Solutions of the Klein-Gordon equation in an infinite square-well potential with a moving wall

0.32 MB·English

by  Michael Koehn

AEI-2013-001
Solutions of the Klein-Gordon equation in an infinite square-well potential with a
moving wall
Michael Koehn∗
Max-Planck-Institut fu¨r Gravitationsphysik, Albert-Einstein-Institut,
Am Mu¨hlenberg 1, 14476 Potsdam, Germany, EU
Employing a transformation to hyperbolic space, we derive in a simple way exact solutions
for the Klein-Gordon equation in an infinite square-well potential with one boundary moving at
constant velocity, for the massless as well as for the massive case.
MSC(2000): 35R37,81Q05,81Q35,35Q40,35Q41
PACSnumbers: 03.65.-w,03.65.Ge,03.65.Pm
Keywords: relativisticquantummechanics;Klein-Gordonequation;movingboundaries;exactsolutions
3
1 Introduction.–Thenon-relativisticsystemofaone- t
0 dimensional infinite square well with a massive particle
2 evolving according to the Schr¨odinger equation is one of
τ
n themostelementaryquantummechanicalsystems,andit ρ
0
a often serves as an approximation to more complex phys-
J ical systems. If, however, the potential walls are allowed

3 to move, as originally in the Fermi-Ulam model for the
acceleration of cosmic rays [1, 2], the situation is much

] more complicated: if one does not choose to rely on an
h
adiabatic approximation, then the system is not sepa-
p x
- rable anymore. Results concerning exact solutions exist ντ
h
only sparsely and have attracted a considerable amount
t
a of attention. For the special case of a non-relativistic
m system with a wall moving at constant velocity, such ex- FIG.1: Theinfinitesquarewellwithmovingwallsasashaded
wedgeinsidetheforwardlightcone,intersectedbyaspacelike
[ act solutions have been obtained first in [3], see [4–6] for hyperboloid of fixed ρ=ρ .
generalizations. 0
1
The relativistic moving-wall system on the other hand
v
6 is a much less common object of study, and there ap- The infinite square well is specified as the domain =
3 pearinterestingsubtleties. Thisarticleconcernstheone- [0,L(t)]. ItisgivenintermsofthelengthfunctionL(Ft)=
4 dimensional Klein-Gordon (KG) particle in an infinite L +ν(t t ), where ν denotes the speed of the receding
0 0
0 squarewellwithaboundarywhichismovingoutwardat wall with−0 < ν < 1. ∂ denotes the boundary of ,
1. constantvelocityν. Wefindaninfinitesetofexactsolu- and t,x R. We have rFescaled the speed of light anFd
0 tionswhichdonotrelyonanadiabaticapproximationon Planck’s∈constant to c = 1 = (cid:126), and we will treat the
3 top of the square-well approximation with definite posi- massless case m=0 first. Since (1) is of second order in
1 tion and momentum configuration. We thereby general- time, the transformation
: ize the solutions presented in the appendix of [7], which
v x
i are valid only for a special case and which are stated x x(cid:48) = (2)
X (cid:55)→ L(t)
without specifying any method on how to obtain them.
r In contrast, we use a transformation to hyperbolic space which is usually applied in the non-relativistic scenario
a
which provides in a simple and new manner a set of gen- (see e.g. [4, 5]) is of not much help in obtaining analytic
eral solutions for the massless as well as for the massive solutions for the relativistic one, although indeed imply-
case, while introducing derivatives of first order into the ing motionless walls for the transformed system.
transformed KG equation. However, by slicing the forward lightcone of two-
Exact solution. – This article concerns the ini- dimensional Minkowski space in terms of hyperboloids
tial/boundary value problem and transforming to hyperbolic coordinates (cf. Fig. 1),
one obtains a separable infinite square well system with

