View metadata, citation and similar papers atSubtleties core.ac.uk in Quantum Mechanical Metastability brought to you by CORE provided by CERN Document Server R. M. Cavalcantia) Departamento de F´ısica, Pontif´ıcia Universidade Cat´olica do Rio de Janeiro, Cx. Postal 38071, CEP 22452-970, Rio de Janeiro, RJ, Brazil
C. A. A. de Carvalhob) Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Cx. Postal 68528, CEP 21945-970, Rio de Janeiro, RJ, Brazil (April 28, 1997)
We present a detailed discussion of some features of quantum mechanical metastability. We analyze the nature of decaying (quasistationary) states and the regime of validity of the exponencial law, as well as decays at finite temperature. We resort to very simple systems and elementary techniques to emphasize subtle aspects of the problem.
I. INTRODUCTION method. Why this method works is the question we try to answer in this paper in a very elementary way.3 Thus, Complex-energy eigenfunctions made their d´ebut in in Section II, we show Gamow’s method in action for a Quantum Mechanics through the hands of Gamow, in very simple potential. Some of the results obtained there the theory of alpha-decay.1 Gamow imposed an “outgo- are used in Section III, where we study the time evolu- ing wave boundary condition” on the solutions of the tion of a wave packet initially confined in the potential Schr¨odinger equation for an alpha-particle trapped in the well defined in Section II. This is done with the help of 4 nucleus. Since there is only an outgoing flux of alpha- the propagator built with normalizable eigenfunctions, particles, the wave function ψ(r, t) must behave far from associated to real eigenenergies. As a bonus, we show the nucleus as (for simplicity, we consider an s-wave)2 that the exponential decay law is not valid for very small times or for very large times. This is the content of Sec- e−iEt+ikr tion IV, where the region of validity of the exponential ψ(r, t) ∼ (r →∞). (1) r decay law is roughly delimited. This boundary condition, together with the requirement Another topic we address in this paper is decay at fi- of finiteness of the wave function at the origin, gives rise nite temperature. This is done in Section V, where we to a quantization condition on the values of k (and, there- study, with the help of an exactly solvable toy model, fore, on the values of E = k2/2). It turns out that such how the decay of a metastable system is modified when values are complex: it is coupled to a heat bath. It is shown that, under suit- able conditions, the decay is exponential, with a decay kn = κn − iKn/2,En=n−iΓn/2. (2) rate Γ given by the thermal average of the Γn’s, It follows that −βn nΓne −Γn t+Knr Γ= . (4) 2 e −βn |ψ (r, t)| ∼ (r →∞). (3) n e n r2 P P Thus, if Γn > 0, the probability of finding the alpha- particle in the nucleus decays exponentially in time. The Although this result appears to be rather obvious, in fact lifetime of the nucleus would be given by τ =1/Γ ,and it is not: the decay of a metastable system is an intrinsi- n n cally non-equilibrium process and, so, there is no a priori the energy of the emitted alpha-particle by n. Although very natural, this interpretation suffers from reason for the decay rate to be given by (4). Finally, in some difficulties. How can the energy, which is an ob- Section VI, we discuss the concept of “free energy of a servable quantity, be complex? In other words, how can metastable phase,” a point where we think there is some the Hamiltonian, which is a Hermitean operator, have confusion in the literature. complex eigenvalues? Also, the eigenfunctions are not The results and ideas presented here are not really normalizable, since