Quantum mechanics on phase space and the Coulomb potential

Physics Letters A 381 (2017) 1129–1133

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Physics Letters A www.elsevier.com/locate/pla

Quantum mechanics on phase space and the Coulomb potential P. Campos a,b,∗ , M.G.R. Martins a , J.D.M. Vianna a,c a b c

Instituto de Física, Universidade Federal da Bahia, Campus Ondina, 40210-340 Salvador, Bahia, Brazil Instituto Federal do Sertão Pernambucano, Campus Petrolina, 56314-520 Petrolina, Pernambuco, Brazil Instituto de Física, Universidade de Brasília, 70910-900 Brasília, DF, Brazil

a r t i c l e

i n f o

Article history: Received 3 September 2016 Received in revised form 31 January 2017 Accepted 2 February 2017 Available online 6 February 2017 Communicated by P.R. Holland Keywords: Symplectic quantum mechanics Coulomb potential Wigner function Levi-Civita transformation

a b s t r a c t Symplectic quantum mechanics (SMQ) makes possible to derive the Wigner function without the use of the Liouville–von Neumann equation. In this formulation of the quantum theory the Galilei Lie algebra is constructed using the Weyl (or star) product with Qˆ = q = q + i2h¯ ∂ p , Pˆ = p  = p − i2h¯ ∂q , and the Schrödinger equation is rewritten in phase space; in consequence physical applications involving the Coulomb potential present some specific difficulties. Within this context, in order to treat the Schrödinger equation in phase space, a procedure based on the Levi-Civita (or Bohlin) transformation is presented and applied to two-dimensional (2D) hydrogen atom. Amplitudes of probability in phase space and the correspondent Wigner quasi-distribution functions are derived and discussed. © 2017 Elsevier B.V. All rights reserved.

1. Introduction There are several alternative ways in order to quantize a micro physical system. One of them refers to the quantum revolution in the twenties of the last century performed by Schrödinger, Heisenberg, Dirac and others, in this standard way we use operators in Hilbert space. Another way is the path integrals, which were conceived by Dirac [1] and formulated by Feynman in 1948 [2,3]. A third way is the formulation of quantum mechanics on phase space (also known as the Moyal quantization or the deformation quantization) which is grounded on Wigner’s quasidistribution function [4] and Weyl’s correspondence between ordinary c-number functions in phase space and quantum-mechanical operators in Hilbert space [5,6]. At the ending of the 1970s Bayen et al. [7,8] laid the groundwork for an alternative description of the phase space formulation of quantum mechanics. The roots of this work are found in earlier works of Weyl [5,6], Wigner [4], Groenewold [9], Moyal [10] and Berezin [11–13] on the physical side and of Gerstenhaber and Schack [14–18] on the mathematical side. Since then, many efforts have been made in order to develop the quantum mechanics on phase space, for a comprehensive treatment of the subject the reader may consult Refs. [19–21]. An extensive collection of important papers and list of references can be found in Refs. [22,23].

The phase space representation of quantum mechanics is less well known but is useful in many branches of physics, for example, in quantum optics [24], nuclear physics [25], atomic physics [26–28], condensed matter [29–31], field theory [32–37], M-theory [38–40], noncommutative geometry [41,42] and the noncommutative field theory models [43–48]. The concept of phase space comes naturally from the Hamiltonian formulation of classical mechanics and plays an important role in the relation between quantum and classical mechanics, i.e. the quantum-classical transition. The quantum mechanics on phase space seems to be a result of a generalization of classical Hamiltonian mechanics, in such a way that the phase space formulation of quantum mechanics should smoothly reduce to the formulation of classical Hamiltonian mechanics as the Planck constant h¯ goes to 0, that is h¯ parameterizes the link between classical and quantum mechanics. The interpretation of phase space representation of quantum mechanics is given by considering the Wigner function f w (q, p ), which both the position and momentum variables are c-numbers. A basic advantage of this representation is that it is possible to perform canonical transformations, just as in classical Hamiltonian mechanics [21]. The stationary Wigner phase space distribution function f w (q, p ) in terms of the wave function  ψ(  q) of the usual timeˆ qˆ , pˆ ψ (q) = E ψ (q), is deindependent Schrödinger equation H fined through the following expression [4,19]

