ОПТИКА И СПЕКТРОСКОПИЯ, 2007, том 103, № 2, с. 340-345
ПРОСТРАНСТВЕННО-ВРЕМЕННАЯ ЭВОЛЮЦИЯ ВОЛНОВЫХ
ФУНКЦИЙ, УПРАВЛЕНИЕ КВАНТОВОЙ ДИНАМИКОЙ =
С ПОМОЩЬЮ КОГЕРЕНТНЫХ ПРОЦЕССОВ
УДК 535.14
SHORT TIME PROPAGATION OF A SINGULAR WAVE FUNCTION:
SOME SURPRISING RESULTS
© 2007 r. A. Marchewka*, E. Granot**, and Z. Schuss***
* Kibbutzim College of Education, Ramat-Aviv, 69978 Tel-Aviv, Israel ** Department of Electrical and Electronics Engineering, College of Judea and Samaria, Ariel, Israel *** Department of Applied Mathematics, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv, Israel
Received October 12, 2006
Abstract—The Schrodinger evolution of an initially singular wave function is investigated. First it is shown that a wide range of physical problems can be described by an initially singular wave function. Then it is demonstrated that outside the support of the initial wave function the time evolution is governed to leading order by the values of the wave function and its derivatives at the singular points. Short time universality appears where it depends only on a single parameter - the value at the singular point (not even on its derivatives). It is also demonstrated that the short-time evolution in the presence of an absorptive potential is different than in the presence of a non-absorptive one. Therefore, this dynamics can be harnessed to the determination whether a potential is absorptive or not simply by measuring only the transmitted particles density.
PACS: 03.65.-w, 42.25.Bs
1. INTRODUCTION
Wave functions with singularity in the space variable can be used to describe many physical phenomena, even when the physics is not genuinely singular. A good example for this is the Gamow theory of alpha decay [1] and the irreversible evolution of quantum particles [2]. In the alpha decay model, the initial localization of the particle is represented by a singular confined wave function, and its tunneling through the barrier is manifested by the well-known exponential decay law.
In most cases, where singularity is used, it is chosen as an approximation to the real physical model. However, using a singular wave function as a limit of regular wave functions not only simplifies the real world, but also reveals new physical effects [3, 4]. In many respects this limit is similar to the singularity arising in phase transitions in statistical mechanics [5].
It seems that the first time a singular wave function was recruited for a real physical problem was for what was termed "Moshinsky's Shutter" in 1952 [6]. Half a century later, the legitimacy of using singular wave functions is still in question, though Hilbert space theory incorporates them into quantum mechanics [7]. It is still unclear whether a singular wave function is merely a simplification of the real world, or can it describe real physical entities.
In this paper we investigate the time propagation of singular wave functions, regardless of their origin. We show that singular wave functions have a generic simple short time evolution. Moreover, we present a new effect: an initially singular wave function (and in general, any sharp boundary wave function) propagates in short times differently, when it is transmitted through
an imaginary or a real potential. Therefore, this effect can be used to distinguish between real and imaginary potentials by merely measuring the transmission coefficient (in contrast to the requirement that both transmission and reflection coefficients be measured).
The structure of this paper is as follows: in Sec. 2 we show, by an example, how a smooth wave function can be approximated by a singular one. In Sec. 3 we derive the generic propagation behavior of a singular wave function in general and in short times in particular. In Sec. 4 we develop a novel classification scheme, which distinguishes between real and imaginary potentials. Section 5 includes a discussion and the conclusions. Finally, in the Appendix, we review the theory of Moshinsky's Shutter and its experimental verification.
2. SINGULAR WAVE FUNCTIONS
AS APPROXIMATIONS TO SMOOTH ONES
As mentioned in the Introduction, singular wave functions can be considered as approximations to analytical wave functions. We clarify here the sense in which the singular wave function can emulate a smooth function with a large gradient. A smooth wave function that approximates a discontinuous one introduces a transition scale B, which characterizes the transition width. This length scale determines a characteristic velocity v ~ B-1 (hereinafter we use the units h = 1, m = 1/2). Therefore, the approximation fails, when the velocity of the particle x/t exceeds the characteristic velocity, i.e., for x that
SHORT TIME PROPAGATION OF A SINGULAR WAVE FUNCTION
341
exceeds xmax ~ t/^. As a specific example, we choose where 9(x) is an arbitrary analytic function. For the
special case of a semi-infinite plane wave 9(x) = e,kx the general solution is [9]
the wave function
Vi (x, t = 0) = If 1-th|
as an approximation to the discontinuous wave function
y2(x, t = 0) = 0(-x)
in the sense that lim y 1 (x, 0) = y2(x, 0). To determine
^ o
the propagation of the two wavefunctions we use the Fourier transform [8] to write
V(x, t)
= f -
m
r I
ikx
e exp
Jim(x-y)2 1 2ht
V.2 n i h tj J " I
= M( x, t, k), where M(x, t, k0) is the Moshinsky function
x - k
M(x, t, k) = 2 expfikx - 1 ik2t ]erfc
Vi(x, t > 0) =
(2it)
1/2
= I dk
and
V2 (x, t > 0) = I dk
2 8( k) + i 4-csch (rck^ /2)
2 8( k) + 20
exp(ikx - ik t)
exp(ikx - ik t).
