An Algebraic Approach to a Charged Particle in a Uniform Magnetic Field

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Heisenberg-Weyl the to group 1) , )Prlmvnme oeetstates coherent number Perelomov 1) ´ cioNcoa,E.9 Unidad 9, Ed. técnico Nacional, 8 iddd México, Mexico. de Ciudad 38, nl v und isBátiz Dios de Juan Av. onal, trnme oeetstates. coherent number ator 73,Cua eMéxico, de Ciudad 07738, . ddCluca,Instituto Culhuacán, idad mncoclao number oscillator rmonic ial,w hwthat show we Finally, . ne ttebgnigo the of beginning the at inger o h adugauge, Landau the for , pligteWy operator Weyl the applying ic Culhuacán, C.P. cisco c cin fteannihilation the of nctions gasmlrt trans- similarity a ng ligteWy operator Weyl the plying a n ttsaegaussian are states ent , cfil ihtwo with field ic iexadLevelut and oiteux to any excited state [4]. These states are called displaced number states or number coherent states and were extensively studied in the middle of the last century. Most of their properties are compiled in references [5–8]. In reference [9], Nieto review these states and gave their most general form. Perelomov generalized the Klauder coherent states to any Lie group by applying the group displacement operator to the lowest normalized state [10]. The Perelomov coherent states haven been applied to many physical problems as can be seen in references [11–13]. Gerry defined the SU(1, 1) number coherent states by applying the Perelomov displacement operator to any excited state and used this definition to calculate the Berry’s phase in the degenerate parametric amplifier [14]. Recently, we have studied the Perelomov number coherent states for the SU(1, 1) and SU(2) groups. In particular, we gave the most general expression of these states, their ladder operators and applied them to calculate the eigenfunctions of the non-degenerate parametric amplifier [15] and the problem of two coupled oscillators [16]. In reference [17], we computed the number radial coherent states for the generalized MICZ-Kepler problem by using the su(1, 1) theory of unitary representations and the tilting transformation. On the other hand, the problem of a charged particle in a uniform magnetic field has been widely studied in classical mechanics, condensed matter physics, quantum optics and relativistic quantum mechanics, among others. The energy spectrum of this problem is known as the Landau levels. The interaction of an electron with the uniform magnetic field is described by means of electromagnetical potentials. However, different gauges give raise to the same electromagnetic field [18]. The coherent states for this problem have been obtained previously by using different formalisms, as can be seen in references [19–24]. The aim of this work is to introduce an algebraic approach to study the problem of a charged particle in an uniform magnetic field and obtain its coherent states. Specifically, the algebraic approach used in this work is the tilting transformation, which offers to graduate students an alternative method to the commonly used analytical approach, in order to study and exactly solve several problems in quantum mechanics. This work is organized as it follows. In Section 2, we give a summary on the Heisenberg- Weyl and SU(1, 1) groups and its number coherent states. In Section 3, we study the problem of a charged particle in a uniform magnetic field in cartesian coordinates with the Landau gauge. We solve this problem and show that its eigenfunctions are the harmonic oscillator number coherent states. In Section 4, we study the Landau levels problem in cylindrical coordinates with the symmetric gauge. We construct the SU(1, 1) Perelomov number coherent states for its eigenfunctions. In Section 5, we contract the SU(1, 1) group to the Heisenberg- Weyl group. We show that, under this contraction the SU(1, 1) Perelomov coherent states are related to the harmonic oscillator coherent states. Finally, we give some concluding remarks.

