Induced Fractional Zero-Point Angular Momentum for Charged Particles of the Bohm-Aharonov System by means of a "Spectator" Magnetic Field

0.14 MB·English

by  Jian-Zu Zhang

Induced Fractional Zero-Point Angular Momentum for Charged
Particles of the Bohm-Aharonov System by means of a
“Spectator” Magnetic Field
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Jian-Zu Zhang

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Institute for Theoretical Physics, East China University of Science and Technology, Box
7
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316, Shanghai 200237, P. R. China

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a Abstract
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An induced fractional zero-point angular momentum of charged particles by the
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Bohm-Aharonov (B-A) vector potential is realized via a modified combined trap. It

1
explores a “spectator” mechanism in this type of quantum effects: In the limit of
v
8 the kinetic energy approaching one of its eigenvalues the B-A vector potential alone
1 cannot induce a fractional zero-point angular momentum at quantum mechanical
6
level in the B-A magnetic field-free region; But when there is a “spectator” magnetic
2
. fieldtheB-Avectorpotentialinducesafractionalzero-pointangularmomentum. The
1
0 “spectator” does not contribute to such a fractional angular momentum, but plays
9 essential role in guaranteeing non-trivial dynamics at quantum mechanical level in
0
the required limit. This “spectator” mechanism is significant in investigating the
:
v B-A effects and related topics in both aspects of theory and experiment.
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a ∗ Revised version of arXiv: 0709.0843
As is well known, quantum states of charged particles can be influenced by electro-
magnetic effects even if those particles are in a region of vanishing field strength [1, 2].
As predicted by Bohm and Aharonov (B-A) [2], experiments [3] showed that in a mul-
tiply connected region where field strength is zero everywhere the interference spectrum
suffered a shift according to the amount of the loop integral of magnetic vector potential
around an unshrinkable loop. Wu and Yang [4] pointed out that the B-A effects is due
to the non-trivial topology of the space wehere the magnetic field strength is vanishing.
The B-Aeffect is purely quantum mechanical one which explores far-reaching consequences
of vector potential in quantum theory. This effect has been received much attention for
years [5, 6, 7]. Recently investigations in this topic concentrated on revealing new types of
quantum phases: The Aharonov-Casher effect [8], the He-McKellar-Wilkens phase [9] and
the Anandan phase [10].
In another aspect a fractional angular momentum originated from the Poynting vector
produced by crossing the Coulomb field of a charged particle with an external magnetic
field has been predicted by Peshkin, Talmi and Tassie for years [6, 11]. There are lots
of works concerning fractional angular momentum in B-A dynamics and their fractional
statistics (see the reviews [12–17] and references therein). Spatial noncommutativity also
leads to fractional angular momentum [18, 19].
Recently Kastrup [20] considered the question of how to quantize a classical system
of the canonically conjugate pair angle and orbital angular momentum. This has been
a controversial issue since the founding days of quantum mechanics [21]. The problem
is that the angle is a multivalued or discontinuous variable on the corresponding phase
space. A crucial point is that the irreducible unitary representations of the euclidean
group E(2) or of its covering groups allow for orbital angular momentum l = h¯(n + δ)
where n = 0, 1, 2, , and 0 δ < 1. The case δ = 0 corresponds to fractional
± ± ··· ≤ 6
zero-point angular momentum. Kastrup investigated the physical possibility of fractional
orbital angular momentum in connection with the quantum optics of Laguerre-Gaussian
laser modes in external magnetic fields, and pointed out that if implementable this would
lead to a wealth of new theoretical, experimental and even technological possibilities.
In this paper the induced fractional zero-point angular momentum of charged parti-
2
cles by the B-A vector potential is realized via a modified combined trap. It explores a
“spectator” mechanism in this type of quantum effects: In the limit of the kinetic energy
approaching one of its eigenvalues the B-A vector potential alone cannot induce a frac-
tional zero-point angular momentum of charged particles at quantum mechanical level in
a region of vanishing B-A field strength; But when there is a “spectator” magnetic field
the B-A vector potential induces a fractional zero-point angular momentum in the same
region. The “spectator” does not contribute to such a fractional angular momentum, but
plays essential role in guaranteeing non-trivial dynamics at quantum mechanical level in
the required limit. This type of quantum effects is so remarkable that in quantum me-
chanics the vector potential itself has physical significant meaning and becomes effectively
measurable not only in shifts of interference spectra originated from quantum phases but
also in physical observables.
