Commun. Theor. Phys. (Beijing, China) 41 (2004) pp. 365–368
c International Academic Publishers
Vol. 41, No. 3, March 15, 2004
Quantum Mechanical Fourier Hankel Representation Transform for an Electron
Moving in a Uniform Magnetic Field∗
FAN Hong-Yi1,2
1
Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
2
Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China
(Received June 18, 2003)
Abstract We find quantum mechanical Fourier–Hankel representation transform for an electron moving in a uniform magnetic field. The physical meaning of Fourier decomposition states of electron’s coordinate eigenstate and the
momentum eigenstate are revealed.
PACS numbers: 03.65.Ud, 42.30.Lr
Key words: Fourier–Hankel transform, electron moving, uniform magnetic field
Fourier–Hankel transform is frequently used in solving
many mathematical physics equations, e.g., heat conduction equation and some electrostatic problems.[1] In this
work we try to recast the Hankel transform into the formalism of two representations’ mutual transform in quantum mechanics. For example, we explain that the mutual
transform between the coordinate eigenstate and the momentum eigenstate of an electron moving in a uniform
magnetic field is such a kind of Fourier–Hankel transform.
Let us first recall the two-variable Fourier transform,
Z
1
F (~κ ) =
f (~r ) e i~κ·~r d 2~r ,
(1)
2π
Z
1
f (~r ) =
F (~κ) e i~κ·~r d 2~κ .
(2)
2π
By introducing polar coordinates for ~r and ~κ, ~r =
(r, θ), ~κ = (κ, φ), ~κ · ~r = kr cos(θ − φ) and expanding
both f (~r ) ≡ f (r, θ) and F (~κ) ≡ F (κ, Φ) in Fourier series
we have
∞
X
f (r, θ) =
fn (r) e inθ ,
(3)
n=−∞
∞
X
F (κ, φ) =
Fn (κ) e inφ ,
(4)
n=−∞
where
2π
1
fn (r) =
2π
Z
1
2π
Z
Fn (κ) =
f (r, θ) e −inθ dθ ,
(5)
0
2π
F (κ, φ) e −inφ dφ .
(6)
0
Substituting Eq. (1) into Eq. (6) and using Eq. (3) to
represent f (r, θ) we obtain
Z 2π
Z ∞
Z 2π
∞
X
1
−inφ
ikr cos(θ−φ)
e
dφ
r
dr
e
fm (r) e imθ dθ
(2π)2 0
0
0
n=−∞
Z ∞
Z 2π
Z ∞
1
=
fn (r)r dr
e inα+iκr cos α dα =
fn (r)Jn (κr)r dr ,
2π 0
0
0
Fn (κ) =
where Jn is the Bessel function
∞
X
(−1)n x n+2k
,
Jn (x) =
k!(n + k)! 2
(8)
k=0
and we have used the generating function of the Bessel
function,[2]
∞
X
e ix sin t =
Jm (x) e imt .
(9)
Fourier–Hankel transform, since the subscript n is an integer. In this work we shall point out that it is possible to construct a one-to-one correspondence between
Eqs. (3) ∼ (7) and their quantum mechanical representation transform. For example, corresponding to Eqs. (3)
and (4) we establish the following Fourier series expansion,
m=−∞
Similarly, we may derive the reciprocal relation of Eq. (7),
Z ∞
fn (r) =
Fn (κr)Jn (κr)κdκ .
(10)
0
Equations (7) and (10) are named as the integer-order
∗ The
(7)
|ηi =
∞
X
|q, rii e iqθ ,
η = |η| e iθ ,
r ≡ |η| ,
(11)
|s, r0 i e isϕ ,
ξ = |ξ| e iϕ ,
r0 ≡ |ξ| ,
(12)
q=−∞
|ξi =
∞
X
s=−∞
project supported by National Natural Science Foundation of China under Grant No. 10175057 and the President Foundation of
the Chinese Academy of Sciences
366
FAN Hong-Yi
where |ηi is the entangled state (the common eigenvector
of X1 − X2 and P1 + P2 ) in two-mode Fock space,[3]
o
n 1
|ηi = exp − |η|2 + ηa†1 − η ∗ a†2 + a†1 a†2 |00i ,
2
η = η1 + iη2 ,
(13)
and |ξi is the common eigenvector of X1 + X2 and
P1 − P2 ,[4]
o
n 1
|ξi = exp − |ξ|2 + ξa†1 + ξ ∗ a†2 − a†1 a†2 |00i ,
2
ξ = ξ1 + iξ2 ,
(14)
|00i is the two-mode vacuum state, a†i , i = 1, 2, are related
to coordinate operator
Xi and momentum
operator Pi by
√
√
Xi = (ai + a†i )/ 2, Pi = (ai − a†i )/ 2 i. |ηi and |ξi are
named entangled states as
√
√
(X1 − X2 )|ηi = 2 η1 |ηi , (P1 + P2 )|ηi = 2 η2 |ηi , (15)
√
√
(X1 + X2 )|ξi = 2 ξ1 |ξi , (P1 − P2 )|ξi = 2 ξ2 |ξi . (16)
It was Einstein, Podolsky, and Rosen[5] who first used
[(X1 − X2 ), (P1 + P2 )] = 0 to explain the quantum entanglement. Similar to Eqs. (5) and (6), the reciprocal
relations of Eqs. (11) and (12) are
Z 2π
1
|η = r e iθ i e iqθ dθ ,
(17)
|q, rii =
2π 0
Z 2π
1
|s, r0 i =
|ξ = r0 e iϕ i e isϕ dϕ .