Γn positive implies Kn positive and, new, but discussions on these matters usually involve 2 therefore, according to (3), |ψn(r, t)| diverges exponen- the use of sophisticated mathematical techniques, such tially with r. as functional5,6 or complex analysis.7–11 For this reason, In spite of such problems (which, in fact, are closely we have tried to make the presentation as clear as possi- related), it is a fact of life that alpha-decay, as well as ble by resorting to very simple systems and elementary other types of decay, does obey an exponential decay techniques — in fact, techniques that can be found in law, with a rate close to that obtained using Gamow’s any standard Quantum Mechanics textbook.12
1 II. DECAYING STATES On the other hand, the derivative of the wave function has a discontinuity at x = a, which can be determined In order to exhibit Gamow’s method in action, we shall by integrating both sides of (7) from a − ε to a + ε,with + study the escape of a particle from the potential well ε→0 : given by: 2λ ϕ0(a)−ϕ0(a)= ϕ (a), (12) 2 1 a 2 V (x)= (λ/a) δ(x − a)forx>0, (5) +∞ for x<0. from which there follows another relation between A and n B: Motion in the region x<0 is forbidden because of the B e−ika cos ka infinite wall at the origin. The positive dimensionless = . (13) constant λ is a measure of the “opacity” of the barrier at A i − 2λ/ka x = a; in the limit λ , the barrier becomes impene- →∞ Combining (11) and (13), we obtain a quantization con- trable, and the energy levels inside the well are quantized. dition for k: If λ is finite, but very large, a particle is no more con- fined to the well, but it usually stays there for a long 2λ cotan ka = i − . (14) time before it escapes. If λ is not so large, the particle ka can easily tunnel through the barrier, and quickly escape The roots of Eq. (14) are complex; when λ 1, those from the potential well. Metastability, therefore, can only which are closest to the origin are given by8 be achieved if the barrier is very opaque, i.e., λ is very large. For this reason, we shall assume this to be the case 1 2nπλ nπ 2 k ≈ −i (nπ λ) (15) in what follows and, whenever possible, we shall retain n a 1+2λ 2λ only the first non-trivial term in a 1/λ expansion. ≡ κn − iKn/2. To find out how fast the particle escapes from the po- tential well, we must solve the Schr¨odinger equation The corresponding eigenenergies are
2 2 3 ∂ 1 ∂ λ 2 1 nπ (nπ) i ψ(x, t)=− ψ(x, t)+ δ(x−a)ψ(x, t). (6) En = kn/2 ≈ − i (16) ∂t 2∂x2 a 2 a (2λa)2 ≡ − iΓ /2. ψ(x, t)=exp(−iEt) ϕ(x) is a particular solution of this n n equation, provided ϕ(x) satisfies the time-independent (Note that Kn κn and Γn n; these results will be Schr¨odinger equation used later.) The imaginary part of En gives rise to an 2 exponential decay of |ψn(x, t)| , with lifetime equal to 1 d2 λ ϕ(x)+ δ(x a)ϕ(x)=Eϕ(x). (7) 2 − 2 − 2(λa) 2 dx a τ =1/Γ ≈ . (17) n n (nπ)3 Denoting the regions 0
2 ∞ 2 III. TIME EVOLUTION OF A WAVE PACKET G(x, x0; t)= e−ik t/2 ϕ∗(k, x) ϕ(k, x0) dk. (27) Z0 We now return to Eq. (8) and write, for the solution The function ϕ(k, x) is the solution of Eq. (7) correspond- in region 2, instead of (10), the sum of an outgoing plus ing to the energy E = k2/2: an incoming wave:
−ikx ikx 1 A(k)sinkx (x
0 2 2 The wave function at a later time t is given by Knx, Knx ≤ Kna ≈ (nπ) /2λ 1, (33) ∞ we can substitute k for κ in sin kx and sin kx0 in the 0 0 0 n ψ(x, t)= G(x, x ; t) ψ(x , 0) dx , (26) integrand of (32). On the other hand, this is not allowed 0 2 Z for e−ik t/2, since the time t is not bounded. where the propagator, G(x, x0; t), can be written as Let us examine the integrals