∂∂t22Ψ(t,x)= ∂∂x22Ψ(t,x)−m2Ψ(t,x) in F sρtasetircvebsouansdtahreiense,wwhteimreet-lhikeehycopoerrdbionliacter.adEiaxlpcliocoitrldyi,ntahtee
Ψ =0 on ∂ (1)
∂F
Ψ(|t ,x)=f(x), (∂ Ψ)(t ,x)=g(x). F transformation is given by
0 t 0
t=ργ x=ργ (3)
t x
subject to the constraint
∗[email protected] (γt)2 (γx)2 =1 (4)
−
2
Λ(ν) Due to the separability of the system in hyperbolic coor-
12 dinates, the general solution is given by
Ψ(ρ,v)=ψ (ρv)+ψ (ρ/v) , (8)
1 2
8 where ρv = t+x and ρ/v = t x, and we can easily
−
choose a solution that matches the Dirichlet boundary
conditions, e.g.
4 1
Ψ (ρ,v)= sin(k ln(v))exp( ik ln(ρ)) (9)
n n n
√nπ −
with
ν
1 1 0 1 1 nπ
− −2 2 kn = . (10)
ln(Λ(ν))
FIG. 2: The position Λ of the boundary in hyperbolic space
Forthemassivecasecorrespondingto∂2Φ=∂2Φ m2Φ
depends only on the speed ν of the moving wall. t x −
with m=0, the above procedure implies the solutions
(cid:54)
Ψ10(5409,x) Φn(ρ,v)=Csin(knln(v))[Jik(mρ)+Yik(mρ)] , (11)
1
4 where J and Y are the Bessel functions of imaginary
ik ik
order ik. Of course we could have also chosen real so-
lutions to the real KG equation. We remark that in a
x general number of dimensions, having a definite direc-
0 1 1 tion of wavepacket propagation is related to complexity
2
of the wavefunction.
We have chosen the integration constants of the solu-
tions(9)suchthattheyarenormalizedto1withrespect
to the KG-like scalar product
FIG. 3: Exemplary plot of the wavefunction Ψ (50,x) with
10 49 (cid:90) ↔
speed ν = 4590 of the moving wall at time t= 5409. (ψ1,ψ2)=iρ dvolψ1∗∂ρψ2 , (12)
which implies that ρ2 =t2 x2. wheredvol= dvv denotesthevolumeelementontheone-
− ↔
The coordinates γt and γx are constrained to the one- dimensional hyperbolic upper half-plane, and where ψ∂ρ
dimensional hyperboloid, such that one of the two coor- φ ψ∂ φ φ∂ ψ. The solutions (9) can be expressed in
ρ ρ
dinatesisredundantandwemayparametrizethehyper- fla≡t coordi−nates as
boloidastheone-dimensionalanalogueofthehyperbolic
upper half-plane through the transformations Ψ (t,x)= 1 sin(cid:20)kn ln(cid:16)t(cid:48)+x(cid:17)(cid:21)
n √nπ 2 t(cid:48)−x ×
v2+1 v2 1 (cid:20) (cid:21)
γt = 2v γx = 2−v v =γt+γx . (5) exp ikn ln(cid:0)t(cid:48)2 x2(cid:1) (13)
× − 2 −
One can easily find a full set of exact solutions in these
coordinates by requiring as an intermediate step that with t(cid:48)(t) t t + L0, 0 < ν < 1. FIG. 3 displays an
≡ − 0 ν
L = νt . In order to obtain solutions for general pa- exemplary plot of one specific such function. We note
0 0
rameters (L ,t ), one may afterwards freely shift the tip that the solutions (13) are normalized to 1 with respect
0 0
ofthelightcone. Thepositionoftherightboundarydoes to the standard KG-invariant scalar product,
not depend on the time-like coordinate ρ, but merely on
(cid:90)
the constant velocity ν. Employing (5), the position of ψ ψ =i dxψ∗∂↔ψ . (14)
therightboundaryinhyperbolicspacecanbedetermined (cid:104) 1| 2(cid:105) 1 t 2
as (cf. FIG. 2)
The respective norms in hyperbolic and in flat space are
(cid:114)1+ν preserved, and we can directly show for (13) that
Λ(ν)= , (6)
1 ν
− Ψn Ψm =δnm . (15)
(cid:104) | (cid:105)
while the transformed massless KG equation reads
Numericalinvestigations.–Wenowinvestigatethe
ρ∂ (ρ∂ Ψ(ρ,v))=v∂ (v∂ Ψ(ρ,v)) . (7) properties of a relativistic quantum wavepacket evolving
ρ ρ v v
3
0.06 0.18 0.06 0.06
0.04 0.12 0.04 0.04
0.02 0.06 0.02 0.02
0 0 0 0
0 10 20 30 0 10 20 30 0 10 20 30 0 1500 3000 4500
(a)Beforereflection (b)Duringreflection (c)Redshiftedwavepacket (d)Late-timebehavior(scale
changed)
FIG. 4: Snapshots of |ψ(t,x)|2 for a one-dimensional Gaussian wavepacket evolving with respect to (1) and redshifted upon
reflection off a wall moving in flat space. The wall is moving at ν = 1 and is represented by the vertical bar in (a)-(c). The
2
scaling in (d) differs from (a)-(c).
0.06 0.18 0.06 0.06
0.03 0.09 0.03 0.03
0 0 0 0
0 0.4 0.8 1.2 1.6 0 0.4 0.8 1.2 1.6 0 0.4 0.8 1.2 1.6 0 0.4 0.8 1.2 1.6
(a)Beforereflection (b)Duringreflection (c)Reflectedwavepacket (d)Late-timebehavior