2 2 H(4) and SU(1, 1) number coherent states 2.1 Heisenberg-Weyl group The annihilation and creation operators of the harmonic oscillator a, a†, together with the number and identity operators a†a, I satisfy the following relations

[a, a†]= I, [a, a†a]= a, [a†, a†a]= a†, [a, I] = [a†a, I] = 0 = [a†,I]. (1) − These equations are known as the Heisenberg-Weyl algebra h(4). The action of these operators on the Fock states is given by

a† n = √n +1 n +1 , a n = √n n 1 , a†a n = n n . (2) | i | i | i | − i | i | i The harmonic oscillator coherent states are deﬁned in terms of these operators as

∞ n † ∗ 1 α α = D(α) 0 = eαa −α a 0 = exp α 2 n , (3) | i | i | i −2| | √ | i 0 n! X where D(α) is the Weyl operator (also called displacement operator), 0 is the ground state | i and α is a complex number given by

mω i α = x + p . (4) ~ 0 √ ~ 0 r 2mω This unitary operator D(α) can be expressed in a disentangled form by using the Weyl identity as follows [25] 2 † ∗ D(α)= e−|α| /2eαa e−α a. (5) The harmonic oscillator number coherent states are deﬁned as the action of the Weyl operator on any excited state n [4] | i 2 † ∗ n, α = D(α) n = e−|α| /2eαa e−α a n . (6) | i | i | i By using the Baker-Campbell-Hausdorﬀ identity 1 1 1 e−ABeA = B + [B, A]+ [[B, A], A]+ [[[B, A], A], A]+ ..., (7) 1! 2! 3! and the commutation relationship of the ladder operators [a, a†] = 1, it can be shown the following properties

A(α)= D†(α)aD(α)= a + α, A†(α)= D†(α)a†D(α)= a† + α∗. (8)

These operators play the role of annihilation and creation operators when they act on the harmonic oscillator number coherent states, since [4]

A† n, α = √n +1 n +1,α , A n, α = √n n 1,α . (9) | i | i | i | − i

3 By using the disentangled form of the Weyl operator D(α) of equation (5), we can prove that the most general form of these states in the Fock space is [9]

∞ † k n ∗ j 1/2 2 (αa ) ( α a) (n j + k)!n! n, α = e−|α| /2 − − n j + k . (10) | i k! j! (n j)!(n j)! | − i j=0 Xk=0 X − − On the other hand, the eigenfunctions of the one-dimensional harmonic oscillator are given by − 1 β2x2 ψn(x)= NnHn(βx)e 2 , (11)

mω β 1/2 where Hn(βx) are the Hermite polynomials and β = ~ , Nn = π1/42nn! . The Weyl operator D(α) can be expressed in terms of the harmonic oscillator position x and momentum p operators as p i ix0p0x ip0xx ix0px ~ (p0xx−x0px) − ~ ~ ~ D(x0,p0x)= e = e 2 e e , (12) as it is shown in reference [25]. Thus, the action of this operator on the harmonic oscillator eigenfunctions of equation (11) is

− 1 β2(x−x )2 D(x , 0)ψ (x)= N e 2 0 H (β(x x )) . (13) 0 n n n − 0 2.2 SU(1, 1) group

The su(1, 1) Lie algebra is generated by the set of operators K+, K−,K0 . These operators satisfy the commutation relations [26] { }

[K ,K ]= K , [K ,K ]=2K . (14) 0 ± ± ± − + 0 The action of these operators on the Fock space states k, n , n =0, 1, 2, ... is given by {| i } K k, n = (n + 1)(2k + n) k, n +1 , (15) +| i | i K k, n =p n(2k + n 1) k, n 1 , (16) −| i − | − i K0 k,p n =(k + n) k, n . (17) | i | i In analogy to the harmonic oscillator coherent states, Perelomov deﬁned the standard SU(1, 1) coherent states as [11]

∞ Γ(n +2k) ζ = D(ξ) k, 0 = (1 ζ 2)k ζs k,s , (18) | i | i −| | s!Γ(2k) | i s=0 s X where k, 0 is the lowest normalized state. In this expression, D(ξ) is displacement operator for this| groupi deﬁned as D(ξ) = exp(ξK ξ∗K ), where ξ = 1 τe−iϕ, <τ< and + − − − 2 −∞ ∞ 0 ϕ 2π. The so-called normal form of the displacement operator is given by ≤ ≤ D(ξ) = exp(ζK ) exp(ηK ) exp( ζ∗K ), (19) + 0 − − 1 −iϕ 2 where ζ = tanh( 2 τ)e and η = 2 ln cosh ξ = ln(1 ζ ) [27]. This expression is the analogue of− equation (5) for the Weyl− operator.| | −| |

