Evolution of piecewise polynomial wave functions

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by  Mark Andrews

The evolution of piecewise polynomial wave functions
Mark Andrews∗
Department of Quantum Science, Australian National University, ACT2601, Australia
(Dated: January 11, 2017)
For a non-relativistic particle, we consider the evolution of wave functions that
consist of polynomial segments, usually joined smoothly together. These spline wave
functions are compact (that is, they are initially zero outside a finite region), but
theyimmediatelyextendoverallavailablespaceastheyevolve. Thesimplestsplines
7
1 are the square and triangular wave functions in one dimension, but very complicated
0
splines have been used in physics. In general the evolution of such spline wave
2
functions can be expressed in terms of antiderivatives of the propagator; in the case
n
a of a free particle or an oscillator, all the evolutions are expressed exactly in terms
J
of Fresnel integrals. Some extensions of these methods to two and three dimensions

0
are discussed.
1

]
h
p
- I. INTRODUCTION
t
n
a
Functions consisting of polynomial segments, usually joined smoothly at their junctions,
u
q will be referred to here as splines. Such splines have been used in a wide variety of contexts:
[
in approximation and interpolation of functions, and in graphical rendering. In physics

1 splines have been widely used to find approximate energy eigenfunctions. [For a survey, see
v
Ref.1.] Here we consider how spline wave functions evolve in time for a free particle, and
8
0 the results extend easily to a harmonic oscillator. In both cases, the exact evolved wave
5
function can be expressed in terms of Fresnel integrals. Simple examples of such spline wave
2
0 functions in one dimension are shown in Fig.1. We also consider the evolution of splines in
. 2D and in 3D. In principle, the method can be applied to evolution under other forces, but
1
0 none are considered here except centrifugal forces.
7
Normalisable spline wave functions must be zero outside a finite region, but they will im-
1
: mediately extend over all space as they freely evolve. Strictly such wave functions represent
v
a particle initially confined to a finite region and then released from the confining forces,
i
X
but in practice they could be applied usefully to a wider range of situations.
r
a There are very few wave functions with compact support for which the exact evolution is
known, either for a free particle or for a harmonic oscillator. One known case is the square
wave function[2], where the evolution has been expressed in terms of error functions with
complex argument (which can also be expressed in terms of real Fresnel integrals), using the
integral over the propagator; but this wave function has infinite mean energy. The simple
procedure discussed here yields the evolution not only of the square wave function but also
a wide class of more physically realistic wave functions. Since splines are used to derive
approximations to wave functions that arise in a variety of physical situations, the method
used here could enable practical procedures to calculate the subsequent evolution when the
forces change.
∗ [email protected]
2
II. SPLINES IN ONE DIMENSION
We take a spline of degree zero to be a discontinuous function consisting of constant
segments; for example, a square wave. A linear spline (degree one) is continuous and each
segment is a straight line; for example, a triangular wave. A second-degree or quadratic
spline is differentiable and made of quadratic segments. Some simple examples of splines of
degree 0, 1 and 2 are shown in Fig.1.
s = 0
s = 1
s = 2
FIG. 1. Some example of simple spline wave functions of degrees 0, 1 and 2.
Thus a 1D spline consists of n−1 polynomial segments meeting at the junction points
x = a with i = 1,2,...,n and a < a < ... < a . Except for splines of degree zero, a spline
i 1 2 n
φ(x) is continuous at the junction points and we write φ ≡ φ(a ). The simplest class of
i i
normalisable splines has the same degree of differentiability at all junctions, including the
end points, and we will refer to these as pure splines. Only the sth derivative of φ(x) can
jump at any junction, including the end points. Thus all derivatives other than the sth must
be zero in the first and last segments. In the first segment, φ(x) = (x−a )sφ /(a −a )s
1 2 2 1
for a < x < a . A regular spline is pure except at the end points.
