ОПТИКА И СПЕКТРОСКОПИЯ, 2011, том 111, № 4, с. 700-703
ТОМОГРАФИЯ КВАНТОВЫХ СОСТОЯНИЙ
УДК 535.14
CHARGED PARTICLE COHERENT STATES IN TOMOGRAPHIC PROBABILITY REPRESENTATION
© 2011 V. I. Man'ko***, and E. D. Zhebrak**
*P.N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia **Moscow Institute of Physics and Technology (State University), Moscow, 141700 Russia e-mail: manko@sci.lebedev.ru, el1holstein@phystech.edu
Abstract—The symplectic tomograms of coherent states of a charged particle moving in a constant uniform magnetic field are obtained in explicit form. The tomograms are shown to coincide with normal probability distributions of two random variables. The means and dispersions of the variables are found and expressed in terms of means and dispersions of the charged particle coordinates and momenta. The characteristic function of the tomographic probability distribution is found. The center of mass tomogram of the coherent state of charge in magnetic field is also found and the relation of the symplectic tomogram and the center of mass to-mogram is established.
1. INTRODUCTION
The charged classical particle moving in a constant uniform magnetic field has the trajectory which is the circle of a fixed radius (in the case of the particle having zero velocity along the magnetic field). The radius is determined by the conserved energy of the particle. This cyclotron motion in the quantum domain was shown [1] to be described by the wave function corresponding to the coherent state of the charged particle in the constant uniform magnetic field. The wave function has the form of Gaussian packet. Such packets were found for the charge moving in the magnetic field by Kennard [2].
The coherent states of the charge moving in the magnetic field introduced in [1] were obtained by using a formalism of constructing the coherent state of harmonic oscillator (electromagnetic field oscillators) applied by Glauber [3] (see also [4, 5]).
The coherent states of a quantum charged particle in the magnetic field correspond to classical cyclotron motion in the following sense. The center of the Gaussian packet describing the coherent state moves exactly along the trajectory ofthe classical charged particle. The quantum fluctuations of the coordinates and momenta of the particle in the coherent states are time-independent and minimize the Heisenberg uncertainty relation [6]. The coherent states of the charge moving in the magnetic field were studied and applied with respect to different problems in [7—13].
For example in [7], these coherent states were observed in terms of states on a von Neumann lattice and Wannier functions. In [9], coherent states of SU(l, 1) group were considered. In [10, 11], the coherent states were generalized for relativistic quantumparticle moving in magnetic field. In [12] using solution to a version of the Stieltjes moment problem, a family of coherent states of a charged particle in a uniform magnetic field is constructed. In [13], the coherent states are constructed for a charged particle in a uniform magnetic field based
on coherent states for the circular motion which have recently been introduced in [14]. In the case of the coherent states for a particle on a circle, the uncertainty relations have been introduced in [15].
Recently, new formulation of quantum mechanics was suggested in [16] where the fair probability distributions are used as alternative of wave functions or density matrices. This formulation of quantum mechanics is called tomographic probability representation of quantum mechanics.
The probability distribution called tomogram describing the quantum state instead ofwave function is related to the wave function by invertable fractional Fourier transform and to the Wgner function [17] of quantum state by the integral Radon transform [18] (see [16]). In the probability representation, the state tomograms satisfy the quantum evolution equation equivalent to Schrodinger equation for the wave function or Moyal equation [19] for the Wgner function.
The aim of our article is to describe the quantum cyclotron motion in the framework of the tomographic probability representation of quantum mechanics.
We will construct the quantum tomograms of charged particle coherent states. We will apply two different schemes ofquantum tomography — symplectic tomography [16] and center ofmass tomography [20] and find the corresponding tomogram of the charge coherent state.
The article is organized as follows. In Sec. 2, the construction of coherent state of charged particle moving in the magnetic field is reviewed. In Sec 3, the symplectic and center of mass tomogram are considered following [20]. In Sec. 4, the tomograms of charged particle coherent states are calculated, the means and the dispersion matrices correspond to the tomographic probability distributions and the characteristic function of these distributions are obtained. The conclusions and perspectives are discussedin Sec. 5.
2. COHERENT STATES OF CHARGED PARTICLE MOVING IN CONSTANT MAGNETIC FIELD
We shall consider a particle of mass m = 1 and charge e = 1 moving in a constant magnetic field with a potential
A = 1 [H x r].
2 J
The Hamiltonian for such a system is
(1)
H = 2 [( - Ax )2 + (py - Ay )2
Й = с = e = 1. (2)
By direct calculation, one can verify that the following operators are invariants
A =1 [( + ipy ) + (y - ix)],
(3)
B = + iPx ) +- /y)j. The following commutation relations hold:
[A, a+] = \b, B+] = 1, [A, B] = [A, B+] = 0. (4) For oppositely charged particles, the lowering and
raising operators A, B and A+, B+ change their places. For simplicity we suppose e > 0. The Hamiltonian of the system
1
(5)
H = A+A + -2
is apparently an integral of motion.
Because of the axial symmetry ofthe electromagnetic field potential (1), the z component of the angular momentum is also an integral of motion and it too may be expressed in terms of our operators (3):
L r = B+B - A+A.
