Quantum Theory of Tunneling

QUANTUM THEORY OF TUNNELING

QUANTUM THEORY OF

TUNNELING

Mohsen Razavy University of Alberta, Canada

V f e World Scientific lflb

New Jersey • London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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QUANTUM THEORY OF TUNNELING Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-018-3 ISBN 981-238-019-1 (pbk)

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface The present book grew out of a lecture course given at the Institute for Advanced Studies in Basic Sciences, Zanjan, Iran in the summer of 1999. The intent at the outset was to present some of the basic results and methods of quantum theory of tunneling without concentrating on any particular application. It was difficult to decide what topics should be treated at length and which ones should be omitted from the discussion. Thus my main area of interest, the quantum theory of dissipative tunneling, was left out completely since even an introductory survey of the subject would have nearly doubled the size of the book. I am indebted to my dear friends and colleagues Professors Y. Sobouti and M.R. Khajepour for giving me the opportunity of lecturing to a group of enthusiastic graduate students and also encouraging me to write this monograph. I have benefitted immensely from discussions with my colleague Professor A.Z. Capri and with Mr. Robert Teshima. Above all, I am indebted to my wife who never failed to support me.

v

Contents Preface

v

1

1

A Brief History of Quantum Tunneling

2 Some Basic Questions Concerning Quantum Tunneling

3

4

2.1

Tunneling and the Uncertainty Principle

2.2

Decay of a Quasistationary State

9 9 11

Semi-Classical Approximations

23

3.1

The WKB Approximation

23

3.2

Method of Miller and Good

31

3.3

Calculation of the Splitting of Levels in a Symmetric Double-Well Potential Using WKB Approximation

35

Generalization of the Bohr-Sommerfeld Quantization Rule and its Application to Quantum Tunneling 4.1

4.2

41

The Bohr-Sommerfeld Method for Tunneling in Symmetric and Asymmetric Wells

45

Numerical Examples

48 vii

viii 5

Contents Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State 5.1

5.2

5.3

5.4

6

53

Solution of the Schrodinger Equation with Radiating Boundary Condition

53

The Time Development of a Wave PacketTrapped Behind a Barrier

57

A More Accurate Determination of the Wave Function of a Decaying State

61

Some Instances Where WKB Approximation and the Gamow Formula Do Not Work

66

Simple Solvable Problems

73

6.1

Confining Double-Well Potentials

73

6.2

Time-dependent Tunneling for a

V

' -

E

'

(2-10)

which is the same as Eq. (2.6).

2.2

Decay of a Quasistationary State

Consider a system with the initial wave function, \I>o> and let (f>(E) be the eigenfunction of the Hamiltonian H of the system H\(E)) = E\4{E)).

(2.11)

Let us denote the complete set of commuting observables of this system of which H is a member by (H, A), then we have A\<j>(E)) = a\(E)),

(2.12)

[\cp(E,a)){(E,a)\^)\2da,

(2.15)

then we can write Co(t) as roo

C0(t)=

roo

e-iEtuj(E)dE = / JEmln

e-iEt£J(E)dE,

(2.16)

J-oo

where in the last relation v

;

ja,(£) 10

for^>£; for -oo < E < Emin

(2.17)

v

'

Since UJ{E) is zero for E < Emin, therefore from Paley and Wiener theorem [7] [8] it follows that

(2.18) For the convergence of this integral as t —> oo, it is necessary for Co(t) to behave as |ln|Co(t)|| -^ Bt2~p as £->• oo, p > 1, (2.19) and since Co(t) -> 0 as t ->• oo, therefore In |Co(i)| is negative and P(t) = |C0(*)|2 -^ exp[-Ctq],

as i ^ oo, q < 1,

(2.20)

where in this relation C is a positive constant. Equation (2.20) shows that asymptotically the decay is not exponential and the probability of finding the system in its initial state P(t) tends to zero slower than an exponential [9] [10] [11]. This argument is quite general and is applicable to different decaying systems. The reason for the breakdown of the exponential decay law after a long time is due to the fact that the part of the system that is decayed moves away from the decaying part so slowly that there is interference between these two parts. This interference is responsible for the behavior shown by

Quasistationary States

13

Eq. (2.20). However if the motion of the decayed part is accelerated, e.g. by the presence of a constant force, then asymptotically, the system can decay exponentially [13]. A model which shows this type of decay is discussed in Section(24.4). Now let us examine the behavior of P(t) at the initial stages of decay, i.e. as * ->• 0. From Eq. (2.16) it follows that f

roo

/•OO

/

\u{E)\dE=

dE \(ct>(E,a)\^o)\2da = (o\^Q) = l.

(2.21)

Since /•OO

/

\u(E)\dE v{r) be the solution of the radial Schrodinger equation for the / = 0 partial wave with the potential barrier V(r),

HMr) = ( - 2 ^

+ V^)

^

=I^r)'

(* = !)

( 2 - 32 )

and let \&o(r) represent the initial wave packet which is localized behind the barrier. To find the time evolution of this wave packet we expand it interms of p(r), roo

*o(r) = /

cp(f>p(r)dp,

(2.33)

*o(r')$(r')dr'.

(2.34)

where cp= J0

Since 4>p{r) is an eigenfunction of the Hamiltonian with the eigenvalue 2

$-, therefore r

o i

*(r,t) = JQ c p «^(r)expU-|J dp.

(2.35)

Substituting cp from (2.34) in (2.35) we get tt(r,t) = ^ ° ° e x p \ - i ^ P{r)dpj™Mr'WP{r)dr.

(2.36)

Quasistationary States

15

V(r)

I I* I 0

a

r _ •

b

Figure 2.1: A particle trapped in the range 0 < x < a can escape to infinity by tunneling Finally by multiplying (2.36) by \&o(r) Eq. (2.14); C0(t) = JQ exp -i£-

a n d integrating

dp ^

tfo(r

over r we find Co(t)

Wp[r )dr

.

(2.37)

Let us consider the specific problem where a wave packet is trapped behind a barrier and then by tunneling it escapes to infinity [16] [17] [18]. suppose that a particle of mass m is initially confined to a segment 0 < x < a behind a rectangular potential of height VQ and width b — a. The wave function is obtained from the solution of the Schrodinger equation and is

{

• ^ y sin(pr) Ae^r + Be-^ Ce*pr

+ De-ipr

for 0 < r < a for a < r < b ,

for

&


(r)dr = 5(p-p).

(2.40)

16

Some Basic Questions

By imposing the boundary conditions, i.e. the continuity of the logarithmic derivative of the wave function at the points x = a and x = b, we can find the constants A, B, C and D in terms of N(p)

A = - ^ y [sin(pa) + ^ cos (pa)] ,

(2.41)

B=j

(2.42)

~ [sin(po) - ^ cos(pa)] ,

H^W+K1-*)*^

(2-43) (2.44)

and

N2(P) = ^ f1 + ^J) { [ sin (H + * C O S M ] 2 e