The exact expressions for Wigner functions are known only for the simplest quantum systems: one-dimensional harmonic oscillators [1-3] and particles in ...
WIGNER
FUNCTION
FOR A PARTICLE
IN DELTA-POTENTIAL
AND IN A BOX
E. A. Akhundova
I and V. V. Dodonov 2
The exact expressions for Wigner functions are known only for the simplest quantum systems: one-dimensional harmonic oscillators [1-3] and particles in magnetic and electric fields [4-6]. In [7-8] explicit expressions for the Wigner function were obtained for multidimensional systems with the most general quadratic Hamiltonlan, and the results [1-6] have been generalized to more complicated cases. The exact formulas for nonquadratic Hamiltonlans were known only for some individual cases. For example, in [91 the Wigner function was found for the ground state of a Morse oscillator, expressed through the McDonald function with purely imaginary indices, and in [10] the Wigner function was given for the bound state in the delta-potential case (expressed through the elementary functions). In [11-13] the most simple "almost quadratic" system was considered: a quantum particle moving in a homogeneous field with potential U(x) = - F x in the half space: x > 0. For this system the expressions for the Wigner function and equilibrium density matrix were obtained in the quasi-classical approximation. The exact formula is known when F = 0 [1113]. (For an arbitrary time-dependent function F(t), but in the case of motion in the whole space - ~ < x < ~ , the exact Wigner fimction was calculated in [14].) In this work we obtain the exact expressions for Wigner functions for two quantum systems and analyze their asymptotics. Let us consider the one-dimensional movement of a quantum particle o f mass m in the field with the potential of the following form: t.J ( ::-'. )::::'(:, ,, O< .'.-::.( ~
) :=ao,,
i...j ( ::
~,~ ~
• .::i0.
(1)
The wave functions and energies of stationary states of this system are known
(2) E[ = r ~ Z n Z h Z / ( 2 m ~ - ' , .
z)
,
r'm,==l.,2,:';, . . . .
The Wigner function of stationary states is defined in the following form: W , .~(. .,
q .) = j y .~ , ( o. .+. . ~ ,
. ,,t, ( (.~ . . . . . . 2 '
;:~>: p ~".....
h
"~c,"~"
(3)
Substituting the expression for the wave functions (2) into integral (3), we have
":" Is.j_n
Wn(p,,ci~= 9
"
a
rrnr~7 + l
;-":" "
r.;~i.r", ~ '
m a 'q
.....
2
......
""
h
"
(4)
Taking into consideration that the functions ~I'n are nonvanishing only in the interval 0 _< x _< a, we find that ( changes in the two intervals 1Institute of Physics, Baku, Azerbaijan Republic. 2Moscow Physicotechnical Institute, Zhukovsky, Moscow Region, Russia. 312
0270-2010/92/1304-0312512.50 9 1992 Plenum Publishing Corporation
2q _< ~ _< ""r
-
q~,
I .,, e~,:7. ~-,j 7-;~ =~ s rn--I
it.n
rrl
-t? z
Sin
2 . LrrtrI~:~,
(18)
.
m
Using the expression [16] O0
( -..
Y m----t
314
i
J
'i m
cos
m>" =
(:2;,.,z +_ rt2 ) ,
2 fR
.,::.
-n
-< x
_< n
(19)
we obtain the final result 2
O (~.,,,!.--~)
2
~ : : p ( -r~ ~ . / ~ ) ,
13
(20)
3
Therefore, the integral in (15) must be expressed by an error function. Note that one can obtain the formula (15) in another way. First, we find the equilibrium matrix in the coordinate representation p(,.',x"
,~)
=
~
~
(21)
(:.:)~I1"(::.,")e-F~:n n
According to (12) the result is expressed again by theta-functions
.I_
P(:"''"
"
=
'~)
2a
,'.." -I,- .k' '
;.C.- ).C9 ,, ,,~_[9~,..
[ .-
O
a
(
:::?a
)
"
-
e
(
~"
_.[.9~,,,i ]
2~'
e
,
(2 2 )
Then, making the transformation (3) under the restriction (5), we obtain again (15). For the complex values of the parameter/3 = it~h, the function (22) transforms into the propagator of the Schr6dinger equation for a particle in a box which was calculated in several ways, for example, in [17-19]. For the free particle in the half-space x > 0 the equilibrium density matrix has the form (see [11-13]) s
Since the expression must be the limit of (22) when a --, oo, we obtain the following result: -1.
X
].im ,:.~ -- ~ 3 [ a
----"-I/" exp (-rt'