3 −ik2 t/2 1 Now we can be more precise on the smoothness of ψ(x, 0); In(t) ≡ dk e 2 2 . (34) (k − κn) + K /4 roughly speaking, the smaller the value of n beyond ZIn n which cn = 0, the better the results above will be. If Kn κn+1 − κn−1, we can safely extend the interval Eq. (37) is valid only inside the potential well. To find of integration to the whole real axis, and carrying out the the wave function outside the well, we must return to integration we find Eqs. (26)–(28) and make x>a:
∞ ∞ −ik2t/2 1 1 −ik2t/2 ikx ∗ −ikx In(t) ≈ dk e 2 2 ψ(x, t)= dk e [e + B (k) e ] A(k) (k − κn) + K /4 2π Z−∞ n Z0 ∞ ∞ iξk a 1 2 dk e iξ /2t × dx0 ψ(x0, 0) sin kx0. (40) = √ dξ e 2 2 2πit (k − κn) + K /4 0 Z−∞ Z−∞ n Z 2π 1 2 Since most of the contribution to the integral in k comes = √ (In + In); (35a) Kn 2πit from the resonances, we may approximate A(k) by (31) 0 ∗ 2 and make an analogous approximation for B (k). Thus, 1 iξ /2t+i(κn−iKn/2)ξ In ≡ dξ e , (35b) (40) becomes −∞ Z ∞ 2 2 iξ /2t+i(κn+iKn/2)ξ ψ(x, t) ≈ dn Jn(x, t); (41a) In ≡ dξ e . (35c) 0 n Z X a −iκn 0 0 0 √ dn ≡ dx ψ(x , 0) sin κnx , (41b) Except for a region of width ∆ξ ∼ t around the point 2πλ Z0 ξ = −κnt, where the phases of the exponentials are sta- ikx −ik2t/2 e tionary, the oscillations of the integrands tend to can- Jn(x, t) ≡ dk e − c.c. . k − κ + iK /2 cel out, giving a very small contribution to the integrals ZIn n n √ (41c) above. If κnt t, such a region is well inside the negative real axis, therefore the second integral can be where In has the same meaning as in Eq. (32), and c.c. neglected in comparison to the first. For the same rea- denotes complex conjugate. son, we can extend the interval of integration of the first Let us concentrate our attention on the integrals integral to the whole real axis, thus obtaining Jn(x, t). Extending the interval of integration to the ∞ whole real axis, and using the same trick as in Eq. (35a), 2π 2 iξ /2t+i(κn−iKn/2)ξ In(t) ≈ √ dξ e we find Kn 2πit −∞ Z 2πi 1 2 2π −i(κ −iK /2)2t/2 2π −iE t Jn(x, t) ≈−√ (J + J ); (42a) = e n n = e n . (36) 2πit n n Kn Kn −x 2 1 iξ /2t+ikn(ξ+x) Substituting this result in Eq. (32), we find Jn ≡ dξ e , (42b) −∞ Z ∞ 2 ∗ −iEnt 2 iξ /2t+ikn (ξ−x) ψ(x, t) ≈ cn e ϕn(x), (37) Jn ≡ dξ e . (42c) n x X Z where ϕn(x)andcn are defined as As in the case of In(t), the second√ integral is negligible in comparison to the first if κnt t,ort 1/n.On 2 a(κn/λ) 2 nπx the other hand, the first integral may be approximated ϕ (x)= sin κ x ≈ sin , (38a) n 2 K n a a by r n r a ∞ 2 √ 2 iξ /2t+ik (ξ+x) −ik t/2+ik x cn = dx ψ(x, 0) ϕn(x). (38b) dξ e n = 2πite n n (43) 0 Z Z−∞ √ Eq. (37) is formally identical to the well known expansion only if κnt − x t. of the wave function in energy eigenfunctions, except for Returning to Eq. (41), we finally obtain the fact that: (1) it is an approximate result and, as such, −iEnt+ikn x ψ(x, t) ≈−2πi dn e . (44) subject to some restrictions, and (2) the energies En are n complex, as a result of which the probability P (t) to find X the particle inside the potential well at time t decreases We see, therefore, that outside the well the wavefunction in time: behaves as a superposition of outgoing waves, in the way a postulated by Gamow. However, the exponential catas- 2 2 −Γnt trophe does not occur here, for Eq. (44) is valid only P (t)= |ψ(x, t)| dx ≈ |cn| e . (39) √ 0 n under the assumption that κnt − x t. Z X
4 IV. BREAKDOWN OF EXPONENTIAL DECAY Thus, Eq. (37) gives
