Pauli Oscillator In Noncommutative Space

Pauli Oscillator In Noncommutative Space

Mebarek Heddar

¹

Mokhtar Falek

¹

Mustafa Moumni

^1,2

Bekir C. L¨ utf¨ uo˘ glu

^3,a

1

Laboratory of Photonic Physics and Nano-Materials (LPPNNM) Department of Matter Sciences, University of Biskra, ALGERIA

2

PRIMALAB, University of Batna1, ALGERIA

3

Akdeniz University, Antalya, TURKEY

a

correspondant author bclutfuoglu@akdeniz.edu.tr August 7, 2021

Abstract

In this study, we investigate the Pauli oscillator in a noncommutative space. More precisely, we calculate the energy spectrum of a non-relativistic charged particle of spin half that is moving under the action of a constant magnetic field and a oscillator potential in a noncommutative space. We obtain a critical value of a parameter of deformation θ and a critical value of magnetic fieldB, this values counteract the normal Zeeman effect and the anoumalous Zeeman effect by the effect of the noncommutative space. Finally we examine the thermal properties of the system.

Keywords: Pauli Equation; Non-commutative space; Harmonic oscillator

1 Introduction

Historically a large amount effort has been devoted to study the physical effect of noncommutative space [1]. This was motivated by the appearance of noncommutative space effect in the very tiny string scale or at very high energy [2, 3], M-theory [4] compactification and quantum Hall effect [5].

Inclusion of noncommutativity in quantum field theory can be achieved in either of two different ways: via Moyal product,∗, on the space of ordinary functions, or by defining the field theory on a coordinate operator space which is intrinsically non-commutative [6]. The equivalence between the two approaches has been nicely described in Ref. [7] A simple insight on the role of noncommutativity in field theory can be obtained by studying the one particle sector, which prompted an interest in the study of noncommutative quantum mechanics (NCQM) [8, 9]. There are a lot of papers that investigate this problem with different potentials, one of this potentials models is the harmonic oscillator model, [10–14], which contains bound states with a non-zero residual energy and which has been explained by the quantum confinement effect, the Schrodinger oscillator is used to describe the confinement of quarks in the mesons and baryons, while the Dirac oscillator is mainly motivated by its use as a quark-confining potential in quantum chromodynamics [1]. Besides of this fundamental Schr¨odinger and Dirac oscillator, there is another interesting non-relativistic oscillator is Pauli

oscillator, ,this the last is our topic in noncommutative space . The energy levels of this system are called Landau levels there several works are made noncommutative geometry to explain the Landau problem (charged particle moving in magnetic field) , one of them use the two dimensional noncommutative quantum mechanics it has been shown that the equation of motion of harmonic oscillator in a noncommutative space is similar to that of a particle in a constant magnetic field and in the lowest landau level [15], and after that they generalize these relation to the relativistic domain,it is shown that the Dirac and Klein-Gordon oscillators in a non-commutative space is similar to the landau problem in a commutative space although an exact map does not exist [16]

the analytic solution of these equation was calculated and the wave function also, in this work we stay in the non-relativistic motion and studied the landau problem of partial with spin in Harmonic Oscillator potential using the Pauli equation this equation which is only the schrodinger equation of a charged particle in a magnetic field plus the spin interaction the previous works interested by the effect of the magnetic field and explain it as a Harmonic Oscillator.

The aim of this work is solve the Pauli oscillator equation exactly in non-commutative space and to see also the effect of the deformation on non-relativistic system with spin.To reach this gaol ,the outline of this paper is as follows: In section 2 2, we expose the analytic solution of the Pauli equation of oscillator in non-commutative space when the energy spectrum are deduced and we investigate the critical case where the effect of magnetic field was eliminate by the non-commutative effect . In the regime of high temperatures, the thermodynamic properties of the system are investigated and discussed in section 3 3. The last section 4 is left for concluding remarks.

