Quantum anticentrifugal potential in a bent waveguide

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by  R. Dandoloff

Quantum anticentrifugal potential in a bent waveguide
Rossen Dandoloff1 and Victor Atanasov2
1Laboratoire de Physique Th´eorique et Mod´elisation (CNRS-UMR 8089),
Universit´e de Cergy-Pontoise, F-95302 Cergy-Pontoise, France∗
2Department of Condensed Matter Physics, Sofia University, 5 boul. J. Boucher, 1164 Sofia, Bulgaria†
We show the existence of an anticentrifugal force for a quantum particle in a bent waveguide.
This counterintuitive force due to dimensionality was shown to exist in a flat R2 space but there
it needs an additional δ-like potential at the origin in order to brake the translational invariance
andtoexhibitlocalizedstates. Inthecaseofthebentwaveguidethereisnoneedofanyadditional
potential since here the boundary conditions break the symmetry. The effect may be observed in
interference experiments which are sensitive to the additional phase of the wavefunction gained in
the bent regions and can find application in distinguishing between straight and bent geometries.
1 PACSnumbers: 03.65.-w,03.65.Ge,42.50.-p
1
0
2 It has been noted that dimensionality plays an impor- expression:
n tant role in quantum mechanics. The two-dimensional
a Euclidean space takes a special place in the quantum
J world as e.g. the fractional quantum staistics appears dr2 =dξ2+dz2+(1−κξ)2ds2 (1)

6 in 2D as well as the quantum anticentrifugal potential
where the curvature κ = 1 is constant and the corre-
2 that was first shown to exist in 2D [1–3] and later in R
3D curved space [4]. Curved 2D surfaces have also been sponding Lam´e coefficients are hz = 1, hξ = 1, and
h] discussed previously[5, 6]. The centrifugal potential cor- hs =(1−κξ).
Now we are ready to write the Schr¨odinger equation
p responding to vanishing angular momentum states is at-
in this coordinate system. The Laplace operator in the
- tractive rather than repulsive which defies classical intu-
t Schr¨odinger equation for the wave function Ψ, has the
n ition. Thisquantumanticentrifugalforceispartoftheso
following form:
a called quantum fictitious forces that appear in two and
u
three space dimensions [7]. This phenomenon is due on
q
one hand to the nonvanishing commutator of the radial ∂2Ψ ∂2Ψ k ∂Ψ 1 ∂2Ψ
[ momentum pr and the unit vector in radial direction (cid:126)rr ∆Ψ= ∂z2 + ∂ξ2 −(1−κξ) ∂ξ +(1−κξ)2 ∂s2 . (2)
1 andontheotherhandtotherenormalizationofthewave
v function so that the Schr¨odinger equation is covariant in In the curvilinear coordinates the wavefunction ac-
1
the curvilinear coordinates. quires the form
8
0 Let us now consider a rectangular waveguide with an 1 Φ
5 edge a which is bent along a semicircle with radius R in Ψ= √ √ (3)
C 1−κξ
. the (x,y) plane. The semicircle with radius R represents
1
0 the axis of the rectangular waveguide. The radii of the In terms of Φ the norm of the wavefunction inside
1 inner and outer walls of the waveguide are R−a/2 and the waveguide equals[8]: (cid:82)a(cid:82)a/2 (cid:82)πR|Φ|2dzdξds = C,
0 −a/2 0
1 R+a/2respectively. IntheCartesian(x,y,z)coordinate
: system the ”lower” and ”upper” walls of the waveguide
v
i are situated at z =0 and z =a respectively.
X y
Now we will introduce a local coordinate system in
r
the bent waveguide. It is depicted in Fig. 1. We will
a
workwiththeprojectionoftherectangularwaveguideon
the (x,y) plane. This projection is along the OZ-axis.