By expanding csch (n k ^ /2) in a power series, we obtain
5 = y 1( x, t > 0) - y2( x, t > 0)~
—-T- i (ik^2 + Ofc 4k3)) dk exp (ikx - ik2t), 48 J
so that
The propagation of this initial wave function was calculated by several method [10, 11], of which we choose that of [11]. The general solution for the initial wave plane is
V(x, t > 0) = I g(k) exp (ikx - ik21)dk,
where g(k) = (2n) 1J dx^ (x)exp(-ikx) is the Fourier
transform of 9(x). Using the Moshinsky function, we get
V(x > 0, t > 0) = 21dkg(k) exp (ikx - ik21)erfc
(x-2kt)
L 2(it)
1/2
^ 2d- I dk exp (ikx - ik21).
Rearranging the integral and setting [12] w(z) = = exp(-z2)erfc(-iz), where
z =
2 tk - x
This value is negligible relative to y1, 2 only when ^x/t <§ 1 (note that the integral goes like exp(ix 2/4t) and • every spatial derivative is equivalent to a multiplication g by the factor ix/2t). Thus, we obtain again that in the regime x < xmax ~ t/£, (that is, at short distances or long periods) y1 is a very good approximation to y2. Of course, this is not surprising, because a small change (in
the L2 norm) in the initial function in the Schrodinger Using the expansion equation makes a small change in the solution. The approximation error is determined by the parameter
2 (-it)
1/2
V(x, t > 0) =
_ exp (ix /41)
w
2 tk - x
L2 (-it)
1/2
g(k)dk.
w (z) = £
, . n
(iz)
r( n /2+1)'
3. GENERIC SHORT TIME PROPAGATION OF A SINGULARITY
3.1. The Free Particle Case
I. The Generic Propagation of a Jump Discontinuity
Consider the initial singular wave function 9(x, t = 0) = 9(x)©(-x),
we get
V(x, t > 0) =
_ exp (ix /41)
XI
n =
n
(iz)
r( n/2+ 1)
g (k) ekydk\y = 0.
n=0
Each term in the expansion should be understood as an in this paper we always assume that the support of the
operator; for example, the k-th term looks like
wave function is finite or semi finite.
lim f
^ о J
(2tk - x)
k/2.
y ^0 J (i41) Г(k/2+ 1 )
- g ( k ) e'kydk =
= lim-
2t 1- x)k
. k/2
y ^ 0( i41 fT( k/2+1 )
2t- x ¿У
( i41 )k/2 Г( k/2+1 )
f g ( k ) e'kydk =
-v( о, У )
y =0-
Thus, the general solution for any semi-infinite wave function is
x, t > 0 ) =
1exp (i 4-)w ( ~'2™$-x )¥( y,0 y=0.
(1)
3.2. Short Time Propagation
Short time propagation corresponds to the large z expansion of w(z), that is, to (h/2m)(t/x2) < 1. Therefore, using [12], the leading term approximation of Eq. (3) is
V(x, t « x2) - JJexp№-b)2/4t-V(b, 0).
(4)
exp ( i ( x - a ) /41 )
x - a
V(a, 0) к
Equation (4) corresponds to an initial wave function with discontinuities at the points a, b. The probability density is
II. A Discontinuous Wave Function with Compact Support
We write a wave function with compact support as
¥x[a, b] = Vx[-~, b] " Vx[-~, a], (2)
where the indicator function %[a, b] is defined by
Vx[a, b]
| у for a < x < b, 0 otherwise.
IV(x, t)|2 ~ П-
n
-2 <Ж
exp
i\ x - ■
( x - b У a + b)a - b
(x - b)(x - a)
( x - a )
-v(b, 0)V* (a, 0)
(5)
Since (1) is invariant to translations, the propagated We ree that; Nereis interference tet^n the t^ ^g^
wave function is, by (2),
x, t > 0 ) =
y(b, 0)y*(a, 0). There is a new type of interference in
a + b)a - b~
time, given by exp
i\ x -
2t
e'(x-b) /4t (-i2td/9y - (x - b)) ( n)|
-22-w ( 2(V |¥( y'0 )l y = b-
(3)
i (x - a) /4t
- w
- i 2t д/dy - ( x - a )
2 (-it)
1/2
If the initial wave function is continuous, but has discontinuous derivatives at the edges, the leading term in the expansion (3) of the probability density gives
v( y, 0 )|
y = a'
IV(x, t)|2 - 4П-
n
lyXbiMÎ , (a, 0)|2
-2 Ш
exp
л x - ■
( x - b ) a + b)a - b
( x - b )2 ( x - a )2
( x - a )
b, 0)y*'(a, 0)
(6)
This expression reveals three main features:
a. The propagation of the wave function depends only on the value of the wave function and its derivatives at the points of discontinuity (or, in general, singularity).
b. There is an interference term that depends on both discontinuities.
c. There is temporal interference that depends on the size of the initial wave function.
As a matter of fact, the representation (3) is also ap- where the tags denote spatial derivatives, i.e., y'(c, 0) = propriate for initial wave functions supported on the en- = Эу(х, 0)/Эх X = c. We see the same type of interference tire line, but have a point of non analyticity. However, in this case as well.
ОПТИКА И СПЕКТРОСКОПИЯ том 103 < 2 2007
SHORT TIME PROPAGATION OF A SING
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