−ωℏ1−(ΓB)2ΓB2. (2.9)

Thus, the intrinsic structure of the Cox’s particle modifies the frequency of the quantum oscillator (in fact, this result was firstly produced in different formalism by Kisel [2]).

 ω⟹~ω=ω1−Γ2B2,ω=eBMc. (2.10)

## 3 Cox’s particle in the magnetic field in the Lobachevsky space

In a special (cylindrical) coordinate system in the Lobachevsky space, analogue of the uniform magnetic field is determined by the relations (we use dimensionless coordinate obtained by dividing on the curvature radius ):

 dS2=c2dt2−ch2z(dr2+sh2rdϕ2)+dz2, Aϕ=−Bρ2(chr−1),Frϕ=−Bρsh% r,
 B3=−Bρshr,B3=−Bρshrch4z,BiBi=B2ch−4z. (3.1)

The wave equation in this case reads

 DtΨ=12Mρ2(\lx@stackrel∘D1\lx@stackrel∗D1+\lx@stackrel∘D21sh2r\lx@stackrel∗D2+\lx@stackrel∘D3\lx@stackrel∗D3]Ψ, (3.2)

where

 D1=iℏ∂r,D3=iℏ∂z,D2=iℏ∂ϕ+ecBρ2(chr−1),
 \lx@stackrel∘D1=iℏ(∂r+chrshr),\lx@stackrel∘D2=iℏ∂ϕ+ecBρ2(chr−1),\lx@stackrel∘D3=iℏ(∂z+2shzchz),
 \lx@stackrel∗D1=11+Γ2B2ch−4z[iℏ∂r−ΓBch−2zshr(iℏ∂ϕ+ecBρ2(chr−1))],
 \lx@stackrel∗D2=11+Γ2B2ch−4z[(iℏ∂ϕ+ecBρ2(chr−1))+iℏΓBch−2zshr∂r],
 \lx@stackrel∗D3=(D3+Γ2B3B3D3)1+Γ2B2ch−4z=iℏ∂z.

Below the notation is used:

 (eBρ2/ℏc)=b,ΓBch−2z=γ(z).

We compute

 12Mρ2\lx@stackrel∘D1g11\lx@stackrel∗D1=−ℏ2ch−2z2Mρ2(1+γ2(z)) ×(∂2r+(chrshr+iγ(z)bchr−1shr)∂r−γ(z)% shr∂r∂ϕ+iγ(z)b), 12Mρ2\lx@stackrel∘D2g22\lx@stackrel∗D2=−ℏ2ch−2z2Mρ2(1+γ2(z)) ×[1sh2r[∂ϕ−ib(chr−1)]2+γ(z)[∂ϕ−ib(chr−1)]1shr∂r], 12Mρ2\lx@stackrel∘D3g33\lx@stackrel∗D3=−ℏ22Mρ2(∂z+2shzchz)∂z.

After using the substitution for the wave function:

 Ψ=e−iEt/ℏeimϕZ(z)R(r), (3.4)

the Schrödinger equation (3.2) gives (the function must be imaginary, )

 [ch−2z1−γ2(z)(∂2r+chrshr∂r−[m−b(chr−1)]2sh2r+bγ(z)) +ϵ+(∂z+2shzchz)∂z]R(r)Z(z)=0,ϵ=Eℏ2/2Mρ2. (3.5)

In this equation, the variables are separated:

 1R(d2dr2+chrshrddr−[m−b(chr−1)]2sh2r)R +1Z(1−γ2(z))ch2z(bγ(z)ch−2z1−γ2(z)+ϵ+(ddz+2shzchz)ddz)Z=0. (3.6)

 (d2dr2+chrshrddr−[m−b(chr−1)]2sh2r+Λ)R=0; (3.7)

the equation for is (remember that )

## 4 Analysis of the equation in the variable z

In equation (3), let us eliminate the first derivative term:

 Z=1chzf(z),U(z)=−bγ−Λ% ch2zch4z−γ2, (d2dz2+ϵ−1−U(z))f(z)=0. (4.1)

Eq. (4.1) can be viewed as the Schrödinger equation in the effective potential field . The corresponding effective force is

 Fz=−dUdz =2chzshzΛch4z−2bγch2z+γ2Λ(ch4z−γ2)2. (4.2)

We find the points of local extremum: and the roots of a quadratic equation

 Λch4z−2bγch2z+γ2Λ=0⟹ (ch2z)|1,2=bΛγ±√(b2Λ2−1)γ2. (4.3)

When considering the bound states (for motion in the variable ) we have . This means that the square root in (4.3) is an imaginary number. Consequently, the point of zero force (equilibrium points) except cannot exist. The situation is illustrated in the Fig. 1.

After the change of variables , the differential equation (4.1) reads

 [d2dy2+(321y+121y−1)ddy+ϵ4y(y−1)+bγ−Λy(y−γ)(y+γ)4y(y−1)]Z(y)=0. (4.4)

Note that singular points are located outside the physical range of the variable. Further progress in analytical treatment of eq. (4.4) (with 5 singular points) is hardly possible.

