The propagator for the step potential using the path decomposition expansion

0.12 MB·English

by  James M. Yearsley

The propagator for the step potential using the path
decomposition expansion
9 James M Yearsley
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0 Blackett Laboratory, Imperial College, London SW7 2BZ, UK
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E-mail: [email protected]

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a
Abstract. We present a direct path integral derivation of the propagator in the presence of
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a step potential. The derivation makes use of the Path Decomposition Expansion (PDX), and
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also of the definition of the propagator as a limit of lattice paths.
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p
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1. Introduction
t
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The propagator for the step potential, V(x) = V Θ(−x), is a simple model for scattering, and a
a 0
u useful ingredient in arrival time problems [1]. Expressions for this propagator are known [2], but
q existingderivationsarecomplex,andgivelittleinsightintothestructure. Hereweshowhowthis
[
propagator may be obtained in a natural way via the Path Decomposition Expansion (PDX), in

1 which the propagator is factored across a surface of constant time [3] and the Brownian motion
v definition of the path integral. 1
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This propagator may be viewed as a conditional probability density for a random walk on
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4 the real line. We then make this integral over paths into a concrete object by defining it as
3 the continuum limit of a discrete sum of lattice paths, u(0,T|0,0). We consider a rectangular
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1 lattice with spacing in the time direction of (cid:15), and spacing in the x direction of η, and consider
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propagationforatimeT = 2(cid:15)n, sowehave2nstepsinourpaths. TheEuclideanpropagatorg is
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then defined as the continuum limit of u/2η where we take (cid:15),η → 0, n → ∞, keeping (cid:15)/η2 = m
0
: and T = 2(cid:15)n fixed. That is,
v
i
X g(0,T|0,0) := lim (2η)−1u(0,T|0,0). (1)
r η,(cid:15)→0
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2. The Path Decomposition Expansion (PDX)
The PDX is a very useful tool for evaluating discontinuous potentials, such as the step potential.
A typical path from x < 0 to x may cross x = 0 many times, but the set of paths may be
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partitioned according to their first and last crossing time. We therefore split every path into
three parts: (A) a restricted part that starts at x and does not cross x = 0, but that ends on
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x = 0 at time t , (B) an unrestricted part from x = 0 to x = 0 that may cross x = 0 many times
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and, (C) a further restricted part from x = 0 to x that does not re-cross x = 0, see Fig. 1. The
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1 This contribution is based on [4].
t
X
t=T 1
C
t
2
B
t
1
A
t=0 X
X
0
Figure 1. A typical path from x to x .
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propagator can therefore be written as [3];
¯h2 (cid:90) T (cid:90) t2 ∂gr
g(x ,T|x ,0) = dt dt (x ,t|x,t )| g(0,t ;0,t )
1 0 4m2 2 1 ∂x 1 2 x=0 2 1
0 0
∂g
r
× (x,t ;x ,0)| .
1 0 x=0
∂x
Where t and t are the first and last crossing times and the g are the restricted propagators for
1 2 r
theregionsx < 0andx > 0,wherethepotentialisconstant. Theyarethereforeeasilycomputed
by the method of images. Note that when x and x are both positive, there is an additional
1 0
contribution from paths which never cross x = 0 [3]. The calculation of the propagator therefore
reduces to the much simpler task of computing g(0,T|0,0), the propagator along the edge of a
step potential.
3. Lattice Paths
We first switch to working with the Euclidean propagator g by means of a Wick rotation and
we specialise immediately to the case of x = x = 0. That is, we wish to calculate
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(cid:90) x(T)=0
g(0,T|0,0) = Dx e−SE/¯h (2)
x(0)=0
where S is the Euclidean action given by
E
(cid:32) (cid:33)
(cid:90) T mx˙2
S = dt +V Θ(x) . (3)
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2
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Onthelatticetheconditionalprobabilityu(0,T|0,0)isgivenbyasumofpaths,eachweighted
by e−V0τ, where τ is the length of time spent in x < 0. This may be written as
u(0,T|0,0) = 1 (cid:88)n n e−2k(cid:15)V0, (4)
22n k
k=0
where n is the number of paths spending a time 2k(cid:15) < T in the region x < 0. Expresions for
k
thesen areknownand are infactindependentofk andareequaltothe Catalannumbers[4,6].
k
(cid:32) (cid:33)
1 2n
C = (5)
n
n+1 n
where 2n is the total number of time steps. This is the central observation that allows this
propagator to be computed in closed form.
The sum can therefore be performed and, using the very useful asymptotic form for the C
n
22n
C ≈ √ , (6)
n
πn3/2
we find
(cid:32) (cid:33)
1 1−e−2(cid:15)(n+1)V0
u(0,T|0,0) ≈ √ .
π2(cid:15)n3/2 V
0
Taking the continuum limit and Wick rotating back to real time then yields the final result,
(cid:18) m (cid:19)1/2 (1−e−iV0T)
g(0,T|0,0) = −i , (7)
2πi V T3/2
0
whichistheexpressionforthepropagatoralongtheedgeofasteppotential. Thefullpropagator
from x to x may now be obtained using the PDX [4].
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4. Summary
The PDX, together with some elementary combinatorics, allows for a simple derivation of the
propagatorforthispotential. Asimilarmethodallowsforthecomputationofthepropagatorfor
a delta function potential, by counting the number of crossings of the origin [4]. It is interesting
to note that the method described reduces the computation of a path integral to a problem in
combinatorics, and this suggests that other propagators may be computed in this way.
Acknowledgments
The author would like to thank the organisers of DICE 2008, especially Hans-Thomas Elze, for
the opportunity to to take part in this conference. He is also indebted to Jonathan J Halliwell
for his support, and for many useful discussions.
References
[1] Halliwell JJ 2008 Phys. Rev. A 77 062103.
[2] de Carvalho TO 1993 Phys. Rev. A 47 2562.
[3] Auerbach A and Kivelson S 1985 Nucl. Phys. B 257 799.
[4] Yearsley J M 2008 J. Phys. A 41 285301.
[5] See, for example, Hartle JB 1988 Phys. Rev. D 37 2818.
[6] A full proof of this can be found in many textbooks on combinatorics, see for example, Stanley RP 1999
Enumerative Combinatorics vol 2 (Cambridge: CUP)