(18)
2π 0
1
|s, r i =
4π
0
=
=
=
Z
2π
Z
0
1
4π 2
Z
2π
e isϕ dϕ
0
1
2π
Z
1
2π
Z
∞
Z
2π
0
0
e rr /2[ e
r dr
i (ϕ−θ)
2π
∞
2π
Z
0
∞
d 2 η (η∗ ξ−ηξ∗ )/2 X
e
|q, rii e iqθ e isϕ dϕ
π
q=−∞
Z
0
∞
X
− e i (ϕ−θ) ]
|q, rii e iqθ dθ
q=−∞
e irr
0
sin α
e isα dα
0
∞
Z
r dr|s, rii
0
2π
∞
X
Jm (rr0 ) e i(m−s)α dα =
m=−∞
Using Eqs. (12), (17), and (21) we can also prove that the
reciprocal relation of Eq. (23) is
Z ∞
|q, rii =
r0 dr0 |s = q, r0 iJq (rr0 ) .
(24)
0
0
Z
0
r dr|q = s, rii
0
Before we establish a relation corresponding to Eq. (7) we
must construct a link between |ηi and |ξi. Note that both
|ηi and |ξi are complete and orthonormal,[3,4,6]
Z 2
d η
|ηihη| = 1 ,
hη|η 0 i = πδ(η − η 0 )δ(η ∗ − η 0∗ ) ,
π
Z 2
d ξ
|ξihξ| = 1 ,
hξ|ξ 0 i = πδ(ξ − ξ 0 )δ(ξ ∗ − ξ 0∗ ) , (19)
π
so they are qualified to be two mutual conjugate quantum mechanical representations. By “mutual conjugate”
we mean that |ηi and |ξi belong to the eigenstates of
(P1 + P2 , X1 − X2 ) and (P1 − P2 , X1 + X2 ) respectively, which includes the canonical commutative relation
[(X1 − X2 ), (P1 − P2 )] = 2i [(X1 + X2 ), (P1 + P2 )] = 2i,
moreover, the overlap between hη| and |ξi reveals a Fourier
transform
h1
i
1
(20)
hη|ξi = exp (η ∗ ξ − ηξ ∗ ) ,
2
2
∗
∗
since (η ξ − ηξ ) is a pure imaginary. It then follows
Z 2
Z 2
∗
∗
1
d ξ
d ξ
|ξihξ|ηi =
|ξi e (ηξ −η ξ)/2 ,
(21)
|ηi =
π
2
π
Z 2
Z 2
∗
∗
d η
1
d η
|ξi =
|ηihη|ξi =
|ηi e (η ξ−ηξ )/2 .
(22)
π
2
π
Now we construct a Fourier–Hankel transform in the context of quantum mechanical representations which is similar to Eq. (7), substituting Eq. (22) into Eq. (18) and then
using Eq. (11) we have
∗
∗
1
d2η
|ηi e (η ξ−ηξ )/2 e isϕ dϕ =
π
4π
Z
Vol. 41
Since |s, r i and |q, rii, as two quantum vector states, can
play the role in performing Hankel transform, we hope to
find a concrete physical system for which some basic quantum mechanical representations can embody this kind of
transforms.
We consider an electron moving in a uniform magnetic field (UMF). The quantum motion of an electron in
a UMF was first investigated by Landau.[7] The electron
(with mass m) in the presence of UMF is described by the
Hamiltonian (in the units of h̄ = c = 1, where h̄ is the
∞
Z
r dr|q = s, riiJs (rr0 ) .
