# Zeeman Effect in Phase Space

Authors: R. A. S. Paiva; R. G. G. Amorim; S. C. Ulhoa; A. E. Santana; F. C. Khanna
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

The two-dimensional hydrogen atom in an external magnetic field is considered in the context of phase space. Using the solution of the Schrödinger equation in phase space, the Wigner function related to the Zeeman effect is calculated. For this purpose, the Bohlin mapping is used to transform the Coulomb potential into a harmonic oscillator problem. Then, it is possible to solve the Schrödinger equation easier by using the perturbation theory. The negativity parameter for this system is realised.


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## Body

## 1. Introduction

Since the early years of the development of quantum mechanics, its formulation in phase space has been troublesome. The seminal paper by Wigner in 1932 addressed such a problem in an attempt to deal with the superfluidity of helium [1]. The Wigner function, fWq,p, was introduced as a Fourier transform of the density matrix, ρq,q′. Then, the phase space manifold (Γ), which has a symplectic structure, is described by the coordinates q,p [1–4]. The Wigner function is identified as a quasiprobability density since fWq,p is real but it is not positively defined. However, the integrals ρq=∫fwq,pdp and ρp=∫fwq,pdp are distribution functions.In the Wigner formalism, each operator,A, defined in the Hilbert space, H, is associated with a function, aWq,p, in Γ. This procedure is precisely specified by a mapping ΩW:A⟶aWq,p, such that the associative algebra of operators defined in H leads to be an algebra in Γ, given by ΩW:AB⟶aW⋆bW, where the star product, ⋆, is defined by
(1)aW⋆bW=aWq,pexpiℏ2∂←∂q∂→∂p−∂←∂p∂→∂qbWq,p.As a consequence, a noncommutative structure inΓ is obtained that has been explored in different ways [2–25]. Recently [26–30], unitary representations of symmetry Lie groups have been obtained on a symplectic manifold, exploring the noncommutative nature of the star product and using the mapping ΩW [26–28]. As a result, using a specific representation of a Galilei group, the Schrödinger equation in phase space is obtained. On the other hand, the scalar representation of the Lorentz group for spin 0 and spin 1/2 leads to the Klein-Gordon and Dirac equations in phase space. In relativist and nonrelativistic approaches, the wave functions are closely associated with the Wigner function [26, 27]. This provides a fundamental ingredient for the physical interpretation of the formalism showing its advantage in relation to other attempts.In recent years, the two-dimensional physical systems have been investigated due to both experimental and theoretical interests. For example, it is possible to cite the fractional Hall effect in a tilted magnetic field [31, 32], superconductivity in two-dimensional organic conductors induced by magnetic field [33], investigations of graphene [34, 35], etc. Particularly, a two-dimensional model of hydrogen atom was considered in several contexts; for instance, it is possible to describe highly three-dimensional anisotropic crystals [36], semiconductor heterostructures [37–39], and astrophysical applications [40–42]. In addition, the hydrogen atom in a uniform magnetic field can present a nonclassical behavior with the increase of strength of magnetic field [43–46]. In this paper, the objective is to investigate the two-dimensional hydrogen atom in a constant magnetic field in phase space picture, i.e., the Schrödinger equation in phase space is used to study the Zeeman effect. The Zeeman effect is applied in a variety of systems, including intense laser lights, comic rays, and the study of intergalactic and interstellar medium [47, 48]. The importance to study the Wigner function for such an effect is in order to obtain information about the chaotic nature of such systems.In Section2, a review about the formalism of quantum mechanics in phase space and its connection with Wigner function is presented. In Section 3, the Hamiltonian of hydrogen atom in an external magnetic field and Bohlin mapping are discussed. The solution for the Schrödinger equation for the Zeeman effect in phase space is solved by perturbative method in Section 4. In Section 5, a summary and concluding remarks are presented.