*

Corresponding author at: Instituto de Física, Universidade Federal da Bahia, Campus Ondina, 40210-340 Salvador, Bahia, Brazil. E-mail addresses: [email protected] (P. Campos), [email protected] (M.G.R. Martins), jdavid@fis.unb.br (J.D.M. Vianna). http://dx.doi.org/10.1016/j.physleta.2017.02.005 0375-9601/© 2017 Elsevier B.V. All rights reserved.

 f w (q, p ) =

e

ip ξ h¯

ξ ξ ψ † (q + )ψ(q − )dξ, 2

2

(1)

1130

P. Campos et al. / Physics Letters A 381 (2017) 1129–1133

where all integral runs from −∞ to −∞. The Wigner function is identified as a quasi-distribution in the sense that f w (q, p ), where (q, p ) are the coordinates of a phase space manifold  , is real but not positive definite, and as such cannot  be interpreted as probability. However, the integrals ρ q f w (q, p ) dp and ρ ( p ) = = ( )  f w (q, p ) dq are (true) distribution functions. In the Wigner formalism, each operator, say A, defined in the Hilbert space, H, is associated with a function, say a w (q, p ), in  . Then there is an application  w : A → a w (q, p ), such that, the associative algebra of operators defined in H turns out to be an associative (but not commutative) algebra in  , given by  w : A B → a w  b w , where the star-product  is defined by

 a w  b w = a w (q, p ) exp

← −−→ ←−− → ∂ ∂ ∂ ∂ − b w (q, p ) , 2 ∂q ∂ p ∂ p ∂q

ih¯

(2)

and the arrows over the vector fields ∂q , ∂ p denote that a given vector field acts only the function standing on the left or on the right side of the vector field. Studies of the representation of the Galilei group in a manifold with phase space content have been developed since long ago [49–56]. This type of representation, called symplectic unitary representation, has been used by several authors [5,10,19,21]; in particular Oliveira et al. [57] in order to explore the algebraic structure of the Wigner formalism have considered unitary representations based on operators of the type a w  and shown that the operators

= q = q + Q

ih¯ 2

∂p ,

ih¯

P = p  = p − ∂q ,



− t

K = k = mq  −tp  = m Q P ,

2

(3) (4)

ih¯

∂

j

Li = i jk Q P k = i jk q j pk − i jk q j 2 ∂ qk ih¯

+

2

i jk p j

∂ h¯ 2 ∂2 + i jk , ∂ qk 4 ∂ q j ∂ pk

(5)

and

H = h = =

P2 2m

3 1 

2m

=

3 

1 2m

pi −

i =1

P2 i

i =1

ih¯ ∂ 2 ∂ qi

2 ,

(6)



,

H Q P (q, p ) = E (q, p ) .

(7)  

with

= q, and the Wigner +V Q P = p , Q 2m † function is defined by f w = (q, p )  (q, p ). Eq. (7) is symplec-





,

Here H Q P =

2. 2D hydrogen atom For the 2D hydrogen atom, the potential energy is V c (q) = 

−e 2 q−1 , with q =

P2

tically covariant [58,59] and for a complete understanding of this equation the reader may consult Refs. [57–61]. This approach provides satisfactory interpretation for numerous aspects of the phase space quantum theory and, although associated with the Wigner formalism, has a Hamiltonian, not a Liouvillian, operator as generator translation; from Eq. (7) it follows, for example, that  of time 

 † = E  † or H

f w = E f w . SQM has been H

P, Q P, Q applied to some quantum systems: states of linear oscillator, nonlinear oscillator [57], one dimensional hydrogen atom [62] have

q21 + q22 . There is a great interest in this system

due to its applications in condensed matter physics [66–69] and in atomic and molecular physics [70], in particular, in the branch of atomic spectroscopy, the 2D hydrogen atom was regarded as a simplified model of the ionization process of the highly excited 3D hydrogen atom by circular-polarized microwaves [70]. For simplicity of presentation, we suppose units are selected in such a way that both the charge e and the mass m have unit value. With this choice of units, the corresponding classical Hamiltonian for the system is