2 Pauli oscillator equation in noncommutative space

Pauli equation is a quantum equation, that describes the dynamics of the spin-half particle according to the particle’s spin interactions with an external electromagnetic field [17]. Pauli Hamiltonian is based on the non-relativistic limit of the Dirac Hamiltonian. In the absence of a scalar potential energy, Pauli Hamiltonian of a massive charged particle, i.e. an electron with the mass and charge, is given in the form of [18]:

H_P = (~σ·~Π)²

2m = 1

2m

~Π·~Π +i~σ·(Π~ ×Π)~

(1) Here, ~σ represents the Pauli matrices, while ~Π denotes the kinetic momentum terms which are defined by the minimal coupling of the canonical momentum,~p, and the vector potential,A~ =B~×~r, viaΠ =~ ~p−e ~A. Note that, B~ is the external magnetic field.

2.1 Pauli oscillator in commutative space

We consider an oscillator potential energy, ¹₂mω²r², whereω is the oscillator frequency, and define the Pauli oscillator (PO) Hamiltonian in the form of:

H_{P O}=H_P +mω²

2 r² (2)

In a commutative space, we can examine the dynamic of an electron subject to the PO by solving the following equation:

1 2m

Π~ ·Π +~ i~σ·(~Π×Π)~

+m²ω²r²

Ψ (~r, t, s) =EΨ (~r, t, s) (3) Alternatively, one can express eq.3 explicitly in the following form: When we substitute the operator

−

→Π by its expression, we obtain the PO equation in explicit form:

1 2m

"

~ p−e

B~ ×~r²

+i~σ·h

~ p−e

B~ ×~r

×

~ p−e

B~ ×~ri

+m²ω²r²

#

Ψ (~r, t, s)

=EΨ (~r, t, s) (4)

2.2 Pauli oscillator in noncommutative space

In a noncommutative space, commuting operators are substituted with the noncommutative ones so that eq.2 is modified. Roughly speaking, the nonzero values of the new commutations of the coordinate operators cause additional terms to the commutative case which can be seen as a per- turbative contribution [19]. To determine those terms, we have to state the noncommutative space.

Now, let us rewrite the PO equation in noncommutative space. Where their coordinate operators satisfies the commutation relations [15]:

bxⁱ,by^j

=iθ^ij (5)

Andθ^ij is an antisymmetric tensor of space dimension and plays an analogous role to~in the usual quantum mechanics. To preserve the unitarity of the theory. In the non-commutative space, unlike the commutative space, the product of any two infinitely small differentiable functions is expressed with the Moyal-star product, that is originally defined by [20]:

(f∗g) (x) = exp i

2θµν∂x_µ∂y_ν

f(x)g(y)x=y (6)

In the case of [pbi,pbj] = 0, the noncommutative quantum mechanics H(p, x)∗Ψ (x) = EΨ (x) reduces to usual one described byH(˜p,x)Ψ (x) =˜ EΨ (x) [21]. In other words, if one defines a new operators sets by using the noncommutative coordinate operator in terms of following commuting coordinate operators, namely Bopp’s shift [22, 23]:

x_i −→bx_i = x_i− 1 2~

θ_ijp_j (7a)

pi −→pbi = pi (7b)

fori= 1,2, then, the new operators obey the usual canonical commutation relations [24, 25]. Here, θ^ij is chosen as:

θ_ij =_ijkθ_k, and θ₃=θ (8) where_ijk is the Levi-Civita tensor. Based on this choice, we rewrite the transformations 7a and 7b, in the following compact form:

~

r−→~r+

~θ×~p 2~

withθ12=−θ21= θ3=θ (9)

By taking account of this expression, the PO equation can be put as:

1 2m

"

~

p−e ~B× ~r+ θ~×~p

2~

!!

· ~p−e ~B× ~r+

~θ×~p 2~

!!

+ i~σ· ~p−e ~B× ~r+

~θ×~p 2~

!!

× ~p−e ~B× ~r+

~θ×~p 2~

!!

+ m²ω² ~r+

~θ×~p 2~

!