1 2
Constructing a spline through the set φ . Using the notation φ(j)(x) ≡djφ(x)/dxj, we
j
require that φ(s−1)(x) is continuous at all internal points. Then we can construct the spline
from the φ and the set of derivatives at one end (or any other point) using the Taylor
i
expansions:
(cid:88)s
φ = φ + di φ(i)/i! (1)
j+1 j j+1 j
i=1
(cid:88)s−k
φ(k) = φ(k) + di φ(k+i)/i!, (2)
j+1 j j+1 j
i=1
where d = a −a . If we know φ and φ and all φ(k) except φ(s), then we can calculate
j+1 j+1 j j j+1 j j
φ(s) and all φ(k) except φ(s) . Thus we can go through all segments to the end point.
j j+1 j+1
For example, in the case of a quadratic spline, Eqns. (1) and (2) with s = 2 give
φ(cid:48) = 2d−1 (φ −φ )−φ(cid:48) (3)
j+1 j+1 j+1 j j
φ(cid:48)(cid:48) = d−1 (φ(cid:48) −φ(cid:48)). (4)
j+1 j+1 j+1 j
[Note that φ(cid:48)(cid:48)(x) is constant within each segment, and φ(cid:48)(cid:48) is taken to be φ(cid:48)(cid:48)(x) in the segment
j
with a < x < a .] The set φ and one particular φ(cid:48) determine the spline. If the spline
j−1 j i k
is to represent a function with a distinct maximum, one might choose to force φ(cid:48) to be zero
there and this would determine the spline (which may or may not be pure).
3
Evolvable form of a pure spline. We now show how any pure spline can be expressed in
a form that enables its evolution its evolution in terms of antiderivatives of the propagator.
The derivative of a pure spline of degree j is a pure spline of degree j−1 and therefore the
(s+1)th derivative of a pure spline of degree s is a sum of δ-functions at the junction points
x=a , including the end points. Thus
i
n
(cid:88)
φ(s+1)(x) = h δ(x−a ), (5)
i i
i=1
where the h may be complex. The functions
i
f (x) = |x|xj−1/2j! (6)
j
are successive antiderivatives of δ(x), with j = 0,1,2,...; these functions have f(cid:48)(x)=f (x)
j j−1
and f(cid:48)(x)=δ(x). Integrating Eq.(5) gives φ(s)(x) = (cid:80)n h f (x−a )+c where the c is a
0 i=1 i 0 i 0 0
constant of integration. Then for x > a we have φ(s)(x) = 1 (cid:80)n h +c , and for x < a we
n 2 i=1 i 0 1
have φ(s)(x) = −1 (cid:80)n h +c . Since φ(s)(x) must be zero in both cases, we require c = 0
and (cid:80)n h = 0.2 i=1 i 0 0
i=1 i
Now integrating φ(s)(x) = (cid:80)n h f (x−a ) gives φ(s−1)(x) = (cid:80)n h f (x−a ) + c .
i=1 i 0 i i=1 i 1 i 1
Then for x > a , φ(s−1)(x) = 1 (cid:80)n h (x − a ) + c = −1 (cid:80)n h a + c , and for x<a ,
n 2 i=1 i i 1 2 i=1 i i 1 1
φ(s−1)(x) = −1 (cid:80)n h (x−a )+c = 1 (cid:80)n h a +c . Since φ(s−1)(x) must be zero in both
cases, we requ2ire ci=1=i0 and (cid:80)i n 1h a2= 0i.=1 i i 1
1 i=1 i i
This procedure can be continued and the result is that any normalisable pure spline of
degree s can be expressed in the form
(cid:88)n
φ(x) = h f (x−a ), (7)
i s i
i=1
where h is the jump in φ(s)(x) at x = a , i.e. h = φ(s)(a +(cid:15))−φ(s)(a −(cid:15)), and the h will
i i i i i i
satisfy
(cid:88)n
h aj = 0 for j = 0,1,2,...,s. (8)
i i
i=1
Thus, given the set of φ for a pure spline, we can calculate all φ(k) from Eq.(2). Then
j j
h = φ(s) −φ(s) and the evolvable form of the spline is given by Eq.(7).
j j+1 j
Splines can be used in different ways and the method of construction will vary. There
may be more efficient methods than using Eqs. (1) and (2). For example,
(cid:88)j−1 (cid:88)n
φ = h f (a −a )− h f (a −a )
j i s j i i s j i
i=1 i=j+1
and therefore, using Eq.(8),
(cid:88)j−1
φ = 2 h f (a −a ). (9)
j i s j i
i=1
This equation provides an alternative way to successively calculate the h from the set of φ .
j i
A pure spline can be defined by its degree s, the set of junction coordinates a =
(a ,a ,..a ) and the set h = (h ,h ,..h ), such that h gives the change in φ(s) at x = a .
1 2 n 1 2 n j j j
Given a pure spline, all the splines with the same a,h but lower s (the derivatives of the
given spline) will also be pure, but those with with greater s may not be pure.