1. Dynamics in a Modified Combined Trap – We consider ions constrained
in a modified combined trap including the B-A type magnetic field. The Paul, Pen-
ning, and combined traps share the same electrode structure [22]. A combined trap op-
erates in all of the fields of the Paul and Penning traps being applied simultaneously.
The trapping mechanism in a Paul trap involves an oscillating axially symmetric elec-
tric potential U˜(ρ,φ,z,t) = U(ρ,φ,z)cosΩ˜t with U(ρ,φ,z) = V(z2 ρ2/2)/2d2 where
−
ρ, φ and z are cylindrical coordinates, V and d are, respectively, characteristic voltage
and length, and Ω˜ is a large radio-frequency. The dominant effect of the oscillating
potential is to add an oscillating phase factor to the wave function. Rapidly varying
terms of time in Schrodinger equation can be replaced by their average values. Thus for
Ω˜ Ω √2q V /µd2 1/2 we obtain a time-independent effective electric potential [23]
≫ ≡ | |
V = q2 (cid:0)U U/4µΩ˜2(cid:1)= µω2(ρ2+4z2)/2 where µ and q(> 0) are, respectively, the mass
eff ∇ ·∇ P
and charge of the trapped ion, and ω = Ω2/4Ω˜. A modified combined trap combines the
P
above electrostatic potential and two magnetic fields [24]: a homogeneous magnetic field
B aligned along the z axis in a normal combined trap and a B-A type magnetic field B
c 0
produced by, for example, an infinitely long solenoid with radius ρ = (x2 + x2)1/2 = a.
1 2
Inside the solenoid (ρ < a) B = (0,0,B ) is homogeneous along the z axis, and outside
0,in 0
the solenoid (ρ > a) B = 0. The vector potential A of B is chosen as (Henceforth
0,out c c
3
the summation convention is used) A = B ǫ x /2, A = 0, (i,j = 1,2). The B-A
c,i c ij j c,z
−
vector potential A is: Inside the solenoid A = A = B ǫ x /2, A = 0; Outside
0 0,i in,i 0 ij j in,z
−
the solenoid A = A = B a2ǫ x /2x x , A = 0, (i,j,k = 1,2). At ρ = a the
0,i out,i 0 ij j k k out,z
−
potential A passes continuously over into A . The Hamiltonian of the modified com-
in out
bined trap is H = (p qA /c qA /c)2/2µ+p2/2µ+µω2 (x2 +4z2)/2. This Hamil-
i − c,i − 0,i z P i
tonian can be decomposed into a one-dimensional harmonic Hamiltonian H (z) along the
z
z-axis with the axial frequency ω = 2ω and a two-dimensional Hamiltonian H (x ,x ),
z P ⊥ 1 2
H = H (z)+H (x ,x ). Inside the solenoid the ion’s motion is the same as the one with
z ⊥ 1 2
a total magnetic field B +B .
c 0,in
In the following we consider the motion outside the solenoid. The two-dimensional
Hamiltonian outside the solenoid is [22, 23]
2
1 1 ǫ x 1
H (x ,x ) = p + µω ǫ x +µω a2 ij j + µω2x2, (1)
⊥ 1 2 2µ (cid:18) i 2 c ij j 0 2x x (cid:19) 2 P i
k k
where ω = qB /µc and ω = qB /µc are the cyclotron frequencies corresponding to,
c c 0 0
respectively, the magnetic fields B and B . The Hamiltonian H possess a rotational
c 0,in ⊥
symmetry in (x ,x ) - plane. The z-component of the orbital angular momentum J =
1 2 z
ǫ x p commutes with H . They have common eigenstates.
ij i j ⊥
Dynamics in the Limit of the Kinetic Energy Approaching its Lowest Eigenvalue – In
this limit the kinetic energy is E = µx˙ x˙ /2 = (K2 +K2)/2µ where
k i i 1 2
1 ǫ x
K p + µω ǫ x +µω a2 ij j , [K ,K ] = ih¯µω ǫ (2)
i i c ij j 0 i j c ij
≡ 2 2x x
k k
Here K is the mechanical momenta corresponding to the vector potentials A and A .