· ~r+

~θ×p~ 2~

! #

Ψ (~r, t, s) =i~∂Ψ (~r, t, s)

∂t (10)

For simplicity, we assume the orientation of the magnetic field along the z-axis. We take the noncommutative gauge:

A~= 1 2

B~ ×~r=B

2(−byi+bxj), A0= 0 (11)

After some straightforward calculations, the PO equation 10 reduces to the following form:

"

p²_x+p²_y 2M + p²_z

2m+Meω²

2 x²+y² +mω²

2 z²− ηLZ−eB~ 2m

1 +eB

4~ θ

σz

#

Ψ =i~∂Ψ

∂t (12) where

M =

"

1 m

1 + eB

4c~ θ

² +mω²

θ 2~

²#−1

(13a)

ωe =

"

1 M

1 m

eB 2c

² +mω²

!#¹₂

(13b) η = 1

2 eB

m

1 +eBθ 4~

+mω²θ

~

(13c) Before we explore a solution to eq.12, we briefly discuss the following cases:

In a commutative space (θ= 0) and without an external magnetic field (B= 0), we findM =m, eω=ω andη= 0; thus, PO equation transforms to the form:

"

p²_x+p²_y+p²_z

2m +mω²

2 x²+y²+z²

#

Ψ =i~∂Ψ

∂t (14)

In a commutative space (θ= 0) with the presence of an external magnetic field, we findM =m, eω²=ω²+ ^eB_2m²

andη=_2m^eB; so, PO equation writes in the form:

"

p²

2m +mω² 2 r²+ 1

2m eB

2c ²

x²+y²

−eB

2m(L_z+~σ_z)

# Ψ =i~

∂Ψ

∂t (15)

In a noncommutative space without an external magnetic field, we findM =h

1

m+mω² ₂^θ

~

²i−1

, eω²=

ω²+ mω²₂^θ

~

²

andη= ^mω₂^2θ

~ ; Thus, PO equation transforms to:

"

p²

2m +mω² 2 r²+

mωθ 2~

2 p²_x+p²_y 2m

!

−mω²θ 2~ LZ

# Ψ =i~

∂Ψ

∂t (16)

Note that, in this last case, the terms related to the deformation parameter can be taken as per- turbation terms as mentioned at the very beginning of this subsection.

2.3 Solution

To find the time independent PO equation out of eq.12, we take the wave function:

Ψ (~r, s, t) =e^−iE~tψ_s(~r) (17) Thus we have to solve the following eigenvalue equation:

"

p²_x+p²_y

2M +Meω²

2 x²+y² + p²_z

2m+mω²

2 z²−ηL_Z

#

ψ_s(~r) =

E+eB~ 2m

1 + eB

4~θ

σz

ψs(~r) (18) We introduce the cylindrical coordinate (r, ϕ, z,) and doing so, eq.19 becomes:

"

− ~² 2M

1 r

∂

∂r(r∂

∂r) + 1 r²

∂²

∂ϕ²

+Mωe²

2 r²− ~² 2m

∂²

∂z²+mω² 2 z²−η

i~ ∂

∂ϕ #

ψs(~r)

=

"

E+eB~ 2m

1 +eB

4~θ

σz

#

ψs(~r) (19) We write the two-component spinor as follows:

ψs(~r) =ψs(r, φ) =χsexp(imlϕ)R(r)

√r Z(z), s= +1,−1 (20) where χ^T₊₁ = (1,0), and χ^T₋₁ = (0,1). By substituting eq.20 into eq.19, we obtain two separate equations the azimuthal equation and the radial equation:

~² 2m

∂²

∂z² −mω²

2 z²+Ez

Z(z) = 0 (21)

~² 2M

d² dr²+1

r d dr−m²_l

r² −M²eω²

~² r²

+E

R(r)

√r (22)

with:

E= 2M

~²

E+eB~ 2m

1 +eB

4~θ

s+ηml~−Ez

(23) The azimuthal part is the Schr¨odinger equation for the 1D harmonic oscillator; so we write:

E_z=E_nz =~ω

n_z+1 2

;n_z= 0,1,2 (24)

and its wave eigenfunctions are:

Z(z) =C0exp

−mω 2~

z² Hn_z

rmω

~ z

(25)

whereC0denotes the normalization constant andHn_z pmω

~ z

represents the Hermite polynomials.