4
III. EVOLUTION OF PURE 1D SPLINES
The evolution of an initial wave function φ(x) is given by the propagator equation,
(cid:90) ∞
ψ(x,t) = K(x,x(cid:48),t)φ(x(cid:48))dx(cid:48). (10)
−∞
For a spline, integrating this equation s times by parts,
(cid:90) ∞
ψ(x,t) = (−1)s K (x,x(cid:48),t)φ(s)(x(cid:48))dx(cid:48), (11)
s
−∞
where K (x,x(cid:48),t) = ∂ K (x,x(cid:48),t); that is, K (x,x(cid:48),t) is an antiderivative (or indefinite
j x(cid:48) j+1 j+1
integral) of K (x,x(cid:48),t) with respect to x(cid:48), and K (x,x(cid:48),t) is an antiderivative of K(x,x(cid:48),t).
j 0
Then, since φ(s+1)(x) = (cid:80)n h δ(x−a ), and deleting any constants of integration as in
i=1 i i
Eq.(7),
(cid:88)n
ψ(x,t) = (−1)s+1 h K (x,a ,t). (12)
i s+1 i
i=1
IV. FREE EVOLUTION OF 1D SPLINES
The free propagator[3] is K(x,x(cid:48),t)=χ(x−x(cid:48),t), where
(cid:114)
m ımx2
χ(x,t) := exp[ ]. (13)
2πı(cid:126)t 2(cid:126)t
The integral over χ(x,t) can be expressed in terms of the Fresnel integrals[4] C(z) and S(z)
defined by
(cid:90) z 1
E(z) := C(z)+ıS(z) = exp( ıπu2)du. (14)
2
0
Each of C(z) and S(z) is antisymmetric, real if z is real, and approaches 1 as z → ∞,
2
(cid:112)
as shown in Fig.2. Hence E(z) ∼ ı/2 as z → ∞. In terms of the error function (with
(cid:112) (cid:112)
complex argument), E(z) = ı/2 erf( π/2ız).
�(�)
0.5
�(�)
�→
1 2 3 4 5
FIG. 2. The Fresnel integrals C(z) and S(z).
From Eq.(14), it follows that
(cid:90) x 1 (cid:112)
χ (x,t) := χ(x(cid:48),t)dx(cid:48) = √ E( m/π(cid:126)t x) (15)
0
2ı
0
5
satisfies Schro¨dinger’s equation for free evolution
∂ (cid:126)2 ∂2
ı(cid:126) ψ(x,t) = − ψ(x,t). (16)
∂t 2m∂x2
From the asymptotic behaviour and antisymmetry of the Fresnel integrals, χ (x,t) ∼
0
1|x|/x = f (x) as t → 0.
2 0
These results can be easily understood qualitatively: as t → 0, K(x,x(cid:48),t) ∼ δ(x − x(cid:48))
and then the indefinite integral of the δ-function gives the antisymmetric step function and
χ (x,t) is its free evolution.
0
Example: The square wave function, ψ (x,0) = 1 for |x| < a and zero for |x| > a, is
sq
the simplest example of a spline wave function of degree zero. Its derivative is ∂ ψ (x,0) =
x sq
δ(x+a)−δ(x−a), so that h = 1 and h = −1. Therefore its free evolution is
1 2
ψ (x,t) := χ (x+a,t)−χ (x−a,t) (17)
sq 0 0
as shown in Fig.(3). This result has been found by direct integration over the propagator.[2]
The asymptotic form of free evolutionast → ∞isdeterminedbytheFouriertransform
of the initial wave function.[5] Thus
(cid:112)
ψ(x,t) ∼ m/ı(cid:126)t exp(ımX2/2(cid:126)t)Φ(mX/(cid:126)t), (18)
where X := x−(cid:104)xˆ(cid:105) and Φ(k) is the Fourier transform
1 (cid:90) ∞
Φ(k) = √ exp(−ıkx)ψ(x,0)dx. (19)
2π
−∞
Usually, ψ(x,t) will be well approximated by this asymptotic form for all times beyond
about ma2/(cid:126), where a gives the scale of the initial wave function.
The square wave function with ψ(x,0) = 1 for |x| < a, has the Fourier transform
(cid:112)
2/π sin(ka)/k. Therefore
(cid:114)
2 ıx2 sin(x/aσ)
(cid:0) (cid:1)
ψ(x,t) ∼ exp , (20)
ıπσ 2a2σ x/aσ
where σ = (cid:126)t/ma2, also shown in Fig.(3).