4 The SU(1, 1) Perelomov number coherent states were introduced by Gerry and are deﬁned by the following expression [14]

ζ,k,n = D(ξ) k, n = exp(ζK ) exp(ηK ) exp( ζ∗K ) k, n . (20) | i | i + 3 − − | i By using the BCH formula of equation (7), it has been shown that the similarity transformation of the operators K± are [15]

ξ∗ ξ∗ L = D(ξ)K D†(ξ)= αK + β K + K + K , (21) + + − ξ 0 + ξ − + | | ξ ξ L = D(ξ)K D†(ξ)= αK + β K + K + K , (22) − − − ξ 0 − ξ∗ + − | | These action of these on the SU(1, 1) Perelomov number coherent states is

L+ ζ,k,n = (n + 1)(2k + n) ζ,k,n +1 , L− ζ,k,n = n(2k + n 1) ζ,k,n 1 . | i | i | i − | −(23)i p p Thus, L± act as ladder operators for these number coherent states. Also, the most general form of theses states on the Fock space was calculated as follows [15]

∞ ζs n ( ζ∗)j Γ(2k + n)Γ(2k + n j + s) ζ,k,n = − eη(k+n−j) − | i s! j! Γ(2k + n j) s=0 j=0 p X X − Γ(n + 1)Γ(n j + s + 1) − k, n j + s . (24) × Γ(n j + 1) | − i p − These states are the analogue for the SU(1, 1) group to those given by Nieto for the harmonic oscillator in equation (10).

3 A charged particle in a uniform magnetic ﬁeld in the Landau gauge.

The stationary Schrödinger equation of a charged particle in a uniform magnetic field B~ is given by 1 e 2 HΨ= p + A~ Ψ= EΨ, (25) 2µ c where A~ is the vectorial potential, related to the magnetic field as B~ = A~. This vector ∇ × potential does not describe the magnetic field in a unique way, since the magnetic field remains invariant against gauge transformations A~ A~′ = A~ + g, where g is a time independent scalar field [18]. We can choose the vector potential→ as A~ =∇ 1 B~ ~r and our coordinate system 2 × so that the z-axis is parallel to B~ . Then, B A~ = (y, x, 0). (26) − 2 −

5 This choice is known as the symmetric gauge. If we make the gauge transformation with g = B xy, we obtain the following vector potential − 2 A~′ = B(y, 0, 0). (27) − This choice of the vector potential is known as the Landau gauge. With this gauge the equation (25) becomes [18]

~2 d2ψ mω2 ~2k2 + (y d)2ψ = E z ψ, (28) − 2m dy2 2 − − 2m where the Larmor frequency ω and d are deﬁned as eB ~ck ω = , d = x . (29) mc eB By introducing the harmonic oscillator operators a, a†

mω i mω i a = y + p , a† = y p , (30) ~ √ ~ y ~ − √ ~ y r 2mω r 2mω we can write the equation (28) as follows

1 ~2k2 mω2 Hψ = µ a†a + ψ + ν a + a† ψ = E z d2 ψ, (31) 2 − 2m − 2 where ~mω3 µ = ~ω, ν = d. (32) −r 2 ~2 2 2 If we make the deﬁnition ǫ = E kz mω d2 and in order to diagonalize this Hamiltonian, − 2m − 2 we apply the tilting transformation with the displacement operator as follows [15, 16]

D†(α)HD(α)D†(α)ψ = ǫD†(α)ψ. (33)

From equations (8) the tilted Hamiltonian H′ = D†(α)HD(α) becomes

1 H′ = µ a†a + α 2 + ν(α + α∗)+ a†(αµ ν)+ a(α∗µ ν). (34) | | 2 − − −

If we choose the coherent state parameters y0 = d and py0 = 0 we obtain that the tilted Hamiltonian reduces, up to a constant factor, to that of the one-dimensional harmonic oscillator 1 H′ = µ a†a + α 2 + ν(α + α∗). (35) | | 2 − Thus, the energy spectrum for a charged particle in an uniform magnetic ﬁeld is