The projection of the central axis of the waveguide on
the (x,y)-plane is a circle of radius R. The coordinate s,
whichisthearc-lengthofthesemi-circlewithradiusR,is
measured from the point (−R,0). The coordinate along
the normal to the circle within the (x,y)-plane is ξ ∈
[−a,a] and the coordinate along the OZ-axis is z. The
2 2
lineelementinthesecoordinatesisgivenbythefollowing x
FIG. 1: A projection of the geometry of a bent waveguide
∗Electronicaddress: rossen.dandoloff@u-cergy.fr onto the (x,y) plane.
†Electronicaddress: [email protected]
2
where C = C(a,R). In this way one can read off the and simplify
effective potential from the Schr¨odinger equation in the
coordinate system (ξ,s,z). The Laplacian ∆Ψ becomes: − ∂2Φ0 − 1 Φ =(cid:15)2Φ . (10)
∂µ2 4µ2 0 0
1 ∂2Φ k2 Φ
∆Ψ= +
(1−κξ)12 ∂ξ2 4 (1−κξ)52 Next we introduce
∂2Φ 1 ∂2Φ
+ + (4) ζ =(cid:15)µ (11)
∂z2 (1−κξ)2 ∂s2
and enter the above equation with the ansatz
and the time independent Schrodinger equation
− h¯2 ∆Ψ = EΨ, (where M is the mass of the particle) (cid:112)
2M Φ = ζφ(ζ) (12)
takes the following form: 0
Φ ¯h2 (cid:20) 1 ∂2Φ k2 Φ to obtain a zeroth order Bessel equation
E = +
(1−κξ)12 2M (1−κξ)12 ∂ξ2 4 (1−κξ)52 1
φ(cid:48)(cid:48)+ φ(cid:48)+φ(ζ)=0 (13)
∂2Φ 1 ∂2Φ(cid:21) ζ
+ + . (5)
∂z2 (1−κξ)2 ∂s2
which possess oscillating solutions given by J (ζ) =
0
The boundary conditions require that the wave func- 1 (cid:82)π eiζsinτdτ. In terms of ξ the solutions up to a nor-
2π −π
tion becomes zero on the walls of the wave-guide i.e. malization factor are
at z = 0 and z = a. (for the OZ direction) and for
(cid:115)
ξ = R−a/2 and ξ = R+a/2 (for the radial direction). 1 (cid:114)2ME n2π2
The wave-guide is open at both ends and therefore there Φ0(ξ) = |k| ¯h2 − a2 (1−κξ)
are no boundary conditions for s=0 and s=πR. For a
(cid:32) (cid:114) (cid:33)
similar treatment of a straight wave-guide see ref.[9]. 1 2ME n2π2
J − (1−κξ) . (14)
Equation (5) admits separation of variables. We use 0 |k| ¯h2 a2
the following ansatz
(cid:16)nπ (cid:17) Now we need to impose the boundary conditions which
Φ(s,ξ,z)=eimsκsin z Φ (ξ), (6)
a m will determine the energy eigenvalues. For this purpose
wewouldneedthezeroesofthezerothorderBesselfunc-
where m is an integer quantum number which is the an-
trion since the wavefunction vanishes at the borders of
gular momentum of the quantum particle. Indeed m is
the waveguide.