Free evolution for any degree. The function
ı 1
E (z) := zE(z)+ exp( ıπz2), (21)
1
π 2
(cid:112)
is an antiderivative of E(z) with E (z) ∼ ı/2|z| as z→±∞. If the sequence E (z) is
1 n
defined by
1 ı
(cid:2) (cid:3)
E (z) = zE (z)+ E (z) (22)
n n−1 n−2
n π
for n > 1 with E (z) ≡ E(z), then it can be seen that E(cid:48)(z) = E (z). From E(z) ∼
0 n n−1
(cid:112) (cid:112)
ı/2|z|/z as |z| → ±∞, we have E (z) ∼ ı/2|z|zn−1/n!. Furthermore Eq.(22) ensures
n
that the wave function
1 π(cid:126)t (cid:114) m
χ (x,t) := √ ( )n/2E ( x) (23)
n 2ı m n π(cid:126)t
satisfies Schro¨dinger’s Eq.(16) and χ (x,t) ∼ f (x) = 1|x|xn−1/n! as t→0.
n n 2
Thus for any initial wave function that can be expressed as a linear combination of the
f (x) we have the exact evolution in terms of χ (x,t).
n n
6
������������������
�=�
�=���
|ψ|
�=���
�=�
�=��� ξ/�->
1 2 3
FIG. 3. The evolution of the square wave function: |ψ| is shown. The lower solid line is for the
initial time. The next three dashed curves are for t = 0.1, t = 0.2 and t = 0.5, where the time is in
units of ma2/(cid:126); but each curve is raised for separation. The upper two curves overlap considerably
and show the evolution at time t = 1 (dotted) and the asymptotic form for that time (solid) as
calculated from the Fourier transform of the initial wave function. To allow for the spreading
of the wave functions, each curve is shown with a different length scale: |ψ| is plotted against
ξ = x(1+t2/τ2)1/2 with τ = 1/3. After time t=1 the wave function becomes indistinguishable
from the asymptotic form.
A. Linear splines
From Eq.(23) the evolution of f (x) is
1
1 (cid:114) m (cid:114) (cid:126)t ımx2
χ (x,t):=√ [xE( x)+ı exp( )]. (24)
1 2ı π(cid:126)t πm 2(cid:126)t
Alinearsplinethatiszeroatthetwoendpointsispureanditsfreeevolutionis(cid:80)n h χ (x−
i=1 i 1
a ,t). To find the values of h given the set φ = φ(a ), note that Eq.(9) implies that
i j j j
(φ −φ )/d = (cid:80)j h , where d := a −a , and therefore
j+1 j j+1 i=1 i j j j−1
h = d−1φ −(d−1 +d−1 )φ +d−1 φ . (25)
j j j−1 j j+1 j j+1 j+1
For equal spacing d between the points,
h = d−1(φ −2φ +φ ). (26)
j j−1 j j+1
(cid:112)
The Fourier transform, as defined in Eq.(19), of |x| is − 2/πk−2 (as a generalised
function [6]). Hence the Fourier transform of φ(x) = (cid:80)n h f (x−a ) is
i=1 i 1 i
−1 (cid:88)n
Φ(k) = √ h exp(ıa k), (27)
i i
k2 2π i=1
and this can be used to find the asymptotic form of its evolution for large times, using
Eq.(18). Note that Φ(k) is not singular at k = 0 because, for small k,
n n n
−1 (cid:2) 1 (cid:88) ı(cid:88) 1(cid:88) (cid:3)
Φ(k) = √ h + h a − h a2 +O(k),
2π k2 i k i i 2 i i
i=1 i=1 i=1
and the first two terms are zero, from Eq.(8). When there is some spatial symmetry in the
spline, some of the exponential terms may combine to give sines and cosines.
7
The isosceles triangle is the simplest linear spline. If the length of the base of the
triangle is 2a then a = (−a,0,a). If the height is 1, then the derivative of the spline φ(x)
is φ(cid:48)(x) = 1/a for −a < x < 0 and −1/a for 0 < x < a. Therefore h = a−1(1,−2,1) and
(cid:112)
φ(x) = a−1[f (x+a)−2f (x)+f (x−a)] with Fourier transform 2/πk−2(1−coska). The
1 1 1
evolution of this φ(x) is then ψ(x,t) = a−1[χ (x+a,t)−2χ (x,t)+χ (x−a,t)] as shown
1 1 1
in Fig(4). [A related example is the symmetric trapezium with a=(−a−b,−a,a,a+b); if it
has unit height then h = b−1(1,−1,−1,1).]