i c,i out,i
It is worth noting that the B-A vector potential A does not contributes to the commu-
out,i
tator [K ,K ]. The canonical momenta p are quantized, p = ih¯∂/∂x . They commute
i j i i i
−
each other [p ,p ] = 0. We define canonical variables Q = K /µω and Π = K which
i j 1 c 2
satisfy [Q,Π] = ih¯δ . The kinetic energy E is rewritten as the Hamiltonian of a harmonic
ij k
oscillator E = Π2/2µ + µω2Q2/2. The lowest eigenvalue of the kinetic energy E is
k c Ek0 k
[25] = h¯ω /2.
k0 c
E
In a laser trapping field, using a number of laser beams and exploiting Zeeman tuning,
the speed of atoms can be slowed to the extent of 1 ms−1, see [26]. Ions are the common
4
object in cooling and trapping. In order to experimentally realizing the limit of E
k k0
→ E
through laser cooling in a trap ions are used.
In the limit of the kinetic energy approaching its lowest eigenvalue the Hamiltonian H
⊥
in Eq. (1) has non-trivial dynamics [27, 28, 19]. The Lagrangian corresponding to H is
⊥
1 1 ǫ x˙ x 1
L = µx˙ x˙ µω ǫ x˙ x µω a2 ij i j µω2x x . (3)
2 i i − 2 c ij i j − 0 2x x − 2 P i i
k k
In the limit of E , the Hamiltonian H reduces to H = h¯ω /2+ µω2x x /2. The
k → Ek0 ⊥ 0 c P i i
Lagrangian corresponds to H is
0
1 ǫ x˙ x 1 1
L = µω ǫ x˙ x µω a2 ij i j µω2x x h¯ω . (4)
0 −2 c ij i j − 0 2x x − 2 P i i − 2 c
k k
Constraints – For the reduced system (H ,L ) the canonical momenta are
0 0
∂L 1 ǫ x
p = 0 = µω ǫ x µω a2 ij j . (5)
i c ij j 0
∂x˙ −2 − 2x x
i k k
Eq. (5) does not determine velocities x˙ as functions of p and x , but gives relations among
i i j
p and x , that is, such relations are the primary constraints [29, 28, 19]
i j
1 ǫ x
ϕ (x,p) = p + µω ǫ x +µω a2 ij j = 0. (6)
i i c ij j 0
2 2x x
k k
The physical meaning of Eq. (6) is that it expresses the dependence of degrees of freedom
among p and x . The constraints (6) should be carefully treated [30]. The subject can be
i j
treated simply by the symplectic method in [31, 32]. In this paper we work in the Dirac
formalism. The Poisson brackets of the constraints (6) are
C = ϕ ,ϕ = µω ǫ . (7)
ij i j c ij
{ }
From Eq. (7), ϕ ,ϕ = 0, it follows that the conditions of the constraints ϕ holding at
i j i
{ } 6
all times do not lead to secondary constraints.
C defined in Eq. (7) are elements of the constraint matrix . Elements of its in-
ij
C
verse matrix −1 are (C−1) = ǫ /µω . The corresponding Dirac brackets of ϕ ,x ,
ij ij c i j D
C − { }
ϕ ,p , x ,x , p ,p and x ,p can be defined. The Dirac brackets of ϕ with
i j D i j D i j D i j D i
{ } { } { } { }
any variables x and p are zero so that the constraints (6) are strong conditions. It can be
i j
5
used to eliminate dependent variables. If we select x and x as the independent variables,
1 2
from the constraints (6) the variables p and p can be represented by, respectively, the
1 2
independent variables x and x as
2 1
1 x 1 x
p = µω x µω a2 2 , p = µω x +µω a2 1 (8)
1 c 2 0 2 c 1 0
−2 − 2x x 2 2x x
k k k k
The Dirac brackets of x and x is
1 2
1
x ,x = . (9)
1 2 D
{ } µω
c
We introduce new canonical variables x = x and p = µω x . Their Dirac bracket is
1 c 2
x,p = 1. According to Dirac’s formalism of quantizing constrained systems the corre-
D
{ }
sponding quantum commutation relation is [x,p] = ih¯.