aneigenvalueofthez-componentoftheangularmomen-
ζ :J (ζ )=0 2.4048 5.5201 8.6537 11.7915 14.9309
tum operator (which commutes with the Hamiltonian l 0 l
H = − h¯2 ∆, therefore the s−component of the wave- l 1 2 3 4 5
2M Consequently
function is the eigenfunction of the angular momentum
operator. Note that there are no boundary conditions at
(cid:114)2ME n2π2 (cid:20) ξ (cid:21)
s = 0 and s = πR and none of the other boundary con- R − 1− 0 =ζ (15)
ditions depend on s, hence the commutator [J,H]=0 is ¯h2 a2 R l
not altered by the boundary conditions) : (cid:114)2ME n2π2 (cid:20) ξ (cid:21)
R − 1+ 0 =ζ (16)
¯h2 a2 R l+w
i¯h ∂
J =− (7)
κ ∂s after subtracting the second from the first relation we
obtain for the energy
witheigenfunctionseimsκ.Weareespeciallyinterestedin
the case of zero angular momentum m = 0 since in this ¯h2 (cid:34)(ζ −ζ )2 n2π2(cid:35)
case a quantum anticentrifugal force appears. Now the E = l+w l + . (17)
2M 4ξ2 a2
differential equation for Φ takes the following form: 0
0
Note, for very large energies l >> 1 the asymptotic for
∂2Φ (cid:20)n2π2 κ2 (cid:21) 2ME
− 0 + − Φ = Φ (8) thedifferenceofthezerosoftheBesselfunctionisζl+w−
∂ξ2 a2 4(1−κξ)2 0 ¯h2 0 ζ →wπ.Forl∼1,wehaveζ −ζ <wπwhichreflects
l l+w l
theanticentrifugalphenomenonaffectingthedistribution
The above differential equation is an effective
of the zeroes[1].
Schrodinger equation for Φ with an effective potential
(cid:104) (cid:105) 0 It is possible to calculate the Bohm potential corre-
V = n2π2 − κ2 .Inordertosolve(8)wedenote
eff a2 4(1−κξ)2 sponding to the bent waveguide
µ=1−κξ, (cid:15)2 = 1 (cid:20)2ME − n2π2(cid:21) (9) Q=− ¯h2 ∆R. (18)
κ2 ¯h2 a2 2M R
3
We obtain in the ξ coordinate This phase shift is minimized for n=0 and similarly for
the transverse states in ξ
¯h2 κ2(cid:0)a ξ2+a ξ+a (cid:1)
2 1 3
Q(ξ)= , (19)
2M 4ξ2(1−κξ)2
0 π¯h2κ
∆φ = . (26)
where min 4p
(ζ −ζ )2
a = −2 l+w l (20)
1 κ Inroducing the de Broglie wavelength λ=h/p the above
a = (ζ −ζ )2 (21) reduces to
2 l+w l
(ζ −ζ )2
a = l+w l +ξ2 (22)
3 κ2 0 ¯hλκ
∆φ = . (27)
min 8
ThedependenceoftheBohmpotentialonξ isveryweak
and we can approximate it with an effective one dimen-
sional rectangular barrier which corresponds to the case This is an irreducible amount which is always present
of a very thin waveguide. The height of this barrier is whenthegeometryofthewaveguideiscurvedandcanbe
the value of Q at ξ = 0. Adding the contribution form usedinaninterferenceexperimenttodistinguishbetween
the z coordinate we write the total Bohm potential curved and straight geometries.
¯h2 (cid:20)(ζ −ζ )2 κ2 n2π2(cid:21) The quantum anticentrifugal force on the central line
Q0 = 2M l+w4ξ2 l + 4 + a2 . (23) defined as the mean value of the gradient of the effective
0 potential is given by the following expression:
With the help of the Bohm potential we can translate
theeffectoftheboundaryconditionsintermsofanaddi-
¯h2 dV ¯h2 κ3 ¯h2 1
tionalpotentialaffectingthequantummotion. Itchanges F =− eff = = (28)
the interference picture according to 2M dξ 2M 2 2M 2R3
|ξ=0
where R is the radius of the waveguide. We note the
p(cid:48) =(cid:112)2M(E−Q ) (24) unusualoneover(distance)3 dependenceofthisanticen-
0
¯h2 (cid:20)(ζ −ζ )2 κ2 n2π2(cid:21) trifugal force for m=0. This defies classical intuition.
≈p− l+w l + +
p 4ξ2 4 a2 In conclusion, it should be noted the similarity of the
0
stationary Sch¨odinger equation and the Helmhotz equa-
which yields for the phase shift of the wavefunction the tion for the TE modes in a waveguide. Consequently, in
following the interference picture of an electromagnetic process in
a bent waveguide, one should find a similar pattern of
π
∆φ=l∆p= ∆p. (25) interference as in the quantum case.
κ
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