Triangularwavefunction
t=1
t=0.5
|ψ| t=0
t=0.2
t=0.1 ξ/a->
1 2
FIG. 4. The evolution of |ψ| for the symmetrical triangular wave function with base 2a. The lower
solidlineisfortheinitialtimeandthenextthreedashedcurvesarefort = 0.1, t = 0.2andt = 0.5,
where the time is in units of ma2/(cid:126); but each curve is raised for separation. The upper two curves
overlap considerably and show the evolution at time t = 1.0 (dotted) and the asymptotic form for
that time (solid) as calculated from the Fourier transform of the initial wave function. Each curve
is shown with a different length scale: |ψ| is plotted against ξ = x(1+t2/τ2)1/2 with τ = 1/5.
B. Pure quadratic splines
From Eq.(22) the antiderivative of E (z) is
1
1 ı
E (z) := zE (z)+ E(z) (28)
2 1
2 2π
and a solution of Schro¨dinger’s equation is
1 π(cid:126)t (cid:114) m
χ (x,t) := √ E ( x), (29)
2 2ı m 2 π(cid:126)t
with χ (x,t) → 1x|x| as t → 0. Thus the evolution the quadratic spline φ(x) =
(cid:80)n h2f (x−a ) is4 φ(x,t) = (cid:80)n h χ (x−a ,t) and h can be calculated from Eq.(3).
i=1 i 2 i i=1 i 2 i i
(cid:112)
The Fourier transform of x|x| is −ı 8/πk−3, so that the Fourier transform of φ(x) =
(cid:80)n h f (x−a ) is
i=1 i 2 i
(cid:112) (cid:88)n
Φ(k) = −ık−3 1/2π h exp(ıa k), (30)
i i
i=1
and this can be used to find the asymptotic form of its evolution for large times, using
Eq.(18). Again Φ(k) is not singular at k = 0 because, for small k,
n n n n
−ı (cid:2) 1 (cid:88) ı (cid:88) 1 (cid:88) ı(cid:88) (cid:3)
Φ(k) = √ h + h a − h a2− h a3+O(k) , (31)
2π k3 i k2 i i 2k i i 6 i i
i=1 i=1 i=1 i=1
and the first three terms are zero from Eq.(8).
8
Smooth hump. The simplest pure quadratic spline is a symmetric hump with a =
a(−2,−1,1,2) and
(cid:26)
1−x2/2a2 if |x| < a
ψ(x,0) = (32)
(2a−|x|)2/2a2 if a < |x| < 2a,
with h=a−2(1,−2,2,−1) and evolution ψ(x,t)=a−2 [χ (x+2a,t)−2χ (x+a,t)+2χ (x−
2 2 2
(cid:112)
a,t)−χ (x−2a,t)].TheFouriertransformoftheinitialwavefunctionis 8/πa−2k−3sinka(1−
2
coska).
C. Impure quadratic splines
If a wave function is initially confined by infinite barriers, its derivative must be finite at
the barriers. This requires a multiple of f (x) centred on each boundary:
1
n
(cid:88)
φ(x) = c f (x−a )+ h f (x−a )+c f (x−a ). (33)
1 1 1 i 2 i n 1 n
i=1
For x > a we have f (x−a ) = 1(x−a ) and f (x−a ) = 1(x2 −2a x+a2). Hence for
n 1 i 2 i 2 i 4 i i
φ(x) ≡ 0 outside (a ,a )
1 n
(cid:88)n
h = 0, (34)
i
i=1
(cid:88)n
h a = c +c , (35)
i i 1 n
i=1
(cid:88)n
h a2 = 2(c a +c a ). (36)
i i 1 1 n n
i=1
Differentiating Eq.(33) gives
n
(cid:88)
φ(cid:48)(x)=c f (x−a )+ h f (x−a )+c f (x−a ),
1 0 1 i 1 i n 0 n
i=1
and the only term that changes suddenly as x crosses a is the first term because f changes
1 0
from −1 to 1. Hence φ(cid:48) = c and similarly φ(cid:48) = −c . Thus c is the change in the slope
2 2 1 1 n n 1
of φ at x = a , while h is the change in the slope of φ(cid:48). The free evolution of φ(x) is
1 1
c χ (x−a ,t)+(cid:80)n h χ (x−a ,t)+c χ (x−a ,t).
1 1 1 i=1 i 2 i n 1 n
Given the set φ and the initial slope c , the constants h can be found, as before, from
i 1 i
Eq.(3) starting with φ(cid:48) = c and then c will emerge as −φ(cid:48) .