The radial part for R(r) is:

d²

dr² −M²ωe²

~²

r²−m²_l −(1/4) r² +E

R(r) = 0 (26)

We consider the following transformation:

R(r) =e^−ξ2ξ^kW(ξ), ξ=r a

²

, a²= ~ Mωe =

~ h1

m 1 + _4c^eB

~θ²

+mω² ₂^θ

~

²i¹₂ h1

m eB

2c

²

+mω²i¹₂ (27) Thus, eq.26 reduces to the following form:

ξ d²

dξ² +

2k+1 2−ξ

d dξ +n

W(ξ) = 0 (28)

where:

k= 1 2

|ml|+1 2

, n= ε

4 −1

2(|ml|+ 1), ε= ~

MωeE (29)

We note that eq.28 is the confluent hypergeometric equation and we choose its regular solution:

W(ξ) =CF(−n;|ml|+ 1;ξ) (30)

We combine the results from eq.27 and eq.30 to build the radial solution:

R(r) =Ce^−r2/2a2r^|ml|+1/2F

−n;|ml|+ 1,r² a²

(31) Then, we obtain the wave function from eq.20 using both the azimuthal (eq.25) and the radial (eq.31) solutions (N is the normalization constant):

ψn,m_l,n_z(~r, t, s) =N χse^imlϕexp− r²

2a² +mω

2~z² r² a²

|^ml2 |

×

×F

−n;|ml|+ 1,r² a²

Hn_z

rmω

~ z

, s=±1 (32) Before we investigate the quantization of our solutions, we mention that the contribution of noncommutative deformation appears in the parameter a derived in eq.27 and we would like to present the value of this parameter in the three limit cases:

• In a commutative space, θ= 0, without an external magnetic field,B= 0, a²= ~

mω (33)

• In a commutative space, θ= 0, with the presence of an external magnetic field,

a²= ~ mω

"

1 + eB

2mω

2#−1/2

(34)

• In a noncommutative space, without an external magnetic field,B= 0,

a² = ~ mω

"

1 + mωθ

2~

²#−1

(35)

Now we use eqs. 23, 24 and 29, to write energy eigenvalues of the system:

En,m_l,n_z

~ω =

v u u t

1 m

eB 2cω

² +m

! "

1 m

1 + eB

4c~ θ

² +m

ωθ 2~

²#

(2n+|m_l|+ 1)

+

nz+1 2

− eB

2mω

1 + eBθ 4c~

(s−ml)−mωθ 2~ ml

(36) Here also, we write the three limit cases:

• In a commutative space (θ= 0) without an external magnetic field (B= 0):

En,m_l,n_z =~ω

(2nρ+|ml|+ 1) +

nz+1 2

(37) We put 2n_ρ+|ml|+n_z=nso:

E_n,ml,nz =~ω

n+3 2

(38)

• In a commutative space (θ= 0) with the presence of an external magnetic field:

En,m_l,n_z =~ω



 s

1 + eB

2mω 2

(2n+|ml|+ 1) +

nz+1 2

− eB

2mωc(s−ml)



 (39)

• In a noncommutative space, without an external magnetic field (B= 0):

En,m_l,n_z =~ω



 s

1 +

m²ω²θ 2~

²

(2n+|ml|+ 1) +

nz+1 2

+mωθ

2~ ml



 (40) We observe that the effect of the noncommutative space is able to counteract the effect of the normal Zeeman effect when the coefficient ofLz (orml in eq.36) vanishes and this give us a critical value for the magnetic field. In order to determine these critical values, we solveη= 0 in 13c and we find:

Bc=−2~ eθ



1± s

1− mωθ

~ ²



 (41)

where we ignore the nonphysical second root. At the critical value, the energy eigenvalue function reduces to:

En =~ω

~ω

n+3 2

−mωθ 2~

s

(42)

The effect of the noncommutative space is also able to counteract the effect of the magnetic field in the case of anomalous Zeeman effect when the coefficient ofσz (ors in eq.36) is equal to zero.

Here we obtain an other critical magnetic field:

B_c⁰ =−4c~

eθ (43)

In this case, the energy eigenfunctions become:

En,m_l,n_z =~ω



 s

1 +m²ω² θ

2~ ²

(2n+|ml|+ 1) +

nz+1 2



+~ω mωθ

2~

ml (44)

In order to get an upper bound for theθ parameter, we use thes−states of the energies from 36 and we expand them up to the first order inθ:

En,0=E_n,0,0^θ=0 +~ω



(2n+ 1)eB 4~

s 1 +

eB 2mω

²

−(eB)²s 8mω~



θ (45) with:

E_n,0,n^θ=0_z =~ω



(2n+ 1) s

1 + eB

2mω ²

+ (nz+1 2)