1 The evolution of the wave function requires some P (t) ≈ e−βn−Γnt, (50) 17 Z time to reach the regime of exponential decay; typi- n cally, a time corresponding to many oscillations inside the X potential well (i.e., t 1/n). To be more precise, even and, as already discussed in Sec. IV, after a time of the if this condition is satisfied, the decay is not strictly ex- order τ1 =1/Γ1, the decay would be dominated by the ponential, but a sum of exponential decays, one for each decay of the “false vacuum” — the lowest lying reso- resonance [Eq. (39)]. However, since the lifetime τn is, in nance. It follows that the decay rate is almost insensitive 3 general, a rapidly decreasing function of n (τn ≈ τ1/n to the temperature. in our example), the decay becomes a pure exponential Is this conclusion correct? 19 one after a time of the order of τ1. In the literature one finds the statement that, in such On the other hand, the exponential decay does not a situation, the probability P (t) decays as last forever. After some sufficiently long time, it obeys 7–11,16 1 apowerlaw. To see this, note that for t →∞, P (t)=e−hΓit, hΓi = Γ e−βn. (51) Z n the integral (30) is dominated by small values of k.One n finds, then, for x 5 whose solutions are by (51), instead of (52). (In some sense, that is why one can observe a phenomenon like the Stark effect. In −βE 1 4(Γ1 + e Γ2) the presence of a constant external electric field, the po- λ = (1 + e−βE)Γ 1 1 . ± ± − −βE 2 2 s (1 + e ) Γ ! tential which binds an electron to an atom becomes un- bounded from below, and so the atomic energy levels (57) become metastable, with very large lifetimes if the ex- ternal electric field is small compared to the field of the Expanding the square root, we find (since Γ1,Γ2 Γ) nucleus. The role of the heat bath is played here by the λ ≈ (1 + e−βE)Γ, (58a) quantized electromagnetic field; it is the coupling of the + atom to it that causes the excited states of the atom, oth- −βE Γ1 + e Γ2 erwise stationary, to decay to the ground state. Even for λ− ≈ ≡hΓi. (58b) 1+e−βE a highly excited atom, for which the natural lifetime is relatively high (some milliseconds), the situation is well The corresponding eigenvectors are described by saying that the atom is overdamped, since the ionization (decay) rate is very small if the external a+ 1 a− 1 electric field is small compared to the intratomic field.) ≈ , ≈ −βE . (59) b+ −1 b− e (B) On the other hand, if the system is underdamped (Γ Γ1,Γ2), although the decay is still described by Eq. If nj(0) = nj0 (j =1,2), then, at time t,wehave (37), Eq. (50) is possibly not valid. The reason is that, −βE under such a condition, the system does not “know” the n(t) e n10 − n20 1 1 ≈ e−λ+t temperature of the heat bath. It would have decayed be- n2(t) 1+e−βE −1 fore it could thermalize. In practice, as already argued, n10 + n20 1 the excited states would depopulate much sooner than + e−λ−t. (60) 1+e−βE e−βE the “ground state,” and so the decay rate would be very insensitive to the temperature of the heat bath. Therefore, even if the system is initially not in thermal equilibrium with the heat bath, it thermalizes in a time of the order 1/λ+ 1/λ−. In other words, after a tran- VI. THERMODYNAMICS OF METASTABLE sient time of the order of 1/λ+, both levels decay at the SYSTEMS: A BRIEF DIGRESSION same rate, λ−, equal to the thermal average of Γ1 and Γ2, and their populations soon approach the Boltzmann distribution, Finally, we would like to make a brief digression on a point where we think there is some confusion in the lit- n2(t) −βE erature. It concerns the thermodynamics of metastable ≈ e . (61) systems. As an example of such a system, let us con- n1(t) sider a particle interacting with the potential defined in (2) Γ Γ1,Γ2 (underdamping): now, Eq. (55) may Section II. Its partition function is defined as be approximated by Z≡Tr e−βH 2 ∞ ∞ λ − (Γ1 +Γ2)λ+Γ1Γ2 =0, (62) 2 = dk e−βk /2 dx ϕ∗(k, x) ϕ(k, x), (65) 0 0 whose