Quantum Tunneling Using Discrete-Time Operator Difference Equations

PHYSICAL REVIEW LETTERS

VOLUME 55, NUMBER 9

Quantum

Tunneling

26 AUOUST 198S

Using Discrete-Time Operator Difference Equations Carl M. Bender

Department

of Physics,

Washington

University,

St L.ouis, Missouri 63130

and

Fred Cooper, Los

3 lamos

'

James E. O'Dell,

~

L. M. Simmons, Jr.

and

'

National Laboratory, Los Alamos, New Mexico 87545 (Received 8 July 1985)

Canonical discrete-time operator difference equations are introduced as an alternative approach to the numerical solution of a quantum field theory. We apply these techniques to the solution of tunthe operator Heisenberg equations of motion describing the problem of quantum-mechanical neling. Our numerical solutions accurately depict the time evolution of (q") and of a local probability measure. PACS numbers:

03.65.—w, 03.70. + k, 05.50. +q, 11.10.Ef

As an alternative approach to the numerical so1ution of a quantum field theory it has recently been proposed' that the operator field equations be solved directly by time stepping of the operator on a Minkowski lattice. The canonical operator difference equations can be either implicit, as derived from a finite-element approach, or explicit, as derived from a finite-difference These techniques are approach. canonical in the sense that the iteration scheme exactly relations at preserves the equal-time commutation every lattice point. Furthermore, they lead to a consistent formulation of fermions and of gauge invari-

ance. "

Quantum mechanics provides a convenient laborato6 and for the ry for the study of these techniquess of them into practical computational development tools. In one-dimensional quantum mechanics one obtains operator difference equations on a time lattice as an approximation to the operator Heisenberg equations of motion and these equations provide a nonperturbative method for the study of quantum processes in real time. It seems natural to apply these new computational ideas to quantum tunneling. Certain aspects of this problem have recently been studied. Using the finite-element method Bender et al. investigated tunneling in a quartic potential in a one-finite-element approximation and obtained good results for short times. In a second paper Cooper, Milton, and Simmons studied the long-time behavior of the motion of a wave packet in the classically allowed region of an inverted parabolic potential. That study compared the numerical accuracy of various implicit and explicit quantum differencing schemes. The purpose of this paper is to extend the development of operator-difference techniques to a practical computational scheme for quantum problems. We use an explicit differencing scheme to study the motion of an initial Gaussian wave packet in a degenerate double-well potential.

The continuum

Hamiltonian

is H

= —,' p2+

V (q),

where

V(q)

= 4q'(q —P)'/P'.

This potential is symmetric about q = —,P and has apharmonic wells centered at q = 0 and proximately = p. Near q = 0 the approximate potential is q whose eigenvalues are E„=J8 V, „„(q)= 4q ' (n + —, ). The normalized ground-state wave function of + pp „is &&

@o(q) = (to/m)' 4exp(

—q'to/2),

(2)

to = J8.

We choose @o as the initial coordinatewhere space wave packet, centered at q =0 in the left well when t =0. If to/2 P /4, the barrier height, the particle represented by the wave packet @o is classically confined and the wave function tunnels with a frequency determined by the energy splitting of the almost degenerate ground state and first excited state. We study this tunneling process in the Heisenberg picture by directly solving the Heisenberg equations of

(

motion,

q(t) =p(t),

p(t)

=- dv(q)

on a discrete time lattice and by measuring matrix elements of appropriate operators in the false vacuum state represented by (2). The recurrence relations that we use to express the quantum operators at time step n + 1 in terms of those at time step n are

q„+ t = q„+ hp„——, h 2 V' (q„), ' , = p„——, h [ V'(q„) + V'(q„, ) ], '

(4a) (4b)

where h is the time-lattice spacing At and q„and p„are the Heisenberg operators on the discrete time lattice. q„and p„are accurate approximations to the position

1985 The American Physical Society

901

PHYSICAL REVIEW LETTERS

VOLUME 55, NUMBER 9

operators q (t) and p (t) at time and momentum t = nh, with error of order h . This differencing method the scheme shares with the finite-element property that it is canonical; that is, it exactly preserves relation [q„,p„] = i at the equal-time commutation every time step (it is a unitary scheme). 2 The equal-time commutation relation for n = 0 (that is, t = 0) allows us to define a Fock space of states Ik) such that apIO) = 0, where q11

——

(2t0) 'i (ap+ap )

(sa)

—i ( —,' c0)'j2(ap —ap).

(Sb)

vent the algebraic complexity.

(j Iqplk& (OIqp

= (2') 'j'(~jp, k+1+~&~k, +1),

IO) can

(~j hj k+1+~k ~kj +1) (loa) (lob) ( p1) (~j l1j, k+1 ~k ~k j+1) (pp) 'k 1~j,k ~D, where D= —, (3 +1). The matrix (2~)

1

representation of IO) is in (2). To study tunneling in the Heisenberg we calculate

r, (t) = &OIqt(t) IO) = JI 1clp(x, t)x'1ltp(x,

(6a) t)dx,

1ltp(x, t) is the wave

where

@p given picture

(6b)

packet that evolves from

0) = @p(x) under the action of the Hamiltonian. Another quantity of interest is the fractional proba-

1tjp(x,

bility, as a function of time, for the wave packet to remain in the left well; this directly measures the tunneling time. In the Schrodinger picture such a probability measure is PL

(t) = Jt

P/2

1Itp(x, t) tpp(x,

t)dx.