## 5 Solution of the radial equation

Let us turn to the radial equation (3.7) for the function . It is solvable in hypergeometric functions – see more detail [4]. Below we will write done only final results on energy spectrum. There exist only finite series of bound states, defined by relations

 m<2b, m+∣m∣2+n+1/2≤b, Λ−1/4=2b(m+∣m∣2+n+1/2)−(m+∣m∣2+n+12)2; (5.1)

obeys the restriction In usual units the last relation can be written as:

 Λ−14=ρ2Λ0−14,m<2B,m+n+1/2≤eBℏcρ2, ρ2Λ0−14=2eBℏcρ2(m+∣m∣2+n+1/2)−(m+∣m∣2+n+1/2)2,n=0,1,…,NB.

In the limit of vanishing curvature, we obtain the known result in the flat space

 E−P22M=eBℏMc(m+∣m∣2+n+1/2).

## 6 Cox’s particle in the electric field, Minkowski space

Schrödinger equation for Cox’s particle in the electric field has the form (see Section 13)

 (Dt−cΓ2EiEiμ+ΓEjDj2(1+Γ2EiEi))Ψ=−12M\lx@stackrel∘Dkgkj[Dj+Γ2Ej(EiDi)+μΓEj1+Γ2EiEi]Ψ;

the notation is used:

 A0=−eEz,Ei=(F01,F02,F03),g11E1=E1,g22E2=E2,g33E3=E3, iℏ∂t−eA0=Dt,iℏ∂k=Dk,iℏ√−g∂∂xk√−g=\lx@stackrel∘Dk,μ=Mc.

Let us use cylindric coordinates

 dS2=c2dt2−dr2−r2dϕ2−dz2,E3=E,E3=−E,E3E3=−E2. (6.2)

First, we get (let it be )

 (Dt−cΓ2EiEiμ+ΓEjDj2(1+Γ2EiEi))=iℏ∂t+eEz+cγ2μ+γD32(1−γ2). (6.3)

Next, we consider the Hamiltonian

 H=12M[\lx@stackrel∘D1D1+\lx@stackrel∘D21r2D2+\lx@stackrel∘D3(D3+μγ1−γ2)]. (6.4)

In explicit form, the extended Schrödinger equation looks as follows (to allow for the imaginary character of , we make formal change )

 (iℏ∂t+eEz−Mc2γ22(1+γ2)+γ2(1+γ2)ℏc∂z)Ψ =−ℏ22M(∂2r+1r∂r+∂2ϕr2+∂2z−(Mc/ℏ)γ1+γ2∂z)Ψ.

With the substitution and the notation

 M2c2ℏ2=1λ2,2Mℏ2W=w,2Mℏ2eEz=ν,

we get

 1Z(z)(∂2z+νz+w−1λ2γ21+γ2)Z(z)+1R(r)(∂2r+1r∂r−m2r2)R(r). (6.5)

After separation of the variables ( stands for the separation constant) we derive

 (∂2r+1r∂r−m2r2+w⊥)R(r)=0, (∂2z+νz+w′)Z(z)=0, (6.6)

where

 w′=w−w⊥+1λ2γ21+γ2.

In fact, (6.6) coincide with the well known equations for an ordinary particle in the uniform electric field. Equation in the variable looks as a one-dimensional Schrödinger equation in the potential of the form :

 (d2dz2+w′+νz)Z(z)=0. (6.7)

The form of the curve says that any particle moving from the right must be reflected by this barrier in vicinity of the point (we assume that electric force acts in positive direction of the axis ).

Mathematic solutions of the equation (6.7) can be expressed in Airy function. Indeed, let us change the variable

 νz+w′=ax,(d2dx2+a3ν2x)Z(x)=0;

let it be (for definiteness )

 a3ν2=−1,a=−ν2/3,x=νz+w′−ν2/3=−ν1/3z−w′ν2/3; (6.8)

then we arrive at the Airy equation

 (d2dx2−x)Z(x)=0; (6.9)

to the turning point there corresponds the value .

Eq. (6.9) can be related to the Bessel equation. Indeed, let us introduce the variable

 ξ=23x3/2,x=32ξ2/3,

then Airy equation gives

 (13ξddξ+d2dξ2−1)Z=0.

Applying the substitution , we arrive at the Bessel equation [8]

 (d2dξ2+1ξddξ−1−1/9ξ2)f(ξ)=0 (6.10)

with two linearly independent solutions

 f1(ξ)=J+1/3(iξ),f2(ξ)=J−1/3(iξ). (6.11)

Thus, general solutions of Airy equation can be constructed as linear combinations of

 Z1(x)=ξ1/3J+1/3(iξ),Z2(x)=ξ1/3J−1/3(iξ),

where

 iξ=i2√ν3(z+w′ν)3/2.

With the use of the known relation [8, 8]

and with the notation , one expresses two independent solutions of the Schrödinger equation as follows

 Z1=ξ1/3J