(23)
0
Planck constant, and c is the light velocity.)
1
1
H=
(Π2x + Π2y ) = Π+ Π− +
Ω,
2m
2
1
Π± = √
(Πy ∓ iΠx ) ,
2mΩ
(25)
where the kinetic momenta Πj = pj + eAj , (j = x, y),
and the vector potential is A = (−By/2, Bx/2, 0) in the
symmetric gauge, B is the intensity of the uniform magnetic field along the z-axis, Ω = eB/m is the cyclotron
frequency, and the kinetic momenta satisfy
[Πx , Πy ] = −imΩ ,
[Π− , Π+ ] = 1 .
(26)
Johnson and Lippmann[8] found that the guiding center
coordinates (x0 , y0 ) of electron’s orbit cannot be deter-
No. 3
Quantum Mechanical Fourier–Hankel Representation Transform for an Electron Moving in a Uniform Magnetic Field
mined simultaneously, since
i
[x0 , y0 ] =
.
(27)
mΩ
The coordinates of electron, due to the presence of UMF,
are
Πx
Πy
,
y = y0 −
.
(28)
x = x0 +
mΩ
mΩ
Based on the guiding centers and kinetic momenta we
have constructed the complete and orthonormal coordinate representation[9−12]
|ξiB = e −|ξ|
2
ξ = ξ1 + iξ2 ,
/2+ξΠ+ +ξ ∗ K+ −Π+ K+
|00iB ,
(29)
where the subscript B means the existence of magnetic
field, and the ladder operators are
r
mΩ
K± =
(x0 ∓ iy0 ) , [K− , K+ ] = 1 .
(30)
2
The vacuum state is annihilated by Π− |00iB = 0,
K− |00iB = 0. Although the commutators in Eqs. (26) and
(27) are not equal to zero, equation (28) leads to [x, y] = 0,
which indicates that x and y possess common eigenstates.
Comparing the common eigenvectors |ξi of X1 + X2 and
P1 − P2 in Eq. (14) with |ξiB in Eq. (29) and based
on EPR’s original argument,
we can√ make the follow√
ing correspondence:
mΩ
x
→
X1 , mΩ y0 → P1 , and
0
√
√
Πy / mΩ → X2√
, Πx / mΩ → P2 , which
√ indicates the
correspondences mΩ → X1 +X2 , and mΩ y → P1 −P2 .
367
This observation clearly reveals the close resemblance of
|ξiB to the EPR states |ξi. Thus we conclude that |ξiB
is also an entangled state in respect of the guiding-center
coordinates and the kinetic momenta. Note that the electron and the magnetic field are two entities, and it is the
presence of the UMF that induces such an entanglement.
Using Eqs. (26), (27), and (29) we see
(Π− + K+ )|ξiB = ξ|ξiB ,
(Π+ + K− )|ξiB = ξ ∗ |ξiB .(31)
It then follows from Eq. (28) that |ξiB is really the common eigenvector of the electrons coordinates (x, y),
r
r
2
2
x|ξiB =
ξ1 |ξiB , y|ξiB = −
ξ2 |ξiB . (32)
mΩ
mΩ
|ξiB can conveniently describe the orbit track of the electron. Further, the electron’s canonical momentum operators are
1 Πx 1
px = Πx + eB y0 −
= (Πx + mΩy0 ) ,
(33)
2
mΩ
2
Πy
1
1
(34)
= (Πy − mΩx0 ) .
py = Πy − eB x0 +
2
mΩ
2
They possess the common eigenvector
|ηiB = e −|η|
2
/2+ηΠ+ −η ∗ K+ +Π+ K+
η = η1 + iη2 .
|00iB ,
(35)
Indeed, from
(Π− −K+ )|ηiB = η|ηiB ,
(Π+ −K− )|ηiB = η ∗ |ηiB , (36)
we see
r
r 2 −1
mΩ
(Π− − Π+ + K− − K+ )|ηiB =
η2 |ηiB ,
(37)
px |ηiB = 2
mΩ
2
r
r
2 −1
mΩ
py |ηiB = 2
(Π− + Π+ − K− − K+ )|ηiB =
η1 |ηiB .