## 2. Symplectic Quantum Mechanics

In this section, the representation of the Galilei group inHΓ is presented. This procedure leads us to the Schrödinger equation in phase space. Then, a connection between this representation and the Wigner formalism is established.Using the star operator,A^=a⋆, the position and momentum operators, respectively, are defined by
(2)Q^=q⋆=q+iℏ2∂p,(3)P^=p⋆=p−iℏ2∂q.The operators given in equations (2) and (3) satisfy the Heisenberg commutation relation,
(4)Q^,P^=iℏ.In addition, the following operators are introduced:(5)K^=mQ^i−tP^i,L^i=∈ijkQ^jP^k,H^=P^22m=12mP^12+P^22+P^32.From this set of unitary operators, after simple calculations, the following set of commutation relations are obtained:(6)L^i,L^j=iℏ∈ijkL^k,L^i,K^j=iℏ∈ijkK^k,L^i,P^j=iℏ∈ijkP^k,K^i,P^j=iℏmδij1,K^i,H^=iℏP^i,with all other commutation relations being null. This is the Galilei-Lie algebra with a central extension characterized by m. The operators defining the Galilei symmetry P^, K^, L^, and H^ are then generators of translations, boost, rotations, and time translations, respectively.Defining the operators(7)Q¯=q1,P¯=p1,for boost, operators Q¯ and P¯ transform as
(8)exp−ivK^ℏ2Q¯exp−ivK^ℏ=2Q¯+vt1,(9)exp−ivK^ℏ2P¯expivK^ℏ=2P¯+mv1.This shows thatQ¯ and P¯ transform as position and momentum variables, respectively. These operators satisfy Q¯,P¯=0. Then, Q¯ and P¯ cannot be interpreted as observables. Nevertheless, they can be used to construct a Hilbert space framework in phase space. Then, we define the functions ϕq,p in HΓ that satisfy the condition
(10)∫dqdpϕ∗q,pϕq,p<∞.The wave functionψq,p,t=q,p∣ψt associated with the state of the system is defined, but does not have the content of the usual quantum mechanics state.The time evolution equation forψq,p,t is derived by using the generator of time translations, i.e.,
(11)ψt=e−iH^t/ℏψ0.Then, this leads to(12)iℏ∂tψq,p;t=H^q,pψq,p;t,or
(13)iℏ∂tψq,p;t=Hq,p⋆ψq,p;t,which is the Schrödinger equation in phase space [26].The association ofψq,p,t with the Wigner function is given by [26]
(14)fwq,p=ψq,p,t⋆ψ†q,p,t.This function satisfies the Liouville-von Neumann equation [26].


## 3. Two-Dimensional Hydrogen Atom in an External Magnetic Field and Bohlin Mapping

The Hamiltonian for the two-dimensional hydrogen atom in a constant and uniform magnetic fieldB=Bz^ is given as [44, 49, 50]
(15)H=P‐eA22m−kx2+y21/2,where m and e represent the electron mass and charge, respectively, A is the magnetic potential vector and k is a constant. In a two-dimensional case, equation (15) is written as
(16)H=P22m−kx2+y21/2+mω2x2+y2+ωLz,where ω=eB/2mc is the frequency and Lz is the angular momentum in the z-direction. Here, the constant ωLz will be neglected once the energy is defined up to a constant.In order to solve the Schrödinger equation for this Hamiltonian, the Bohlin mapping is used.


### 3.1. Bohlin Mapping

Bohlin mapping is defined by [51–53]
(17)x+iy=q12−q22+i2q1q2,or
(18)x=q12−q22,(19)y=2q1q2.Defining(20)Px+iPy=p1+ip22q1+iq2leads to
(21)Px=p1q1+p2q22q12+q22,(22)Py=p2q1−p1q22q12+q22.Substituting equations (18), (19), (21), and (22) in equation (16) leads to the Hamiltonian
(23)H=12p12+p22q12+q22−kq12+q22+B28q12+q223.Usingℏ=ω=e=m=1 and taking the hypersurface given by H=E leads to
(24)12p12+p22+B28q12+q223−Eq12+q22−k=0,which is the Hamiltonian to be used in the next section. It should be noted that the Bohlin transformation is a canonical transformation [54].


## 3.1. Bohlin Mapping

Bohlin mapping is defined by [51–53]
(17)x+iy=q12−q22+i2q1q2,or
(18)x=q12−q22,(19)y=2q1q2.Defining(20)Px+iPy=p1+ip22q1+iq2leads to
(21)Px=p1q1+p2q22q12+q22,(22)Py=p2q1−p1q22q12+q22.Substituting equations (18), (19), (21), and (22) in equation (16) leads to the Hamiltonian
(23)H=12p12+p22q12+q22−kq12+q22+B28q12+q223.Usingℏ=ω=e=m=1 and taking the hypersurface given by H=E leads to
(24)12p12+p22+B28q12+q223−Eq12+q22−k=0,which is the Hamiltonian to be used in the next section. It should be noted that the Bohlin transformation is a canonical transformation [54].