H c = (2−1 p 2 − kq−1 ), where p = ture for the 2

(8)

+ and k is a positive constant. In the SQM pic(q, p ) we have

p 21 Hc

p 22

2 2  1 i ∂



H Q,P = pi − + V (q1 , q2 ) 2 2 ∂ qi 

(9)

i =1

and in consequence it is difficult to determine solutions of the Schrödinger equation (7). To solve this problem we propose a procedure where the connection between the Coulomb problem in a plane in parabolic coordinates and the 2D harmonic oscillator in Cartesian coordinates is used [71]. Specifically, with our notation, let now U = R2 , X = R2 , U∗ = U \ {0}, X∗ = X \ {0} and

T (u ) :=

satisfy the Lie algebra for the Galilean symmetry with a central extension characterized by m. Furthermore Oliveira et al. have introduced a pair of multiplicative operators Q (coordinates) and P (momenta) which allows us to endow H (), the Hilbert space over  , with basis |q, p  in which Q and P are diagonal operators. It follows that in this formulation, called Symplectic Quantum Mechanics (SQM), the time-independent Schrödinger equation in phase space is written as



been obtained in terms of amplitudes of probability in phase space

(q, p ). However, for two and three dimensional Coulomb potential there are some specific difficulties and it is not known the correspondent (q, p ). In this work in order to solve the Schrödinger equation (7) for the 2D hydrogen atom we present a procedure based on the Levi-Civita (or Bohlin) transformation [63–65].

1 2



−u 2

u1 u2

u1

, u ∈ U∗ .

(10)

The Levi-Civita (or Bohlin) transformation [63–65] is given by f : U∗ → X∗ , x = f (u ) = T (u )u. The columns of T (u ) form analytic orthogonal frame for U∗ . A theorem of Hurwitz [72,73] states that square matrix T (u ) satisfies the three properties of which can be shown by straightforward calculations: T (u ) is orthogonal for all u = 0, T (u ) is linear in u, and one of the columns of T (u ) is u. Therefore we have the transformation



1



q u 21 − u 22 = 1 , (11) q2 u1 u2  with q = u 2 , q = q21 + q22 , and u 2 = u 21 + u 22 . Thus according to the Levi-Civita transformation [74,75] the plan (u 1 , u 2 ) is the double covering of the plan (q1 , q2 ). Therefore, the points (u 1 , u 2 ) and (−u 1 , −u 2 ) represent the same point of the plane and the wavefunctions must satisfy ψ (u 1 , u 2 ) = ψ (−u 1 , −u 2 ). We note x = T (u )u =

2

here the rather obvious fact that the Levi-Civita mapping, which is simply a transformation to parabolic coordinates, carries the flat q space into a flat u space. The “inverse” transformation is given by u 1 = ±



(q1 +2q) 4

1 2

and u 2 =

q2 , u1

giving the parabolic co-

ordinates u i in terms of the Cartesian coordinates. By develop ing dqt dq = 4dut T t T du, we obtain dq1 2 + dq2 2 = q du 1 2 + du 2 2 . The reader will verify that ∂∂u = 2T t ∂∂q , this can be inverted to give ∂∂q = 2q T ∂∂u . Applied to classical Hamiltonian (8), H c in terms of parabolic coordinates u 1 , u 2 defined by q1 = q2 = u 1 u 2 is

1 2



u 21 − u 22



and

P. Campos et al. / Physics Letters A 381 (2017) 1129–1133

Hc =

1



1

2 u 21 + u 22



P 12 + P 22 −



2k

(12)

u 21 + u 22

where P 1 and P 2 are the canonical momenta conjugate to u 1 and u 2 respectively. Hence the hypersurface in phase space defined by H c = E corresponds to