(38)
mΩ
2
Thus |ηiB corresponds to Eq. (13). |ηiB and |ξiB are mutual conjugate, which are the eigenstates of electron’s orbit
position and momentum, respectively. Similar to Eqs. (17) and (18), we can construct
Z 2π
1
|l, |η|iB =
|η = |η| e iθ iB e ilθ dθ ,
(39)
2π 0
Z 2π
1
|ξ = |ξ| e iϕ iB e ilϕ dϕ ,
(40)
|l, |ξ|iB =
2π 0
where l is an integer. Now we try to endow them with physical meaning. Note that the electron’s angular momentum
operator is
i
mΩ h Πx Πy i mΩ h 1
2
2
2
2
Lz = xpy − ypx =
−y y0 +
− x x0 −
=
(Π
+
Π
)
−
(x
+
y
)
= Π+ Π− − K+ K− , (41)
y
0
0
2
mΩ
mΩ
2 (mΩ)2 x
Operating Lz on the state |ξiB and |ηiB respectively yields
2
∗
∂
Lz |ξiB = (Π+ ξ − K+ ξ ∗ ) e −|ξ| /2+ξΠ+ +ξ K+ −Π+ K+ |00iB = −i
|ξi ,
(42)
∂ϕ B
2
∗
∂
Lz |ηiB = |η|(Π+ e iθ − K+ e iθ ) e −|η| /2+ηΠ+ −η K+ +Π+ K+ |00iB = −i |ηiB .
(43)
∂θ
It then follows that
Z 2π
i
dϕ h
∂ Lz |l, |ξ|iB =
−i
|ξ = r0 e iϕ iB e ilϕ = l|l, |ξ|iB .
(44)
2π
∂ϕ
0
368
FAN Hong-Yi
Hence l is electron’s angular momentum quantum number.
On the other hand, from
r
mΩ
(x + iy) ,
K− + Π+ =
2
r
mΩ
K+ + Π− =
(x − iy) ,
(45)
2
we see
2
x2 + y 2 =
(K− + Π+ )(K+ + Π− ) ≡ R2 ,
(46)
mΩ
and
2
R2 |l, |ξ|iB =
|ξ|2 |l, |ξ|iB .
(47)
mΩ
Therefore, |l, |ξ|iB is the common eigenvector of Lz and
R2 (the square of orbit’s radius). On the other hand, from
Eq. (43) we can guess that |l, |η|iiB is the common eigenvector of Lz and p2x + p2y . By analogy to Eqs. (23) and
(24) we have
Z ∞
|l, |η|iB =
r0 dr0 |l, |ξ| = r0 iB Jq (rr0 ) ,
Z
|l, |ξ|iB =
0
∞
0
r dr|l, |η = r|iB Js (rr0 ) ,
(48)
Vol. 41
which is the realization of the Hankel transform. The reciprocal relation to Eqs. (39) and (40) is
∞
X
|ξiB =
|l, |ξ|iB e iϕl ,
l=−∞
|ηiB =
∞
X
|l, |η|iB e iθl .
If we define a distance operator ρ measured from electron’s
coordinate to the guiding center
1
(Π2 + Π2y ) = (x − x0 )2 + (y − y0 )2 = ρ2 , (50)
(mΩ)2 x
which is proportional to the electron’s energy in the presence of UMF, then
mΩ 2
Lz =
(ρ − R02 ) ,
(51)
2
where R02 is
R02 = x20 + y02 .
(52)
Thus for the fixed energy (fixed ρ2 ), a measurement of the
angular momentum completely determines the outcome of
a measurement of R02 . Although
i 2 2
1
2
2
2
(Π
+
Π
),
(K
+
Π
)(K
+
Π
)
=
(Π+ K+ − Π− K− ) ,
−
+
+
−
x
y
(mΩ)2
mΩ
mΩ
i 2 2
h
2
(K− + Π+ )(K+ + Π− ) =
(Π+ K+ − Π− K− ) ,
[R02 , R2 ] = x20 + y02 ,
mΩ
mΩ
[ρ2 , R2 ] =
h
we do have
[Lz , R2 ] =
h mΩ
2
(49)
l=−∞
i
(ρ2 − R02 ), R2 = 0 ,
(53)
(54)
(55)
which indicates that for a fixed R2 although ρ2 and R02 cannot be simultaneously measured, their difference ρ2 − R02
can have a definite value. This is a sort of angular momentum-radius entanglement, a novel fact unnoticed before.
In summary, by virtue of the EPR entangled states we have recast the Hankel transform into the formalism of two
representations’ mutual transform in quantum mechanics. We found that for an electron moving in a uniform magnetic
field, the electron’s coordinate eigenstate |ξiB and the momentum eigenstate |ηiB as well as their Fourier decomposition
states can embody such a kind of Fourier–Hankel transform. The physical meaning of the Fourier decomposition states
are revealed.
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