FIG. 5: Snapshots of the same |ψ(t,x)|2 for a one-dimensional Gaussian wavepacket, but this time with t and x expressed
through the hyperbolic coordinates ρ and v evolving with respect to (7) and reflecting off an infinite square well with right
√
boundary at Λ(1)= 3 according to (6).
2
according to (7), i.e. m = 0. For this purpose, we com- of the energy H of a photon before a bounce from a
inc
poseoutofpositivefrequenciesω = p aone-dimensional moving mirror to its energy H afterwards. The ex-
ref
Gaussian wavepacket | | pectation value Hˆ , which we call the energy expecta-
(cid:104) (cid:105)
(cid:90) ∞ tion value in the following, may serve as a measure for
Ψ(t,x)=A dpe−c2(p−2p0)2+i(px−ωt) (16) the redshift. The square root in the Hamiltonian can be
−∞ avoided adopting the two-component notation [8] for (1)
which is normalized to 1 with respect to (14) if (especially helpful in higher-dimensional cases),
A= c (cid:16)e−c2p20 +√πp0c erf(p0c)(cid:17)−12 , (17) ψ =φ+χ i∂tψ =φ−χ , (20)
√π
which we adapt to the massless case to obtain
where erf denotes the error function. The norm of the
σ +iσ σ iσ
wavepacket (16) with respect to (14) (approximately 1 Hˆ = 3 2∆+ 3− 2 , (21)
if c is chosen small enough) is preserved during its evo- − 2 2
lution in the moving-wall system. However, its absolute where σ denote the Pauli matrices and ∆ = ∂2 . With
width with respect to the x-coordinate grows with each i ∂x2
the two-component vector
reflection off the moving wall due to successive redshifts.
The redshift upon reflection of a one-dimensional wave Ψ=(cid:0)φ(cid:1) , (22)
χ
off a wall which is moving away at a constant speed is
the inner product (14) is expressible as
given by
(cid:90)
1+z = ffref =γ2(1+ν)2 = 11+νν ≡Λ(ν)2 , (18) (cid:104)Ψ1|Ψ2(cid:105)=2 dxΨ†1σ3Ψ2 (23)
inc
−
with γ = (1 ν2)−12, e.g. 1+z = 3 for ν = 1. The and the energy expectation value as
corresponding−reflection of a classical massless 2particle (cid:90)
Hˆ =2 dxΨ†σ HˆΨ . (24)
in one dimension off a moving wall in terms of the rel- 3
(cid:104) (cid:105)
ativistic Hamiltonian H = √π 2, where π denotes the
x x
momentum variable conjugate to x, implies the relation Furthermore, (21) implies the expected expression
(cid:90) (cid:90)
H ↔
1+z = ref (19) Hˆ2 =2 dxΨ†σ Hˆ2Ψ= i dxψ∗∂(∆ψ) . (25)
H (cid:104) (cid:105) 3 − t
inc
4
FIG. 4 and FIG. 5 illustrate the reflection of the Klein-Gordonequationinaninfinitesquare-wellwithone
wavepacket off the wall, obtained as the numerical so- wall moving at a constant velocity. These solutions are
lution of the massless KG equation, on the one hand in obtained employing a simple transformation to hyper-
flat space with moving boundary conditions acording to bolicspace. Wefurthermoreinvestigatednumericallythe
(1), and on the other hand the corresponding evolution propertiesofa masslessrelativistic wavepacketbouncing
in hyperbolic space with static boundary conditions ac- off the moving walls in flat and off the static walls in
cording to (7). In flat space, the wavepacket gets red- hyperbolic space, and in the former case observed the
shifted and loses energy upon reflection off the moving expected redshift. Although the scope of this article was
wall, while its KG norm (14) is preserved. We remark intendedtobelimitedtoadomainwithonewallmoving
that unlike the Hamiltonian and momentum operators, at constant speed, it would nevertheless be of interest to
the position operator generally mixes positive and nega- generalize these results to domains with more arbitrarily
tivefrequencycomponentsofthewavepacketinrelativis- moving walls.
ticquantummechanics[8],andwereferthereadertothe
investigations in [9, 10] for details on the localization of
relativistic particles.
Acknowledgments
Summaryandconclusions.–Exactsolutionsofthe
Schr¨odinger and Klein-Gordon equations in a domain
with time-dependent boundaries are difficult to obtain Theauthorgratefullyacknowledgesthesupportofthe
[3–5, 7, 11]. With this letter, we contribute an infinite EuropeanResearchCouncilviatheStartingGrantnum-
set of orthogonal exact solutions to the one-dimensional bered 256994.
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