−~eBs

2m (46)

These two relations show that the deviation of then-th energy level caused by the modified com- mutation relations 1 is provided by:

∆E^θ_n,0

~ω =



(2n+ 1)eB 4~

s 1 +

eB 2mω

²

−(eB)²s 8mω~



θ (47) We use the experimental results of the electron cyclotron motion in a Penning trap. Here, the cyclotron frequency of an electron trapped in a magnetic field of strengthBisωc=eB/me(without deformation), so we haveme~ωc=e~B= 10⁻⁵²kg²m²s⁻²for a magnetic field of strengthB= 6T. If we assume that only the deviations of the scale of ~ωc can be detected at the level n = 10¹⁰ and that ∆En <~ωc (no perturbation of then-th energy level is observed) [26], we can write the following constraint:

θ <2.57×10⁻²⁶m² (48)

3 Thermal Quantities

In this section, we examine the thermodynamic properties of the deformed PO at a finite tem- perature. For this, we consider the system to be in equilibrium with a reservoir at a constant temperature,T. Then we use the well-known definition of the partition function:

Z(T) =

∞

X

n=0

e^β(En−E0) (49)

whereβ = _k¹

βT, andkβ is the Boltzmann constant. The obtained energy eigenfunction of the PO can be expressed as:

En=αn+λ (50)

where:

α= 2~ω v u u t

1 m

eB 2cω

² +m

! "

1 m

1 + eB

4c~ θ

² +mω²

θ 2~

²#

(51) and

λ

~ω =



 v u u t

1 m

eB 2cω

² +m

! "

1 m

1 + eB

4c~ θ

² +mω²

θ 2~

²#

(|m_l|+ 1)





+

nz+1 2

− eB

2mω

1 +eBθ 4c~

(s−ml)−mωθ 2~

ml

(52) Since E0 is the fundamental state energy, namelyE0 = λ, we immediately find En−E0 = αn.

Therefore, we express the partition function as:

Z(T) =

∞

X

n=0

e^−β(αn). (53)

In order to calculate the partition function we employ the Euler-Maclaurin formula:

∞

X

n=0

f(n) = 1 2f(0) +

Z ∞ 0

f(x)dx−

∞

X

p=1

1

(2p)B2pf^(2p−1)(0), (54) whereB_2p are the Bernoulli numbers, i.e. B₂ = ¹₆,B₄=−₃₀¹, andf^(2p−1) denotes the derivative of order (2p−1). According to:

Z ∞ 0

f(x)dx = Z ∞

0

e^−β(αx)dx= 1

βα (55a)

f⁽¹⁾(0) = −βα (55b)

f⁽³⁾(0) = −β³α³ (55c)

We find the partition function as (τ=β⁻¹):

Z(τ)'1 2 +τ

α+ α

12τ − α³

120τ³ (56)

Next, we use the partition to evaluate the thermal functions. At first, we obtain the free energy by recallingF=−β⁻¹lnZ.

F =−τln 1

2+ τ α+ α

12τ − α³ 120τ³

(57) Then, we derive the average energy by employingU =τ^2∂_∂τ^lnZ.

U = 120τ⁵−10α²τ³+ 3α⁴τ

60ατ³+ 120τ⁴+ 10α²τ²−α⁴ (58)

Figure 1: The partition functionF versusτ=kβT

Figure 2: The Helmholtz free energyF versusτ=k_βT After that, we evaluate the reduced specific heat from _k^C

B =^∂U_∂τ. We arrive at:

C kB

= 14400τ⁸+ 14400ατ⁷+ 4800α²τ⁶−1780α⁴τ⁴−360α⁵τ³−3α⁸

(60ατ³+ 120τ⁴+ 10α²τ²−α⁴)² (59) Finally, we obtain the reduced entropy according to _k^S

B =−^∂F_∂τ: S

k_B = ln 1

2+ τ α+ α

12τ − α³ 120τ³

+

120τ⁴−10α²τ²+ 3α⁴ 60ατ³+ 120τ⁴+ 10α²τ²−α⁴

(60) The plot of Z partition and the thermal functions is in figure 1 to figure 5

At this point, it is remarkable to note that these predicted expressions can be tested; using the limitθ →0, we obtain the ordinary results for the thermodynamic grandeurs of the usual Pauli oscillator.