## 4. Zeeman Effect in Phase Space

Using equation (24), the equation is written as
(25)12p12+p22+B28q12+q223−Eq12+q22−k⋆ψq1,p1,q2,p2=0.It should be noted that the above equation is obtained from the classical Hamiltonian by means of the star product. Thus, the Bohlin mapping that leads to equation (24) is a classical transformation. This equation is analyzed by the perturbation theory. The equation in phase space is defined as
(26)H^0+H^1ψq1,p1,q2,p2=kψq1,p1,q2,p2,where H^0=1/2p12+p22⋆−Eq12+q22⋆ and H^1=B2/8q12+q22⋆.The equation forH^0 has the form
(27)H^0ψ0q1,p1,q2,p2=k0ψ0q1,p1,q2,p2,where ψ0q1,p1,q2,p2 and k0 represent, respectively, the eigenfunction and eigenvalue of the unperturbed Hamiltonian.Defining the operators(28)a^=W2q1⋆+i12Wp1⋆,a^†=W2q1⋆−i12Wp1⋆,b^=W2q2⋆+i12Wp2⋆,b^†=W2q2⋆−i12Wp2⋆,where W2/2=−E, the star operators q1⋆ and p1⋆ are given by
(29)qi⋆=qi+i2∂∂pi,pi⋆=pi−i2∂∂qi,and perturbed Hamiltonian is
(30)H^=Wa^a^†+b^b^†+1+B28a^+a^†2+b^+b^†23.The unperturbed Hamiltonian is defined as(31)H^0=Wa^a†^+b^b†^+1.Then, the perturbed part is(32)H^1=B28a^+a^†2+b^+b^†23.The equation that is to be analyzed is given as(33)H⋆ψq1,p1,q2,p2=kψq1,p1,q2,p2.The unperturbed equation is(34)H0⋆ψn1,n20q1,p1,q2,p2=kn1,n10ψn1,n20q1,p1,q2,p2,The unperturbed part,H^0, has solutions given by
(35)ψn1,n20q1,p1,q2,p2=ϕn1q1,p1Γn2q2,p2,where ϕn1q1,p1 and Γn2q2,p2 are solutions. The eigenvalue equations are given by
(36)a^ϕn1=n1ϕn1−1,(37)a^†ϕn1=n1+1ϕn1+1,(38)b^Γn2=n2Γn2−1,(39)b†^Γn2=n2+1Γn2+1.Using the relationsa^ϕ0=0 and b^Γ0=0, the ground state solution is
(40)ψ0,00q1,p1,q2,p2=Ne−Wq12+p12Ln1Wq12+p12e−Wq22+p22Ln2Wq22+p22,where Ln1 and Ln2 are Laguerre polynomials; and N is a normalization constant. The eigenvalue solutions, given in equation (34), are
(41)kn1,n20=n1+n2+1W.The excited states are obtained from equation (40) using operators given in equations (36), (37), and (38).Then, the solution for the first-order perturbed Hamiltonian is given by(42)ψn1,n21q1,p1,q2,p2=ψn1,n20q1,p1,q2,p2+∑m1≠n1;m2≠n2∫ψm1,m2∗0q1,p1,q2,p2H^1ψn1,n20q1,p1,q2,p2dq1dp1dq2dp2kn1,n20−km1,m20×ψm1,m20q1,p1,q2,p2.It is to be noted that the following integral needs to be solved(43)I=∫ψm1,m2∗0q1,p1,q2,p2B28a^+a^†2+b^+b^†23ψn1,n20q1,p1,q2,p2dq1dp1dq2dp2,before a solution for equation (42). Using the orthogonality relations
(44)∫ϕn∗q1,p1ϕmq1,p1dq1dp1=δn,m,∫Γn∗q2,p2Γmq2,p2dq2dp2=δn,m,the ground state is
(45)ψ0,01q1,p1,q2,p2=ψ0,00q1,p1,q2,p2+B28W−212−18−2510ψ2,00+−322−3ψ2,20+−3021−36ψ4,00+43ψ4,20−81155ψ6,00.And for excited states, the wave functions are(46)ψ1,01=ψ1,00+B28W89,30ψ1,20+13,47ψ1,40+6,32ψ1,60−89,43ψ3,00−19,33ψ3,20−23,51ψ5,00−10,31ψ5,40−11,83ψ7,00,ψ0,11=ψ0,10+B28W−89,30ψ2,10−13,47ψ4,10−6,32ψ6,10+89,43ψ0,30+19,33ψ2,30+23,51ψ0,50+10,31ψ4,50+11,83ψ0,70.The Wigner function for the hydrogen atom in a constant magnetic field is given by(47)fwq1,p1,q2,p2=ψn1,n2∗1q1,p1,q2,p2⋆ψn1,n21q1,p1,q2,p2.It should be noted that all plots considerq2=p2=1 in order to show a 3D figure, thus q1=q and p1=p. In Figures 1 and 2, the behavior of the Wigner function is presented for order zero with magnetic field assuming values B=1 and B=0.1, respectively, with E=1. In Figures 3 and 4, the behavior of the Wigner function is shown for the first order with magnetic field assuming values B=1 and B=0.1, respectively, with E=10.Figure 1