(7)

In the Heisenberg picture it is more convenient to measure an operator that has most of its support in the left well. We choose L

«) = (OIpj'(p+q')

IO),

which has behavior quite similar to Pt (t). We find that PL (0) = 0.97. The recurrence relations (4) are attractive because they allow, in principle, an exact symbolic iteration by means of an algebraic-manipulation program such as MAcsYMA. One organizes the iterates q„and p„, for n =1, 2, . . . , jV, expressed in terms of the Fock-space in normal-ordered form. It is then operators a and trivial to calculate any matrix element of any polynomial in qp and pp. The replacement t = h/N then produces an approximate Taylor series in t for the maand converging to the trix elements, exact to order where N is the number of exact Taylor series as iterations. Because of the nonlinear nature of the equations (4) the algebraic complexity of the iterates grows very rapidly. With our present MACS YMA techniques it is impractical to calculate more than four or five iterates. one can recognize that finiteAlternatively, dimensional numerical matrices can be used to circum-

a,

N,

902

t,

(9)

j, k = 0, 1, 2,

(qp)jk

The coordinate-space

Because

. . . , it is not difficult to see that be calculated exactly by matrix multiplication with a matrix representation of qo of dimensionality d =1+iV; of course, (OIqp +' IO) =0. From the form of the potential &(q) in (1) and the recurrence relations (4) it then follows that we can calculate (OIq~IO) exactly using the finite-dimensional matrix representations

where

and pp

26 AUGUST 1985

for

2

~

size for this exact calculation grows very rapidly with the number of iterations and the limit imposed by computer memory and computational time is likely to be N Because of the difficulties described above in the exact iteration scheme, we found for practical purposes that a numerical approach was needed. The numerical approach is to approximate the infinite matrices po and qo by sequences of larger and larger D XD matrices given by (10) and to iterate (4) with fixed values of D and h. We have implemented the scheme in FoRTRAN and studied the numerical stability and convergence as a function of D and h. At fixed D we studied convergence of our results as a function of h. We found at all D that for h 0.07 the results were stable to at least two significant fig0.008 we found a four-significant-figure ures; for h stability. Of course, at small D the answers at large t are far from the correct infinite-D limit. We next kept h =0.008 and studied convergence to the infinite-D limit as a function of t (at small t finite matrices are exact. ) We note that the dimension D controls the number of harmonic-oscillator basis states in the t =0 Fock space. We compared larger and larger D x D matrices and found for the potential (1) with p= 2. 5 the following times of validity with four1.6; D = 8, t 3; place accuracy: For D =4, t = = D D 16, t 10; 32, all t examined (t 30). These internally consistent results were then confirmed by comparison with a numerical integration of which gave also a fourthe Schrodinger equation significant-figure agreement with the D = 32 results at all times t 30. Running up to t =30 sufficed to display two complete oscillations in the tunneling. In Figs. 1 and 2 we display our numerical results for iteration of the system (4) using 32&& 32 matrices and h =0.008, extending out to t =30. This calculation is enough to display two complete oscillations in the tunneling. This D = 32 calculation agrees to four signifiof the cant figures with a numerical integration

~7.

~

~

~

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VOLUME 55

PHYSICCAL REVIEW LETTERS

N UMBER 9

2. 5

26 AUCiUST 1985

I.O

2.0

.

I

0

5

~

0

.

20

l5

IO

0

5

I5

IO

f tll e probability

t-

atrix represe

or qo and po as given in (10 and a value va of

for th e wave pack t ~0 1 tio se the sa same parameters

1

=0.008.

h

' 'on is culation under study. We thank M. L Simmo

Sc rodinger equation We also calculated (OI q o) (o q me oscillatory beh avior as show

d s o

h

1

h

oth

ho

1

is work

of

1"hes 'hat d

D

h

-dff 1

-

ional n

'

or stud ying

ourr

ff

eory

For ex ample, in

(1

F @, (t)

1

(11a)

(),

a

1'(@)

t)

t

j =1, 2, . . . , %. Thehese 3)

d

operator e

b e discretized

'

(11b)

e use o

techniqu pair of equations

(3). Th is

. Sharp, Phys.

Rev.

51, 1815 (

harp, Phys. Rev.

31, 383 (1985).

ender, K. A. M' Ph . R . D -der L M SirnrnonsJ

to,

e,

+i t)+4, -t«) —2@, «)

ys. Rev. D

C. M. Bender, K. A.. M Mi}to ilto,

ensional x, t) must e a by

=mj(t),

. ...

CMBdK L tt.

1

h fi ld operators @( atti iscretized on a space latti

e

eoretical D'ivision. unicatio C omputing and Comm iC. M B. d. n dD . H H. Sharp, , Phys. R Lett. 50 1 ys. Rev. (a)

(1983).

40

1

.

to th

requiree puter; for two-si n'

earlier,

t o fEnergy.

amos National Lay the U. S.

PP o

—. 1

t

was

PP o

actical alternative

1

ensive help in e used in th'

. P. Gut schick and M M ""d'' -equation calculation.

n

e

'

50

FIG. 2. Tiime dependenc

FIG. 1. T' Time dependenc

described here ime on a C accuracy one n A e

25

20

Time

Time

to

I

I

I

0

ar

dR

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L S tong, to be

ender, F. Coo e . . u sc ic Phys Revv. D (to be publi h immons, Jr. , Ph ys. , K. A. Mil (o e published). . P. Gutschick . . ie }1 1 egration t h q es in Computerer Methods for Parti al u III d' ons Differential Equati a ssociation fo ' and C in n, ew Brunswick, N . J., 1979), p. 313 ~

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