solutions are λj =Γj (j=1,2). The correspond- Z Z 20 ing eigenvectors are or, alternatively, as a1 1 a2 0 ∞ ≈ , ≈ . (63) −βE b1 0 b2 1 Z = ρ(E) e dE. (66) Z0 The solution of (53) is, therefore, Clearly, the spectral density ρ(E)isgiven,inthiscase, −Γjt 4 nj(t) ≈ nj0 e (j =1,2). (64) by In this case, the levels decouple from each other, and dk δ(0) ρ(E)=δ(0) = √ . (67) each one of them decays with its own decay rate. The dE 2E coupling to the heat bath is so weak that the system is effectively insulated. We conclude, therefore, that there is no sign of metasta- Now we are in position to answer the questions posed bility in the partition function. Now, let us try to define in this section: a “partition function inside the well;” since this is not a (A) If the system is overdamped (i.e., Γ Γ1,Γ2), fundamental concept, more than one definition is possi- it decays as indicated in (51). The analysis of the over- ble. One such definition is inspired by (65); restricting damped case also explains why the decay rate is given the x-integration to the interval [0,a], we have 6 ∞ a 5 −βk2/2 ∗ may be understood by noticing that they are good ap- Z1 = dk e dx ϕ (k, x) ϕ(k, x). (68) 0 0 proximate solutions to the time-dependent Schr¨odinger Z Z equation, although nonuniform ones (i.e., they are not If we make the same approximations we made in Sec- valid in the entire range of values of t and x). tion III, we can reexpress Z as in (66), but now with a 1 In Section V we examined another “well known” re- spectral density given by sult, that the decay rate of a metastable system in con- tact with a heat bath is given by Eq. (51). Although we 1 Γn/2 ρ1(E) ≈ . (69) have used a very simple toy model to discuss this point, π (E − )2 +Γ2/4 n n n we believe it contains the essential physics of the phe- X 16,21 nomenon, at least in the two limiting cases we studied in This kind of spectral density contains some dynami- some detail. The important lesson to be learned here is cal information — resonant levels and decay rates. If the that Eq. (51) is correct (at least in first approximation), latter are small enough, the Lorentzians in (69) can be provided the condition of “overdamping” is satisfied.14,24 approximated by delta functions, and we obtain, there- At low temperatures, where only the lowest lying decay- fore, ing states take part in the process, it is an easy matter to verify if it is so. However, as the temperature increases, −βn −βF1 Z1 ≈ e ≡ e . (70) decaying states of higher energy are excited and begin n X to play an increasingly important role in the overall de- cay. Since the decay rates Γ become larger with n,the If the coupling with the heat bath is strong (in the sense n of Section V), F can be interpreted22,23 asthefreeen- overdamping condition eventually fails to be satisfied by 1 states actively involved in the process of decay. Thus, ergy of the metastable phase. One should not confuse21 this “free energy” with the true equilibrium free energy, one should expect deviations from the decay rate given by Eq. (51). Another source of deviations, not taken into F = −(1/β)lnZ. account in our toy model, is a possible renormalization Another possible definition of the “partition function of the complex energies E , caused by the interaction of inside the well” is inspired in the complex eigenenergies n the system with the heat bath. This may affect Eq. (51) of Gamow’s method: even at low temperatures, for obvious reasons.25 −β(n−iΓn/2) −βF2 Z2 ≡ e ≡ e . (71) n X This definition, although almost identical to (70), pre- ACKNOWLEDGMENTS sents a new and interesting feature: the free energy F2 of the metastable phase is complex! Its real part is