Quantum Behavior of the Reduced System – Now we consider quantum behavior of the
reduced system (H ,L ). By defining the following effective mass and frequency, µ∗
0 0
≡
µω2/ω2, ω∗ ω2/ω , the Hamiltonian H is represented as H = p2/2µ∗+µ∗ω∗2x2/2+
c P ≡ P c 0 0
h¯ω /2. We introduce an annihilation operator A = µ∗ω∗/2h¯ x+i 1/2h¯µ∗ω∗ p and its
c
p p
conjugate one A†. The operators A and A† satisfies [A,A†] = 1. The eigenvalues of the
number operator N = A†A is n = 0,1,2, . Using A and A†, the reduced Hamiltonian
···
H is rewritten as H = h¯ω∗ A†A+1/2 +h¯ω /2.
0 0 c
(cid:0) (cid:1)
Now we consider the angular momentum of the ion. Using Eq. (8) to replace p and
1
p by, respectively, the independent variables x and x , the orbital angular momentum
2 2 1
J = ǫ x p is rewritten as
z ij i j
q 1
J = Φ + µω (x2 +x2), (10)
z 2πc 0 2 c 1 2
where Φ = πa2B is the total flux of the magnetic field B inside the solenoid. Similarly,
0 0 0
using A and A† to rewrite J , we obtain J = qΦ /2πc +h¯ A†A+1/2 . The zero-point
z z 0
(cid:0) (cid:1)
angular momentum of J is = h¯/2+qΦ /2πc. In the above the term [33]
z 0 0
J
q
= Φ (11)
AB 0
J 2πc
is the zero-point angular momentum induced by the AB vector potential. takes
AB
J
fractional values. It is related to the region where the magnetic field B = 0 but the
0,out
corresponding vector potential A = 0.
out
6
6
2. Dynamics in the Case of Bc = 0 – It is worth noting that here Bc, like a
“spectator”, does not contribute to . In order to clarify the role played by B , we
AB c
J
consider the case of B = 0. In this case the modified combined trap is as stable as a Paul
c
trap. The corresponding kinetic energy reduces to E˜ = µx˙ x˙ /2 = K˜2 +K˜2 /2µ where
k i i 1 2
(cid:16) (cid:17)
ǫ x
K˜ p +µω a2 ij j , [K˜ ,K˜ ] = 0. (12)
i i 0 i j
≡ 2x x
k k
IntheaboveK˜ isthemechanicalmomentacorrespondingtotheB-AvectorpotentialA .
i out,i
Unlike the ordinary vector potential, the special feature of the B-A vector potential is that
it doesnotcontributes tothecommutator [K˜ ,K˜ ]. Because K˜ arecommuting, behavior of
i j i
E˜ issimilar toaHamiltonianofafreeparticle. Itsspectrum isacontinuous one. WhenE˜
k k
approaching some constant ˜ (= 0) the Hamiltonian H reduces to H˜ = ˜ +µω2x x /2.
Ek 6 ⊥ 0 Ek P i i
The Lagrangian corresponding to H˜ is
0
ǫ x˙ x 1
L˜ = µω a2 ij i j µω2x x ˜ . (13)
0 − 0 2x x − 2 P i i −Ek
k k
˜
From L we obtain the canonical momenta
0
∂L˜ ǫ x
p˜ = 0 = µω a2 ij j . (14)
i 0
∂x˙ − 2x x
i k k
Now we clarify that the case ˜ = 0 should be excluded. The limit of the kinetic energy
k
E
E = µx˙ x˙ /2 0 corresponds two possibilities: x˙ = 0 or µ 0. In the case x˙ = 0
k i i i i
→ →
the Lagrangian L in Eq. (3) reduces to L˜′ = µω2x x /2. The corresponding canonical
0 − P i i
momenta p˜ = ∂L˜′/∂x˙ = 0. Therefore there is no dynamics. According to the definition
i 0 i
of the frequency Ω the other possibility µ 0 is forbidden.