4 Conclusion

In this work we have exposed an analytic study of the 2D Pauli oscillator in a noncommutative space and with the presence of a magnetic field. Written the Pauli equation of our system in the

Figure 3: The internal energyU versusτ=kβT

Figure 4: The specific heatC versusτ =kβT

Figure 5: The entropyS versusτ=kβT

cylindrical coordinates (r, ϕ, z), we found the explicit solutions of both the energy eigenvalues and the wave functions. The energy spectrum depends, as it should, on the deformation parameter θ which adds contributions coming from the interactions of the noncommutativity with both the angular momentum and the spin. These additional contributions are similar to those coming from the interaction of the system with the magnetic field and thus, we found that the effect of the noncommutative space is able to counteract the Zeeman effect when the magnetic field equal to a critical valueBc=−(2~/eθ)

1±

q

1−(mωθ/~)²

. The same effect was found for the anomalous Zeeman effect and we got another critical value for the magnetic fieldB_c⁰ =−4c~/eθ.

We have validated our spectrum by studying the limit θ−→0 which give us the same results found for the ordinary space. We have also determined an experimental limit for the deformation parameterθby using the results of the electron cyclotron motion in a Penning trap and thus, we have found that for a magnetic field of strengthB= 6T, we have the constraintθ <2.57×10⁻²⁶m². In the regime of high temperatures, we showed that the thermodynamic properties of our sys- tem have been also influenced by the noncommutative parameterθ. The plots of these thermody- namic quantities show that all the thermodynamic quantities decrease with increasingθexcept the Helmholtz free energy.

References

[1] M. Moshinsky and A. Szczepaniak, J. Phys. A. Math. Gen.22, L817 (1989) [2] E. Witten, Nucl. Phys. B460, 335 (1996)

[3] N.Seiberg and E. Witten, JHEP1999, 032 (1999).

[4] T. Banks, W. Fischler, S.H. Shenker and L. Susskind, Phys. Rev. D55, 5112 (1997) [5] L. Susskind, arXiv:hep-th/0101029

[6] M.R. Douglas and N.A. Nekrasov, Rev. Mod. Phys.73, 977 (2001) [7] L. Alvarez-Gaume and S.R. Wadia, Phys. Lett. B501, 319 (2001).

[8] R.P. Malik, A.K. Mishra and G. Rajasekaran, Int. J. Mod.Phys. A13, 4759 (1998) [9] V.P. Nair, Phys. Lett. B505, 249 (2001)

[10] A. Hatzinikitas and I. Smyrnakis, J. Math. Phys.43, 113 (2002) [11] A. Smailagic and E. Spallucci, Phys. Rev. D.65, 107701 (2002) [12] A. Smailagic and E. Spallucci, J. Phys. A35, L363 (2002) [13] A. Kijanka andP. Kosinski, Phys. Rev. D70, 127702 (2004)

[14] J. Ben Geloun, S. Gangopadhyay and F.G. Scholtz, EPL86, 51001 (2009) [15] J. Gamboa, M. Loewe and J.C. Rojas, Phys. Rev. D64, 067901 (2001) [16] B. Mirza and M. Mohadesi. Commun. Theor. Phys.42, 664 (2004)

[17] W. Pauli, Z. Physik43, 601 (1927)

[18] M.A. Ajaib. Int. J. Quan. Found.2, 109 (2016) [19] I. Haouam, arxiv:2012.06986

[20] J.E. Moyal, Proc. Cambridge Phil. Soc.45, 99 (1949) [21] E.U. Saka, K. ˜Aœlker, Phys. Rev. D75, 085009 (2007)

[22] M. Moumni, A. Benslama and S. Zaim, J. Geom. Phys.61, (2011) [23] L. Gouba, Int. J. Mod. Phys. A31, 1630025 (2016)

[24] J. Gamboa, F. Mendez, M. Loewe and J.C. Rojas. Mod. Phys. Lett. A16, 2075 (2001) [25] J. Gamboa, M. Loewe, F. Mendez and J.C. Rojas. Int. J. Mod. Phys. A17, 2555 (2002) [26] L.N. Chang, D. Minic, N. Okamura and T. Takeuchi, Phys. Rev. D65, 125027 (2002)