A method for the solution of the time‐dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two‐dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave‐packet dynamics of chemical reaction rates.

1.
J. H.
Weiner
 and
Y.
Partom
,
Phys. Rev.
187
,
1134
(
1969
).
Google Scholar
Crossref
Search ADS
 
2.
E.
 
Madelung
,
Z. Physik
 
40
,
322
(
1926
);
Google Scholar
Crossref
Search ADS
 
L.
 
de Broglie
,
Compt. Rend.
 
183
,
447
(
1926
).
Google Scholar
 
For a recent discussion of this analogy see
H. E.
Wilhelm
,
Phys. Rev. D
1
,
2278
(
1970
).
Google Scholar
Crossref
Search ADS
 
3.
J. H.
Weiner
 and
Y.
Partom
,
Phys. Rev. B
1
,
1533
(
1970
).
Google Scholar
Crossref
Search ADS
 
4.
It is important to distinguish between the (inertial) mass of each particle of the continuum and the (statistical) density of the continuum. For the purpose of computing the trajectory of a generic particle of the continuum, its inertial mass m must be employed as indicated in Eq. (2.3). For the purpose of the weight to be assigned a portion of the continuum in computing averages, its statistical density ρ(x,t) must be employed as indicated in Eq. (2.9).
5.
When limits of integration are omitted, as in Eq. (2.7), integration over the entire plane −∞<xa<∞,α = 1,2 is understood.
6.
T.
Levi‐Civita
,
Atti Pontif. Accad. Sci. Nuovo Lincei
88
,
102
(
1934–1935
).
Google Scholar
 
7.
M. L.
Wilkins
,
Methods Computational Phys.
3
,
211
(
1964
).
Google Scholar
 
8.
See, for example, A. Messiah, Quantum Mechanics (North‐Holland, Amsterdam, 1964), Vol. 1, p. 216.
9.
This result may appear surprising in view of the familiar phenomenon of a reflected wave packet and a transmitted wave packet when dealing with potentials of finite range. The quadratic potential, however, extended to infinite range.
10.
It is noted that for large Nα, Δyα,i→(Nαρα,i−1)−1, i.e., the particle spacing becomes inversely proportional to the density. This was the scheme employed for the particle spacing in Ref. 1.
11.
The major part of the computations were performed with a time increment δt = 0.006 (units in which m = ℏ = 1 were employed). This value of δt gave accurate results for all quantities although a small, increasing error in X̄1(t) was noted. A change to δ = 0.003 reduced this error to negligible size.
12.
M. H.
Bramball
 and
B. M.
Caspar
,
Am. J. Phys.
38
,
1136
(
1970
).
Google Scholar
Crossref
Search ADS
 
13.
A.
Goldberg
,
H. M.
Schey
, and
J. L.
Schwartz
,
Am. J. Phys.
35
,
177
(
1967
).
Google Scholar
Crossref
Search ADS
 
14.
E. A.
 
McCullough
,Jr. and
R. E.
 
Wyatt
,
J. Chem. Phys.
 
51
,
1253
(
1969
);
Google Scholar
Crossref
Search ADS
 
E. A.
 
McCullough
,Jr. and
R. E.
 
Wyatt
,
54
,
3578
,
3592
(
1971
). ,
J. Chem. Phys.
Google Scholar
Crossref
Search ADS
 
Y. R. L.
Park
,
C. T.
Tahk
, and
D.
Wilson
,
J. Chem. Phys.
53
,
786
(
1970
).,
J. Chem. Phys.
Google Scholar
Crossref
Search ADS
 
This content is only available via PDF.
© 1971 American Institute of Physics.
1971
American Institute of Physics
You do not currently have access to this content.