Wigner function zero order, Zeeman effect,E=1 and B=1.Figure 2
Wigner function zero order, Zeeman effect,E=1 and B=0.1.Figure 3
Wigner function first order, Zeeman effect,E=10 and B=1.Figure 4
Wigner function first order, Zeeman effect,E=10 and B=0.1.Comparing the graphics given in Figures1–4, the negative part of the Wigner function increases with larger values of energy and magnetic field.The Wigner function to the first order for magnetic field valuesB=1 and B=0.1 is shown in Figures 5 and 6 for E=1. The behavior of the Wigner function to the first order with magnetic field value B=0.5 is shown in Figures 7 and 8, for E=1 and E=10, respectively. The correction of the first order of eigenvalue of equation (33) is given by
(48)δkn1,n21=∫ψn1,n20H^1ψn1,n2∗0dq1dp1dq2dp2.Figure 5
Wigner function first order, Zeeman effect,E=1 and B=1.Figure 6
Wigner function first order, Zeeman effect,E=1 and B=0.1.Figure 7
Wigner function first order, Zeeman effect,E=1 and B=0.5.Figure 8
Wigner function first order, Zeeman effect,E=10 and B=0.5.Performing calculations forH^1 leads to
(49)kn1,n21=n1+n2+1W+B28δ,where
(50)δ=n1+1n1+2n1+3+n1+1n1+22+n1−1n1n1+12+n1+1n1n1+1+n13n1+13+n1+1n12+n1+1n1n1−1+n1n1−12+n1n1−1n1−2+3n1+1n1n2+1+3n1+1n1n2+3n12n2+13n12n2+3n1−1n1n2+1+3n1−1n1n2+3n2+1n2n1+1+3n2+1n2n1+3n22n1+13n22n1+3n2−1n2n1+1+3n2−1n2n1+n2+1n2+2n2+3+n2+1n2+22+n2−1n2n2+12+n23n2+13+n2+12n24+n2+1n2n2−1+n2n2−12+n2n2−1n2−2.Then, the eigenvalue is(51)W=kn1,n21−B2/8δn1+n2+1.With the relationW2/2=−E, the eigenvalues associated to the Zeeman effect in phase space are given by
(52)En1,n2=−12kn1,n21−B2/8δn1+n2+12,EN=−12kn1,n21−B2/8δN2,where N=n1+n2+1. Note that if B⟶0, the known results are obtained [55].Using the Wigner function, the negative parameter for the system is calculated. The results are presented in Tables1 and 2. It is to be noted that when the magnetic field increases the negativity parameter also increases. In addition, for a given value of the magnetic field, the negativity parameter increases when the sum n1+n2 increases. This result is presented in the graphics above.Table 1
Negativity parameter,B=1.
n1,n2
ηψ
0, 0
0.14345
0, 1
0.32645
1, 0
0.32645
1, 1
0.45786
2, 0
0.45786
0, 2
0.45786Table 2
Negativity parameter,B=0.1.
n1,n2
ηψ
0, 0
0.0034
1, 0
0.0562
0, 1
0.0562
1, 1
0.0635
2, 0
0.0635
0, 2
0.0635


## 5. Concluding Remarks

The Zeeman effect in phase space for the Schrödinger equation, which endows the Galilean symmetry, is analyzed. The Wigner function is calculated numerically and presented in the panels for several parameters. Such a function has a clear interpretation in the classical limit and can be projected in the momenta or coordinate space for experimental purpose. The modulus of the Wigner function is also finite that allows a calculation its negativity. The results are presented in Tables1 and 2. It indicates a direct relation between the magnetic field and the discrete parameter N=n1+n2. The increase of the magnetic field is related to the departure from the classical behavior since the negativity parameter increases accordingly.



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