1 2







(13)

hE =

2

+

P 22



−E



u 21

+ u 22



consider H ho = q ( H c − E ) to study the 2D hydrogen atom in the phase space quantum mechanics picture, i.e. we determine the amplitudes of probability (u 1 , u 2 ) and hence the Wigner function f w (u 1 , u 2 ) = (u 1 , u 2 )  † (u 1 , u 2 ). For this, however, it is necessary to analyze the star product. In fact it is a direct calculation to show, using



2

u1 −u 2

u2 u1

(15)

,

← − that P 1 = u 1 p 1 + u 2 p 2 , P 2 = −u 2 p 1 + u 1 p 2 and hence that ∂q · − → ← − − → ← − − → ← − − → ∂ p − ∂ p · ∂q = ∂u ·∂ P − ∂P · ∂u . Furthermore the Poisson brackets   give u i , u j = 0, u i , P j = δi j , P i , P j = 0, with (i , j ) referring to the (u 1 , u 2 ) directions, respectively. Let us consider then the Hamiltonian of a 2D hydrogen-like  atom in phase space T ∗ R2 − {0} and symplectic form d P ∧ du given by: H ho = q ( H c − E ). Now, in the SQM, the position and momentum operators are written as U = u  = u + i2h¯ ∂ P and P = P  = P−

ih¯ ∂ , 2 u

H ho  =



respectively. Then we have that

1

P1 −

2

− E u1 +

ih¯ 2

ih¯ 2

2 +

∂u 1

2 ∂P1



P1 −

 P2 −

ih¯ 2 ih¯ 2

2

1

P2 −

2

− E u2 +

2 ∂u2

+ W u1 +

∂u1

− ∂z −

4

z∂z2

−

K h¯ W

f ( z ) = 0,

1 2

(19)

+ W u2 +

ih¯ 2 ih¯ 2

where K is K 1 or K 2 . Moreover, setting f ( z) = e − 2 L ( z), this yields the differential equation



ih¯ 2

ih¯ 2

2 (16)

∂u 2

2 ∂P2

− 2k.



2 − 2K 1

∂P1

(17)

f 2 = 0,

(18)



2 ∂P2

f 1 = 0,

− 2K 2

where W = −2E, and we introduce two separation constants K 1 and K 2 such that K 1 + K 2 = 2k. These separated equations are quite similar to the differential equation for the one-dimensional harmonic oscillator in phase space whose quantum-mechanically acceptable solutions are known to be expressible in terms of Laguerre polynomials multiplying a Gaussian exponential [21,22]. Furthermore, the energy E appears as a factor in precisely the place where the classical frequency of the oscillator usually appears, and the separation constants appear in the position usually occupied by the energy eigenvalue. In present paper the equations (17) and (18) are solved analytically, but it may be approached algebraically (annihilation and creation operators) [57]. Performing the algebra, the imaginary part of the equations above is null, that is ( P ∂u − W u ∂ P ) f = 0, f is seen to depend on only one variable,

K h¯ W

1 2

−

1

 L ( z) = 0

2

(20)

which is the Laguerre’s differential equation.   It is satisfied by Laguerre polynomials L n ( z) = n1! e z ∂zn zn e −z , for n = K 1 − 12 = h¯ W

2

0, 1, 2, .... The Laguerre polynomials do not themselves form an orthogonal set. However, the related set of functions f n ( z) = z e− 2 L n ( z) is orthonormal for the interval 0 ≤ z < ∞, that is, ∞ −z e L m ( z) L n ( z) dz = δn,m . Observe that the eigenfunctions are 0 not positive definite, and are the only ones satisfying the boundary conditions, L ( z = 0) finite and limz→∞ L ( z) → 0. Therefore, the requirement of the boundary conditions of the functions leads to the following restrictions on the values of the separation constants:

K1 h¯ W

1

1 2

= n1 + , 2

K2 h¯ W

1

1 2

= n 2 + , n 1 , n 2 = 0, 1, 2, · · · 2

(21)

Consequently, the frequency and the energy levels (with k = 1) are required to present the following forms

 W n1 ,n2 =

2

1

2

h¯

n1 + n2 + 1

(22)

and

E n1 ,n2 = −

To solve the equation H oh  = 0, let us assume that has the form (u , P ) = f 1 (u 1 , P 1 ) f 2 (u 2 , P 2 ). In consequence we have two differential equations



z

(14)