→
Eq. (14) gives the reduced primary constraints
ǫ x
ϕ˜ = p˜ +µω a2 ij j = 0. (15)
i 0i 0
2x x
k k
Here the special feature is that the corresponding Poisson brackets are zero,
C˜ = ϕ˜,ϕ˜ 0. (16)
ij i j
{ } ≡
FromEq.(16), ϕ˜ ,ϕ˜ 0,itfollowsthattheconditionsoftheconstraintsϕ˜ holdingatall
i i i
{ } ≡
times lead to secondary constraints ϕ˜(2) = µω2x . The Poisson brackets ϕ˜(2),ϕ˜ = 0,
i − P i { i j}
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ϕ˜(2),ϕ˜ (2) = 0, and ϕ˜(2),H˜ = 0, so that persistence of the secondary constraints ϕ˜(2)
{ i j } { i 0} i
(3)
in course of time does not lead to further secondary constraints ϕ˜ .
i
Because of C˜ 0,theinverse matrix ˜−1 does notexist. The Diracbrackets ϕ˜,x ,
ij i j D
≡ C { }
ϕ˜,p , ϕ˜(2),x , ϕ˜(2),p , x ,x , p ,p , and x ,p cannot be defined.
i j D i j D i j D i j D i j D i j D
{ } { } { } { } { } { }
According to Dirac’s formalism of quantizing constrained systems, there is no way to
establish dynamics at quantum mechanical level. This means that the B-Avector potential
alonecannotleadtonon-trivialdynamicsatquantummechanicallevelintherequiredlimit,
thus does not contribute to the energy spectrum and angular momentum at all.
It is clear that though the vector potential A of the “spectator” magnetic field B
c,i c
does not contribute to , it plays essential role in guaranteeing non-trivial dynamics
AB
J
at quantum mechanical level in the limit of the kinetic energy approaching one of its
eigenvalues. This example reveals that, unlike ordinary vector potential, the physical role
played by the B-A vector potential is subtle. This needs to be carefully analyzed at
quantum mechanical level.
3. Dynamics in the Case of B0 = 0 – In order to further clarify the essential
difference between A and A in the region of B = 0 we consider the case of B = 0.
o c 0,out 0
In this case the modified combined trap reduces to a combined trap. The Hamiltonian
H (x ,x ) in Eq. (1) reduces to Hˆ (x ,x ) = (p +µω ǫ x /2)2/2µ+µω2x2/2. Its kinetic
⊥ 1 2 ⊥ 1 2 i c ij j P i
energy is Eˆ = (Kˆ2 +Kˆ2)/2µ where
k 1 2
Kˆ p +µω ǫ x /2, [Kˆ ,Kˆ ] = ih¯µω ǫ . (17)
i i c ij j i j c ij
≡
In Eq. (17) Kˆ is the mechanical momenta corresponding to the vector potentials A . The
i c,i
commutation relations between Kˆ ’s are the same as the ones between K ’s in Eq. (2).
i i
ˆ ˆ
The eigenvalues of E is = h¯ω (n+1/2), which are just the Landau levels of charged
k kn c
E
particles in an external magnetic field.
In the following we consider the limit of Eˆ approaching the lowest eigenvalue ˆ =
k k0
E
h¯ω /2. The Lagrangian corresponding to Hˆ is
c ⊥
Lˆ = µx˙ x˙ /2 µω ǫ x˙ x /2 µω2x x /2. (18)
i i − c ij i j − P i i
In the limit of Eˆ ˆ , the Hamiltonian Hˆ reduces to Hˆ = h¯ω /2+µω2x x /2 which
k → Ek0 ⊥ 0 c P i i
8
is the same as H . The Lagrangian corresponds to Hˆ is
0 0
Lˆ = µω ǫ x˙ x /2 µω2x x /2 h¯ω /2. (19)
0 − c ij i j − P i i − c
ˆ ˆ ˆ
For the reduced system (H ,L ) the canonical momenta are pˆ = ∂L /∂x˙ = µω ǫ x /2.
0 0 i 0 i c ij j
−
It leads to the following constraints
ϕˆ = p +µω ǫ x /2 = 0. (20)
i i c ij j
The Poisson brackets of ϕˆ are the same as ones of the constraints ϕ in Eq. (7):
i i
Cˆ = ϕˆ ,ϕˆ = µω ǫ . (21)
ij i i c ij
{ }
From Eq. (21), ϕˆ ,ϕˆ = 0, it follows that the conditions of the constraints ϕˆ holding at
i i i
{ } 6
all times do not lead to secondary constraints.
