VOLUME 85, NUMBER 4 PHYSICAL REVIEW LETTERS 24 JULY 2000
Comment on “Quantum Wave Packet
Dynamics with Trajectories”
In a recent Letter [1], Lopreore and Wyatt (LW) con-
cluded that the usual view of particle tunneling within the
Bohm interpretation (the quantum potential lowers the bar-
rier so that the Bohm particles can travel classically over
it [2,3]) is “misleading and incorrect for smooth barrier
penetration.” They claim that the Bohm particles feel only
a significant quantum force during a short boost phase just
after the launching of the wave packet (far from the bar-
rier), and only those particles that acquire enough kinetic
energy (KE) during this boost phase can pass over the bar-
rier and contribute to the tunneling transmission. Con-
sequently, they propose a decoupling approximation that
consists of ignoring the quantum force after a short ini-
tial decoupling time. In this Comment, we argue that the
results of LW correspond to a particular case in which tun-
neling is not significant. Thus, their conclusions about the
tunneling mechanism are not valid.
For comparison with LW results, we consider the scat-
tering of Gaussian wave packets, jCt , by an Eckart po-
tential V x V
0
sech
2
ax 2 x
b
. If the transmission
coefficient of the eigenstate jC
k
is T k , the total trans-
mission probability T can be decomposed into pure tun-
neling (T
tun
) and over-the-barrier (T
OB
) components:
T T
tun
1 T
OB
Z k
B
0
jak j
2
T k dk
1
Z `
k
B
jak j
2
T k dk , (1)
where ak C
k
j Ct , and k
B
q
2mV
0
¯ h
2 is the wave
vector associated with the top of the barrier.
Figure 1 shows T , T
tun
, and T
OB
as a function of the
barrier width. For wide barriers, T
tun
tends to be negli-
gible and T converges to T
OB
and to the result given by
the LW decoupling approximation. The case analyzed by
LW corresponds to this limit, i.e., T
OB
¿ T
tun
. However,
for thin enough barriers, tunneling tends to dominate the
total transmission, and the decoupling approximation fails
(it underestimates T ). The increase of transmission due to
tunneling can only be accounted for by particles that have
KE lower than the barrier height V
0
after the boost phase.
Since Bohm trajectories are classical, we must conclude
that the quantum potential lowers the classical barrier, as
is usually claimed [2,3] and has been clearly shown for
Hamiltonian eigenstates [4,5]. The transmission coeffi-
cient calculated under the decoupling approximation de-
pends on the barrier height but not on the barrier width
(see Fig. 1). The decoupling approximation is reasonable
for thick barriers but underestimates the transmission by
FIG. 1. Transmission coefficient of a Gaussian wave packet
(central energy of 0.03 eV, spatial dispersion of 5 nm) impinging
upon an Eckart barrier as a function of the barrier width at half
maximum (BWHM) for a fixed barrier height, V
0
0.04 eV.
The bold line corresponds to the total transmission coefficient
(T ), the dashed line to the tunneling component (T
tun
), the dotted
line to the over-the-barrier component (T
OB
), and the continu-
ous line to LW decoupling approximation. Inset: transmission
coefficient versus central energy (wave vector) for two barrier
widths: BWHM 1.3 nm (squares) and BWHM 18.8 nm
(circles). The continuous line corresponds to the decoupling
approximation.
orders of magnitude when tunneling dominates (inset of
Fig. 1).
In conclusion, the interpretation of tunneling given by
LW is misleading and incorrect. Tunneling requires that
the quantum potential lowers the classical barrier as pre-
viously claimed [2,3]. The decoupling approximation is
valid only when quantum effects (tunneling and over-the-
barrier resonances) are unimportant.
Support by the DGES project PB-97-0182 is
acknowledged.
Jordi Suñé and Xavier Oriols
Departament d’Enginyeria Electrònica
Universitat Autònoma de Barcelona
08193-Bellaterra, Spain
Received 24 September 1999
PACS numbers: 34.10.+x
[1] C. L. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82, 5190
(1999).
[2] D. Bohm and B.J. Hiley, The Undivided Universe (Rout-
ledge, London, 1993).
[3] P. R. Holland, The Quantum Theory of Motion (Cambridge
University Press, Cambridge, England, 1993).
[4] X. Oriols, F. Martín, and J. Suñé, Phys. Rev. A 54, 2594
(1996).
[5] X. Oriols, F. Martín, and J. Suñé, Solid State Commun. 99,
123 (1996).
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