2 with (u 1 , u 2 ) in place of (x, y) and E takes the place of ω2 . We

1



z∂z2 + (1 − z) ∂z +

is an auxiliary Hamiltonian that depends parametrically on E and is of the form of the a 2D oscillator in plane (x, y) if E < 0

T t (u ) :=



1



that is, H c = E is equivalent to the condition h E = 2k, where

P 12

1

z = 2H = h2¯ W 2 u 2 + W − 2 P 2 , and so the real part of the equah¯ tion reduces to a simple ordinary differential equation:

z

P 12 + P 22 − E u 21 + u 22 = 2k 1

1131

1 2h¯ 2



n1 + n2 + 1 2

−2 ,

(23)

respectively, when the separation constants are eliminated from Eq. (21). The spectrum of states divides itself into degenerate submanifolds, each characterized by a particular value of n = n1 + n2 . The degeneracies of these submanifolds are dn = n + 1. This pattern of degeneracy is a direct manifestation of the unitary symmetry of the isotropic oscillator [76–80]. The n-th irreducible representation of SU (2) has dimension n + 1 [81], so each appears exactly once in the isotropic oscillator spectrum. Observe that the three-dimensional rotation group [82], whose irreducible representations are always of odd dimension (dl = 2l + 1), cannot adequately reproduce the observed degeneracies in the isotropic oscillator spectrum. It is well known that SU (2) is the double covering of the S O (3) group. This means that S O (3) and SU (2) are locally isomorphic. However, SU (2), and not S O (3), is the correct symmetry group of the isotropic oscillator [76–80,83]. Under the inversion in u space, one has ψ (−u ) = (−1)n1 +n2 ψ (u ). Since q j are the even functions of u j , the inversion causes no effects in the physical q j space, and hence only the states with n1 + n2 = 2n correspond to the functions of the hydrogen atom. Therefore, the analytic expressions for f n1 and f n2 have the forms

f n1

   H n1 ,n2 u¯ 1 , P¯ 1 (−1)n1 = × exp − π h¯ h¯    2H n1 ,n2 u¯ 1 , P¯ 1 L n1 , h¯

(24)

1132

f n2

P. Campos et al. / Physics Letters A 381 (2017) 1129–1133

   H n1 ,n2 u¯ 2 , P¯2 (−1)n2 = exp − × π h¯ h¯    2H n1 ,n2 u¯ 2 , P¯2 L n2 , h¯

(25)





1 h¯



(i )

H n1 ,n2  n1 ,n2 = 4h¯

and (i )



(i )

H n1 ,n2 f n1 ,n2 = 2

u¯ 2i + h¯ P¯ i2 ≡ H n1 ,n2 , (i )

(26)

u¯ i =

1

2

2

n1 + n2 + 1

u i , P¯ i =



Pi.

L n1

1

(−1)n1 +n2 ×

    −1  × exp H n1 ,n2 u¯ 1 , P¯ 1 + H n1 ,n2 u¯ 2 , P¯ 2 h¯       2H n1 ,n2 u¯ 1 , P¯ 1 2H n1 ,n2 u¯ 2 , P¯ 2 L n1 L n2 ; h¯ h¯ L 0 = 1, L 1 = 1 −

2H n1 ,n2 h¯

, L2 =

(π h¯ )2

4H n21 ,n2 2 h¯

−

8H n1 ,n2 h¯

(28)

†

(n)

f w (u , P ) =

1

(π h¯ )2

 L n1

(1)

2H n1 ,n2



h¯

 L n2

(30) (2)

2H n1 ,n2



h¯

×

(31)

 −1  (1) (2 )  H n1 ,n2 + H n1 ,n2 h¯

 −1  (1) (2) × exp H n1 ,n2 + H n1 ,n2 h¯  (1)   (2) 

exp

L n1

2H n1 ,n2 h¯

L n2

2H n1 ,n2 h¯

(33)





4

2n1 + 1

h¯

n1 + n2 + 1 L n1

2



2n2 + 1

4

L n2



n1 + n2 + 1

h¯

(1)

2H n1 ,n2





L n2

h¯



(2)

2H n1 ,n2 h¯

(34)



n1 + n2 + 1



× ×  ×

 −1  (1) (2) exp H n1 ,n2 + H n1 ,n2 h¯ or, in this case, (n)

f w (u , P ) ∼ n1 ,n2 .