By the similar procedure of treating the constraints (6), we find that the reduced
system (Hˆ ,Lˆ ) has non-trivial dynamics at quantum mechanical level in the limit of
0 0
Eˆ ˆ . The constraints (20) are strong conditions which can be used to eliminate
k k0
→ E
dependent variables. We select x and x as the independent variables. The variables p
1 2 1
andp canberepresented by, respectively, x andx asp = µω x /2, p = µω x /2.The
2 2 1 1 c 2 2 c 1
−
Dirac brackets of x and x is x ,x = 1/µω . We introduce new canonical variables
1 2 1 2 D c
{ }
x = x and p = µω x . Their Dirac bracket is x,p = 1. The corresponding quantum
1 c 2 D
{ }
commutationrelationis[x,p] = ih¯. UsingtheseresultstheorbitalangularmomentumJ =
z
ǫ x p canberepresentedbythecanonicalvariablesxandpasJˆ = (p2/2µ+µω2x2/2)/ω .
ij i j z c c
The zero-point angular momentum can bereadout fromthis harmonic-like “Hamiltonian”,
ˆ = h¯/2. We note that in this case there is no fractional zero-point angular momentum.
0
J
The above results elucidate that A are essentially different from A : the A alone can
c 0 c
lead to non-trivial dynamics at quantum mechanical level in the limit of the kinetic energy
approaching its lowest eigenvalue.
4. Gauge Transformation–Asiswellknown,wecanperformagaugetransformation
χ so that the resulting vector potential A′ = A + χ = 0. A suitable gauge function
out out ∇
[34] is χ = B a2tan−1(x /x )/2. In the Schr¨odinger equation the corresponding gauge
0 2 1
−
transformation is = exp(iqχ/ch¯). Under this gauge transformation the Hamiltonian
G
9
H (x ,x ) in Eq. (1) is transformed into H H −1 = H ′ = (p +µω ǫ x /2)2/2µ+
⊥ 1 2 ⊥ → G ⊥G ⊥ i c ij j
µω2x2/2. Here H ′ is the same Hˆ .
P i ⊥ ⊥
In the limit of the kinetic energy approaching its lowest eigenvalue the corresponding
reduced constraints are the same ϕˆ in Eq. (20). Under the gauge transformation the
i
G
angular momentum J = ǫ x p is transformed into J J −1 = J′ = x p x p +
z ij i j z → G zG z 1 2 − 2 1
qΦ /2πc. Using the constraints ϕˆ in Eq. (20) to represent p and p by, respectively, the
0 i 1 2
independent variables x and x , the first term in J′ reads x p x p = µω (x2 +x2)/2.
2 1 z 1 2 − 2 1 c 1 2
Thus we obtain
q 1
J′ = Φ + µω (x2 +x2). (22)
z 2πc 0 2 c 1 2
J′ is the same J in Eq. (10). This result shows that the fractional zero-point angular
z z
momentum induced by the B-A vector potential is a real physical observable which cannot
be gauged away by a gauge transformation.
In summary, this paper explores a “spectator” mechanism in B-A effects. It is clari-
fied that the B-A vector potential alone cannot lead to non-trivial dynamics at quantum
mechanical level in the limit of the kinetic energy approaching one of its eigenvalues. In
such a limit the B-A vector potential alone cannot induce a fractional zero-point angular
momentum. When there is a “spectator” magnetic field the B-A vector potential induces a
fractional zero-point angular momentum. The induced effect essentially depends upon the
participation of a “spectator” magnetic field. The “spectator” vector potential does not
contribute to the fractional angular momentum, but plays essential role in guaranteeing
non-trivial dynamics at quantum mechanical level in the required limit. The “spectator”
mechanism is significant in both aspects of theory and experiment. In the theoretical as-
pect, it is revealed that, unlike ordinary vector potentials, the physical role played by the
B-A vector potential is subtle. This needs to be carefully analyzed at quantum mechanical
level. In the experimental aspect, existence of a “spectator” magnetic field is necessary for
inducing the fractional angular momentum by the B-A vector potential. As an example,
the modified combined trap provides a realistic way to realize this “spectator” mechanism.
The author would like to thank M. Peshkin for the helpful discussions and Referee for
the valuable comments. This work has been supported by the Natural Science Foundation
10