Mean position of a parti le submitted to a

Description

Mean position of a parti le submitted to a potential barrier D. Mer ier, V. Régnier ? Abstra t A one-dimensional Klein-Gordon problem, whi h is a physi al model for a quantum parti le submitted to a potential barrier, is studied numeri ally : using a variational formulation and a Newmark numeri al method, we ompute the mean position and standard deviation of the parti le as well as their time evolution. Key words Klein-Gordon equation, Newmark method, mean and standard devia- tion, kernel smoothing, linear regression. AMS 35A15, 65M06, 65M12, 81Q05, 81Q10. 1 Introdu tion It has been well-known for a few years now that in quantum me hani s a parti le an limb up a step even if it has not enough energy a priori and it will be ree ted then with a delay. In lassi al me hani s it would just try and go ba k to its position. Everything happens here as if the parti le ould go through a wall ( f. Fig. 5.1 in [17?) ! This phenomenon is alled tunnel ee t and has been a subje t of interest for physi ians and mathemati ians : the delay has been measured by the physi ists A.

• iennes


• tion


• initial boundary


• tly supported initial


• exa tly


• gaussian wave


• tral theory


• mer ieruniv-valen


• delay whi


• numeri al

Sujets

• Mehmeti
• maths

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

∗
∗
osition
of
a
een
W
Mon
particle
(cf.
submitted
[13])
to
all
a
has
p
frequency
oten
J.
tial
T
barrier

D.
phenomenon
Mercier,
ph
V.
ysicists
R?gnier
ximately
p
of
in
gathered


A
w
one-dimensional
Applications
Klein-Gordon
de
problem,
9,
whic
go
h
in
is
eect
a
of
ph
:
ysical
b
mo
and
del
that
for

a
is
quan
of
tum
erluminal
particle

submitted
e
to
(cf.
a
and
p
terested
oten
de
tial
Institut
barrier,
V
is
du
studied
59313
n
denis.mercier@univ-v
umerically
particle
:
a
using
Fig.
a
!
v

ariational
has
form
sub
ulation
terest
and
and
a
dela
Newmark
een
n
the
umerical
Haib
metho
Nim
d,
It
w
is
e
to

of
the
that
mean
enden
p
shap
osition
barrier.
and
on
standard
elo
deviation

of
in
the
to
particle
t
as
tunnel
w
In
ell

as
E.
their
ere
time
1993
ev
Lab
olution.
et
Key
V
w
Sciences
ords
hniques
Klein-Gordon
Univ
equation,

Newmark
br?sis,
metho
Houy
d,
ALENCIENNES
mean

and
;
standard
the
devia-

tion,
through
k
w
ernel
(cf.
smo
5.1
othing,
[17])
linear
This
regression.
is
AMS
tunnel
35A15,
and
65M06,
b
65M12,
a
81Q05,

81Q10.
in
1
for
In
ysicians
tro
mathematicians

the
It
y
has
b
b
measured
een
y
w
ph
ell-kno
A.
wn
el
for
G.
a
tz
few
[16]).
y
seems
ears
it
no
appro
w
equal
that
the
in

quan
their
tum
and

it

indep
a
t
particle
the

e

the
b
The
up
orkshop
a
Sup
step
V
ev

en
some
if
t
it
tributions
has
Cologne
not
1998
enough
describ
energy
transien
a
phenomena
priori
the
and
eect
it
[20]).
will
particular
b
M.
e
h

F.
then
Lo
with
w
a
in
dela
in
y
(see
Mean
.
In
oratoire

Math?matiques

ses

de
it

w
des
ould
et
just
ec
try
de
and

go
ersit?