1 ~2k2 E = n + ~ω + z . (36) 2 2m 6 The wave function ψ is obtained by applying the displacement operator D(α) to the harmonic oscillator wave functions ψ′. Thus, from equation (13)

(y−y )2 ′ − 0 y y0 ψ = D(α)ψ = N e 2λ2 H − . (37) 1 n λ

~c 1/2 In this expression N1 is a normalization constant and λ is the magnetic length λ = eB . The eigenfunction for the general problem Ψ is obtained by adding the free particle term ei(kxx+kzz) to equation (37). Therefore, we have showed that the eigenfunctions of a charged particle in an uniform magnetic field are the harmonic oscillator number coherent states. The treatment developed in this section can be also applied to the problem of a charged particle in a pure electric field or in a magnetic and electric field. With a proper choice of the coherent states parameters it can be shown that the harmonic oscillator number coherent states are the eigenfunctions of these problems.

4 A charged particle in a uniform magnetic ﬁeld in the symmetric gauge and its SU(1, 1) number coherent states

In the symmetric gauge (equation (26)) the Schr¨odinger equation of a charged particle in a uniform magnetic ﬁeld is

~2 eB e2B2 Hψ = − 2ψ + L ψ + (x2 + y2)ψ = Eψ. (38) 2µ ∇ 2µc z 8µc2

If we consider the wave function ψ(r) = U(ρ)eimφeikz (m = 0, 1, 2, ...) in cylindrical coordinates, the Schr¨odinger equation for a charged particle in an external magnetic ﬁeld remains

d2 1 d m2 e2B2 2mE eBm + ρ2 + k2 U(ρ)=0. (39) dρ2 ρ dρ − ρ2 − 4~2c2 ~2 − ~c −

eB By performing the change of variable x = 2~c ρ in the above equation we obtain q d2 1 d m2 + +(λ x2) U(x)=0, (40) dx2 x dx − x2 − where we have introduced the variable λ deﬁned as 4µc ~2k2 λ = E 2m. (41) eB~ − 2µ − The su(1, 1) Lie algebra for this problem is well known and the generators for its realization are given by [28] 1 d K = x x2 +2K 1 , (42) ± 2 ± dx − 0 ± 7 1 d2 1 d m2 K = + + x2 . (43) 0 4 −dx2 − x dx x2 Moreover, by deﬁning y = x2, the normalized wave functions are

2n! −y/2 m/2 m Un(y)= e y Ln (y), (44) s(n + m)! which are the Sturmian functions for the unitary irreducible representations of the su(1, 1) Lie algebra. Also, the Bargmann index k is k = m/2+1/2, and the other group number is just the radial quantum number n. Therefore we can construct the SU(1, 1) Perelomov number coherent states for this problem by substituting equation (44) into equation (24). By interchanging the order of summations and using the properties 48.7.6 and 48.7.8 of reference [29] we obtain

n ∗ n 2 m + 1 n 2Γ(n + 1) ( 1) ( ζ ) (1 ζ ) 2 2 (1 + σ) ψ = − eimφ − −| | n,m Γ(n + m + 1) √π (1 ζ)m+1 s − 2 2 − ρ (ζ+1) ρ σ e 2(1−ζ) ρmLm , (45) × n (1 ζ)(1 σ) − − where we have defined 1 ζ 2 σ = −| | . (46) (1 ζ)( ζ∗) − − These are the SU(1, 1) Perelomov number coherent states of a charged particle in a magnetic field. As a particular case of this result we can see that for n = 0 these states reduces to the standard Perelomov coherent states, presented in reference [30]. These states are significant in quantum optics, since a particular case of them are the eigenfunctions of the non-degenerate parametric amplifier [15].