(6)
The operators (3) can be expressed in the following form:
A = [y - Уо - i (x - xo )], B = [xq - iyo ], where
(7)
xo = 2 x + Py,
yo=2y
A\а,р) = а|а,ß)> B\а,р) = р|а,ß),
(8)
In this case we have two lowering operators A and B the coherent states will carry two parameters:
а,в) = exp I"-2 (|a|2 + |ß|2) £
nlO n2
a П1в
nhn2=0
% П2
(9)
where a,p are arbitrary constant complex numbers. Here,
I \ A )'("T|0O
\ni,n^ =i, , 0,0, Vn^i
A| 0,0 = B| 0,0 = 0 are solutions of the Schrodinger equation
i dt ~ H) y = 0, and at the same time they are eigen-
states of the Hamiltonian H and the angular momentum L :
(10)
n2)=(«1+2) i % «2),
AK «2) = («2 - «l)| «1, «2). There are two unitary displacement operators
D (a) = exp (aA+ - a*A), D-1 (a) AD (a) = A + a,
V 7. (11)
D (p) = exp (p B+ - P*B), D-1 (P)PD (p) = B + p
which commute with each other:
[D (a), D (P)] = 0.
Coherent states | a, p) may be constructed explicitly by acting upon the vacuum
ice*
Z = x + iy
(12)
I0,0 =
Vn
with operators (11)
|a,p) = D (a) D (p)| 0, 0). (13)
Wave function corresponding to this coherent state vector has the following Gaussian form
-e 2 e 2 . (14)
V (x, у ) = «1 e \n
are the well- known "coordinates" of the center of the orbit of a particle moving in a constant magnetic field.
It is apparent from (6) that the eigenvalue of the invariant B determines the coordinates of the center of the orbit in the xy plane, and the eigenvalue of the operator A determines the current coordinates of the center of the packet.
The operators A, B have the normalized eigenvectors called coherent states | a, p) that obey the Schrodinger equation. The following formulas hold:
3. SYMPLECTIC AND CENTER OF MASS TOMOGRAPHY
According to approach with tomographic probability distributions determining the quantum states, one can construct different kinds of quantum tomogram — sym-plectic and center of mass tomograms. Since we study the motion of charge in constant magnetic field, we fo-cuse on the case of systems with two degrees of freedom. On these cases, the symplectic ws (X1, X2,^1;v2,v 2) and center of mass wcm (X,^1,v2,v2)tomograms are defined using the density operator p of the quantum states as
W ( Xi, X2,Vl,Vi,V2,V 2 ) = Trp5(( -Ц1#1 -ViPi) (( -Ц2^2 -V2_p2),
Wem (X,^1,V1,^2,V2 ) = = Trp5 (( - Ц141 -V1P1 - ц2q2 - v2jp2).
(15)
ОПТИКА И СПЕКТРОСКОПИЯ том 111 № 4 2011
702
MAN'KO, ZHEBRAK
Both tomograms have probability distribution functions depending on extra real parameters ^ 2, Vi, v2. The symplectic tomogram depends on two random variables Xi and X2. The center of mass tomogram depends on one variable X . The tomograms are nonegative and normalized functions, i.e.,
JV (Xi, X2,^i,Vi,^2,v2)dXidX2 = 1, (17)
JWcm (X, ^i,Vi, ^2,V2)X = i. (18)
Also the tomograms have the property ofhomogene-ity, namely
and
\k ik 21
Wcm {kX,k^i,kVi,X^2,^V2) =
= 1 Wcm (X,H-1,V1,H-2,V2)•
|A,|
(19)
(20)
x e
P =~T1 K (Xl> X2,Vl,Vl,V2,V2 4n J
i(Xi + X2-^1<?1-v1jP1-^2^2-V2 ¡2d
(21)
'JdXldX 2d^d 2d v 2.
The center of mass tomogram also provides the density operator
P = \Wcm (xX,^l,Vi,^2,V2)x 4n J
x ei((-^-v^-^-v2¡>2)dxd^ldvld^2dv2.
(22)
Wcm (X,^i,Vi,^2,V2) =
= jws (((i^2,V2)S (( - Y - 72 ws (xi, x22,v2) =
= jwcm (( fcl^l, ^lvl, k2V2, k2V 2)x
x ei(-kivi2-k2v2)dkidkidY.
(23)
(24)
l
4n |vlv2|
Ws (Xb X2,^l,Vi,^2,V2) =
'±lx 2 - 'hx + lEly 2 _ iX2 y
jV (x, y)e 1 Vl 2V2 v2 dxdy
(25)
The center of mass tomogram is expressed in terms of the wave function as
Wcm Kl,Vi,^2,V2) =
l
4n2 Viv 2I
ikX-ik(Yl+Y2)
jV (x, y )e2vi
iui 2 iYi iu2 2 iY2
■f1 x--1 x y--2 y
vi 2v2
dxdy
(26)
dkdY1dY2.
The tomogram provide the posibility to reconstruct the density operator p. The symplectic tomogram yields the density operator in the form
In the case ofpure state p v = | y) (y |, the expressions for the tomograms in terms of wave functions read
In next section, we apply the obtained formulation to the case of quantum charged partic
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