V
k
et
to
Hainaut-Cam
its
Le
p
t
osition.
,
Ev
V
erything
Cedex
happ
FRANCE,
ens
:
here

as

if
13−λu
F
F
3F(u) =−λu
n
v
ersing
a
prop
a
of

step.
gular
prew
p
v
oten
existence
tial
on
barrier.
v
T.
the
Hartman



in
repulsiv
[19]
Gaussian
the
t
tunnelling
m
time
([2]).
of
sion
Gaussian
℄
w
time
a
V.
v
initially
e
equation

e
k
but
ets
W
for
of
one-
self-in
dimensional
problem

at
p
w
oten
F.
tial
net
barriers
y
based

on
similar
the
net
time-dep
simpler
enden
ersiv
t
its
Sc
een
hr?
him
dinger
nor
equation.
Reed
An
that
analytic
t-hand
expression
y
is
in
giv
a
en
with
in
t-hand
terms
v
of
using
the
the
w
eects
a
a
v
problem
e
net
n
innite
um
t.
b
equa-
er.
and
It
e
is
ed
easy
in
to
a
adapt
as
his
an

in
to
our
obtain
app
an
in
expression
in
for
D.
the
er,
dela
o
y
es
of
is

the
The
and
main
The
obstacle
F.
for
an
a
In
rigorous
osition
form
is
ulation
en

Simon
from
sho
the
a

(nonlinear)
that
ets
T.
,
Hartman
holds.

v
phase
the
shifts
y
for
Klein-Gordon
mono
transmission

more
hromatic
nonlinear
w
v
a
also
v
the
es.
a
Clearly
repulsiv
,
.


h
lo
w
sup
a
and
v
oten
es
that

a
exhibit
a
ph
ork
ysically
o
measurable
hes
transien
p
t
t
features.
dieren
Instead
on
signals
ramied
ha
generally
ving

a
in

Mehmeti
frequency
the
ha
e
v

e
ork
to
in
b
result
e
generalized
studied.
R?gnier
That
in
is
of
wh
Analogous
y
(analogous
F.
and
Ali

Mehmeti
in
and
a
V.

R?gnier
v
used
star-shap
w
ork
a
thesis
v
this
e
geometry

t
k
hes)
ets
a
with
dis-
a
Our
narro
study
w
olution
fre-
esp

p
band
v
together
p
with
has
a
b
solution
Mehmeti
form
who
ula
to
inspired
doing
b
tum
y
the
J.
a
M.
momen

wn
h
giv
and

F.
and
E.
ha
Lo
e
w
wn
in
for
their
Klein-Gordon
2003
with
pap
righ
er
side
([6]
k
and
e
℄

F
still
or
W

pro
y
ed
and
℄
to
same

ert
v
for
er
one-dimensional
the
problem
relativistic
with

and
w
a
e
general

e
hose
righ
the
side
one-dimensional
.
Klein-Gordon
e
equation
pro
(as
e
w
global
e
of
will
solution
do
the
tra
eness
,
a
problem
In
a
particular
ysical
w
del
w-frequency
a
of
tum
erluminal
submitted
in
this
the
pap
is
er
ph
as
mo
w
for
ell)
quan
and
particle
form
to
ulated

our
to
results
p
in
tial
terms
Note
of
this
the
is
energy
transmission
o
on
w
simple
to
w
ensure
:
the
w
ph
semi-
ysical

measurabilit

y
one
of
oin
the


results
transien
partial
t
tial
phenomena.
tions
W
net
e
orks,
ga
spaces
v
more
e
on
an

analytic
b
expression
found
of
℄
the
Ali
dela
solv
y
explicitly
whic
w
h
v
is
equation
in
a

ed
with
w
the
in

seminar
of
1982
Haib
The
el
w
and
then
Nim
b
tz.
V.
More
for

y
tly
teger
(cf.
alue
in
℄
w
℄
e

got
ts
in
as
terested

in
transmis-


y
ts)
.
ear
It
a
is
w
w
y
ell-kno
the
wn
of
that
a
the
elets
solu-
a
tion
ed
of
w
the
in

Mercier's
linear
([23
w
In
a
pap
v
the
e
is
equation
(only
with
w


supp
but
orted
w
initial
v

are
and
p
v
e.
anishing
aim
initial
to
v
the
elo
ev

of
y
particle
is
ecially
also
mean

osition
supp
the
orted
ariance
in
its
a
osition.
set
idea
dep
b
ending
suggested
on
y
time
Ali
:
to
the
R?gnier
supp
w
ort
ts
of
thank
the
for
solution
so.
at
quan
time

t
ne