5 SU(1, 1) contraction to the Heisenberg-Weyl group.

In this section, we will contract the su(1, 1) Lie algebra to the h(4) algebra of the harmonic oscillator. The proceeding developed here is analogue to that presented by Arecchi in reference [31] for the su(2) algebra. Thus, we deﬁne the following transformation

h+ c 00 0 K+ h 0 c 0 0 K  − =    − . (47) h 0 0 1 1 K 0 − 2c2 0 h  000 1  K   I     I        These new operators h satisfy the following commutation relationships [h , h ]= cK , [h , h ]=2c2K , [~h, h ]=0. (48) 0 ± ± ± − + 0 I In the limit c 0 this transformation becomes singular. However, the commutation relationships are well→ deﬁned and become [h , h ]= h , [h , h ]= h , [~h, h ]=0, (49) 0 ± ± ± − + 0 I 8 which is nothing but the h(4) algebra with the deﬁnition

† † lim h0 = n = a a, lim h+ = a , lim h− = a. (50) c→0 c→0 c→0 Also, in order to contract the displacement operator D(ξ) to the Weyl operator D(α) the coherent state parameters must satisfy ξ ξ∗ lim = α, lim = α∗. (51) c→0 c c→0 c To obtain the relationship between the contraction parameter c and the group number k we apply the h operator to an arbitrary su(1, 1) state n, k 0 | i 1 1 h n, k = K K n, k = n + k n, k . (52) 0| i 0 − 2c2 I | i − 2c2 | i If we demand that this eigenvalue must vanish for the lowest state 0,k we obtain | i 1 lim k =0. (53) c→0 − 2c2 Thus, in the limit c 0, c = 1 and the su(1, 1) irreducible unitary representations → 2k contract to the h(4) irreducible unitaryq representations. The relationship between the states of both groups can be obtained by deﬁning the state

, n = lim n, k . (54) |∞ i c→0 | i With this deﬁnition we obtain

† 1 1 a a , n = lim K0 n, k = lim n + k n, k = n , n . (55) |∞ i c→0 − 2c2 | i c→0 − 2c2 | i |∞ i In a similar way we obtain

a† , n = √n +1 , n +1 a , n = √n , n 1 . (56) |∞ i |∞ i |∞ i |∞ − i Therefore, the Perelomov number coherent states contract to the harmonic oscillator number coherent states, since

2 2 k ξ 2 ∗ 1/2c αa† −|α|2 αa† α = lim ζ,n,k = lim 1 ζ e K+ n, k = lim 1 c αα e 0 = e e 0 . | i c→0 | i c→0 −| | | i c→0 − | i | i (57) In our problem, this implies that the SU(1, 1) Perelomov number coherent states of a charged particle in a magnetic ﬁeld of equation (45), under the contraction of the SU(1, 1) group, reduce to the number coherent states of the harmonic oscillator of equation (37). In reference [32], the authors studied the contraction of the SU(1, 1) group to the quantum harmonic oscillator. Moreover, they shown that one advantage of working with SU(1, 1) is that its representation Hilbert space is inﬁnite-dimensional, thus it does not change dimension in the contraction limit, as it happens for the SU(2) case.

9 6 Concluding remarks

We applied the generalized number coherent states theory to study the problem of a charged particle in the Landau and symmetric gauge. We showed that for the Landau gauge, the eigenfunctions for the Landau level states can be represented in terms of the harmonic oscillator coherent states. For the symmetric gauge we study the eigenfunctions of this problem in cylindrical coordinates and we constructed the SU(1, 1) Perelomov number coherent states in a closed way. We show that under a contraction of the SU(1, 1) group, the Perelomov number coherent states are reduced to the number coherent states of the harmonic oscillator, related to the Heisenberg-Weyl group. It is important to note that the tilting transformation method used in this work has been applied to more novel problems, as the non-degenerate parametric ampliﬁer [15], the problem of two coupled oscillators [16], and the generalized MICZ-Kepler problem [17].

Acknowledgments

This work was partially supported by SNI-M´exico, COFAA-IPN, EDI-IPN, EDD-IPN, SIP- IPN project number 20170632.

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