(35)

Remark that the states ( s) then the star product  project onto their space. Therefore, the ground state (n = 0) and the degenerate excited state (n = 2) Wigner function for the 2D Coulomb potential are given by †

(0)

f w (u , P ) = 0,0  0,0 ∼ 0,0 =

(36)

and

+ 1, · · · (29)

f w (u , P ) = n1 ,n2 (u , P )  n1 ,n2 (u , P ) , where



⎫ ⎧

2 ⎨ −1  h¯ 2 2 2 ⎬ 1 exp P + uj ⎭ ⎩ h¯ 2 j h¯ (π h¯ )2 j =1

To physically interpret the formalism, we associate (u , P ) with the Wigner function, f w (u , P ), which is given by the expression [22,38,57], f w = (u , P )  † (u , P ). In the SQM, the Wigner function satisfies also the phase space Schrödinger equation the probability density both in configuration [22] and determines  space, ρ (u ) = d P (u , P )  † (u , P ) = dp (u , P ) † (u , P ),  and in momentum space, ρ ( P ) = du (u , P )  † (u , P ) =  dq (u , P ) † (u , P ). Therefore the general Wigner function for our problem is (n)



(27)

A remark about the origin of these multiplying factors is relevant. In the phase space Schrödinger equation for the harmonic oscillator, the frequency is the multiplier of u 2 , while the quantized energy is the constant term in a differential equation similar to Eq. (17); but in the present paper it is E, which multiplies u 2 , and the separation constant which is quantized. Thus, the energy constant E enters into the scaling factor for the function argument, in the place where the classical frequency ordinarily appears. The consequence is that the energy eigenvalues enter into the argument of the eigenfunction, as well as into the indices of the Laguerre polynomials which comprise part of the eigenfunctions. The final solution of the phase space Schrödinger equation is given by

n1 ,n2 = f n1 f n2 =

(i )

f n1 ,n2 ,

n1 + n2 + 1

(n)

2

n1 + n2 + 1

2ni + 1

(32)

n1 ,n2 ,

2

f w ∼ exp −4

− 1

2

with i = 1, 2, we obtain the analytic expression for the n-th Wigner function, i.e.

and the argument variables are defined by



n1 + n2 + 1

i =1

where

H n1 ,n2 u¯ i , P¯ i =

2 

.

By developing the exponential function in the Taylor series and utilizing the properties of the star product, and the following expressions

1

(2 )

f w (u , P ) ∼ 1,1 =

 1−



2 h¯ 2



3h¯ 2 3h¯

P 12 + P 22 +

2

(π h¯ )2



3h¯ 2

u 21

×

(37)

×



u 22

× h¯ 2 3h¯ ⎫ ⎧

2 ⎨ −1  3h¯ 2 2 2 ⎬ . exp Pj + uj ⎭ ⎩ h¯ 2 3h¯ j =1 1−

In this case, the Wigner functions agree with the solutions found for the amplitudes. This is consistent, since they obey the same (0) eigenvalue equation. f w is positive everywhere in phase space (2) and f w is positive at the origin. The n-th state Wigner functions, Eq. (35), are invariant under rotations around the origin, and can also be written in terms of the Hermite polynomials [84]. 3. Conclusion In summary, in this paper we have presented the results for the 2D hydrogen atom within the formulated symplectic quantum mechanics. The 2D Coulomb problem has been transformed to the 2D harmonic oscillator in Levi-Civita coordinates. We have calculated the energy levels as well as the Wigner quasi-distribution functions of the 2D hydrogen atom. A detailed analysis of the results obtained here, together with the treatment of other fundamental physical problems, will be given a further communication. In particular, we would like to note that according to Gosson

P. Campos et al. / Physics Letters A 381 (2017) 1129–1133

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