1 1 n n
The truncated quadratic is a simple example with
φ(x) = (1−x2/a2) for |x| < a (37)
or φ(x)=2a−2[af (x + a) −f (x + a) + f (x − a) + af (x − a)]. Its Fourier transform is
1 2 2 1
(cid:112)
8/π(sinka − kacoska)/a2k3 and its free evolution is 2a−2[aχ (x + a,t) − χ (x + a,t) +
1 2
χ (x−a,t)+aχ (x−a,t)], as shown in Fig.(5).
2 1
9
�����������������������������
�=�
�=�
�=���
|ψ| �=���
�=��� ξ/�->
1 2 3
FIG. 5. The evolution of |ψ| for a truncated quadratic wave function. The lower solid curve is for
the initial time. The dashed curves are for t = 0.1, t = 0.2 and t = 0.5 and t = 1.0, where the
time is in units of ma2/(cid:126); but after the first, each curve is raised. The upper solid curve shows
the asymptotic form, also at time t = 1.0. Each curve is shown with a different length scale, with
ξ = x(1+t2/τ2)1/2 with τ=1/4.
V. SPLINES IN TWO DIMENSIONS
A simple generalisation to two dimensions is given by the product φ (x)φ (y) of two 1D
1 2
splines. For example, if φ (x) is the smooth hump in Eq.(32) and φ (y) is the truncated
1 2
quadratic in Eq.(37), then the product is the biquadratic spline shown in Fig.(6). Its
evolution is the product of the evolutions given for φ (x) and φ (y). Clearly, we need to
1 2
consider more general wave functions.
FIG. 6. The product of a pure quadratic hump φ (x) and an impure truncated quadratic φ (y).
1 2
Bilinear splines generated from a given set of complex values φ(i,j) can be related to a set
of 1D linear splines parallel to the x-axis and another set parallel to the y-axis, intersecting
at all the junction points. The set φ(i,j) completely defines the bilinear spline
(cid:88)
φ(x,y) = h f (x−a )f (y −b ). (38)
ij 1 i 1 j
i,j
As shown in the Appendix , h(i,j) is determined by φ(i,j) and its eight nearest neighbours:
d−2h = +φ −2φ +φ
ij i−1,j+1 i,j+1 i+1,j+1
−2φ +4φ −2φ
i−1,j i,j i+1,j
+φ −2φ +φ . (39)
i−1,j−1 i,j−1 i+1,j−1
10
This can be applied to any junction (even on the boundary). Then the evolution is easily
calculated by replacing f (u) by χ (u,t) in Eq.(38).
1 1
Except for wave functions with rectangular boundaries, there will be artifacts that arise
through approximating the boundary by one that lies on the grid. Also the boundary is
constrained to not allow any segment to have three sides on the boundary, because the only
bilinear function that is zero on three sides is zero throughout.
Example: thebilinearsplinegeneratedbytheaxiallysymmetrictruncatedquadraticφ(r) =
1 − r2/a2 for r < a with grid-spacing a/5 is shown in Fig.7, which also shows the effects
that come from boundary approximation.
FIG.7. Thebilinearsplinewithagridspacingofa/5thatcontainstheaxiallysymmetrictruncated
quadratic φ(r) = 1−r2/a2 for r < a.
In three dimensions it is well known that the free evolution of spherically symmetric wave
functions ψ(r) can be reduced to the free evolution in one dimension of ψ(r)/r. Examples
will be given in Section VI. For two dimensions the equivalent reduction introduces an
additional term, proportional to r−2, in the Hamiltonian and the method used above cannot
be easily applied. This means that circular boundaries are not easily dealt with.
The evolution of a bilinear spline φ(x,y) in the form given in Eq.(38) is then
(cid:88)
ψ(x,y,t) = h χ (x−a ,t)χ (y −b ,t). (40)
ij 1 i 1 j
i,j
As in the derivation of Eq.(27), the Fourier transform of φ(x,y) is
(cid:88)
Φ(k ,k ) = (2π)−1(k k )−2 h exp(ıa k +ıb k ) (41)
x y x y i,j i x j y
i,j
and the asymptotic form of the evolved wave function is
m ım(x2 +y2) mx my
(cid:2) (cid:3)
ψ(x,y,t) ∼ exp Φ( , ), (42)
ı(cid:126)t 2(cid:126)t (cid:126)t (cid:126)t
where we have assumed that the wave function is centered so that (cid:104)xˆ(cid:105) = 0 and (cid:104)yˆ(cid:105) = 0. As
an example the evolution of the bilinear spline in Fig.7 is shown in Fig.8.