essen- tially equal to F1, and so has thermodynamical content, We thank Pavel Exner for pointing out Ref. 5, and but its imaginary part provides dynamical information: Gernot Muenster for pointing out Refs. 9 and 10. This if the Γn’s are small, it is easy to show that work had financial support from CNPq, FINEP, CAPES i and FUJB/UFRJ. Im F ≈− hΓi, (72) 2 2 where hΓi is the thermal average of the Γn’s, as defined in (51). That is the reason why definition (71) is so pop- ular in the literature, although it is no more fundamental than definition (68). a) E-mail: rmc@fis.puc-rio.br b) E-mail: [email protected] 1 VII. CONCLUSION G. Gamow, “Zur Quantentheorie des Atomkernes,” Z. Phys. 51, 204-212 (1928); “Zur Quantentheorie der Atom- zertr¨ummerung,” Z. Phys. 52, 510-515 (1928). In Section III we showed that decaying states, although 2 We use units such thath ¯ = m =1. plagued by the exponential catastrophe, give a fairly 3After this paper was written, we became aware of a paper good description of the decay of a metastable state, pro- by Barry Holstein, in which the content of Sections II and vided some conditions are satisfied. In fact, the main III is discussed using a more general potential. See B. R. result of this paper is that one can compute the decay Holstein, “Understanding alpha decay,” Am. J. Phys. 64, rate solving the time independent Schr¨odinger equation 1061-1071 (1996). 4 0 subject to the “outgoing wave boundary condition,” Eq. Normalizable in the sense that hψk|ψk0 i = δ(k − k ). (10). This is far from being a trivial result, since the 5 E. Skibsted, “Truncated Gamow Functions, α-Decay and corresponding eigenstates are nonphysical. The “effec- the Exponential Law,” Commun. Math. Phys. 104, 591- tiveness” of the decaying states in describing the decay 604 (1986). 7 6 A. Bohm, M. Gadella and G. Bruce Mainland, “Gamow references therein. vectors and decaying states,” Am. J. Phys. 57, 1103-1108 (1989). 7 G. Beck and H. M. Nussenzveig, “On the physical inter- pretation of complex poles of the S-matrix – I,” Nuovo Cimento 14, 416-449 (1960). 8 H. M. Nussenzveig, Causality and Dispersion Relations (Academic Press, New York, 1972), Ch. 4. 9 C. B. Chiu, E. C. G. Sudarshan and B. Misra, “Time evolu- tion of unstable quantum states and a resolution of Zeno’s paradox,” Phys. Rev. D 16, 520-529 (1977). 10 L. Fonda, G. C. Ghirardi and A. Rimini, “Decay theory of unstable quantum systems,” Rep. Prog. Phys. 41, 587-631 (1978). 11 G. Garc´ıa-Calder´on, J. L. Mateos and M. Moshinsky, “Res- onant Spectra and the Time Evolution of the Survival and Nonescape Probabilities,” Phys. Rev. Lett. 74, 337-340 (1995); “Survival and Nonescape Probabilities for Reso- nant and Nonresonant Decay,” Ann. Phys. (N.Y.) 249, 430 (1996). 12 See, for example, E. Merzbacher, Quantum Mechanics (John Wiley & Sons, New York, 1961), Ch. 7. 13 V. I. Gol’danskii, L. I. Trakhtenberg and V. N. Fleurov, Tunneling Phenomena in Chemical Physics (Gordon and Breach, New York, 1988), pp. 11-17. 14 A. J. Legget, “Quantum mechanics at the macroscopic level,” in Chance and Matter, Les Houches session XLVI (1986), edited by J. Souletie, J. Vannimenus and R. Stora (North-Holland, Amsterdam, 1987), pp. 395-506, in partic- ular p. 469. 15 L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1958), §19. 16 A. Patrascioiu, “Complex time and the Gaussian approxi- mation,” Phys. Rev. D 24, 496-504 (1981). 17 It can be shown9,10 that initially the decay is slower than exponential; more precisely, dP/dt =0whent=0. 18 Here a comment is in order: Garc´ıa-Calder´on, Mateos and Moshinsky11 argue that the “nonescape” probability P (t) decays as t−1 when t →∞, in contrast to Eq. (46). How- ever, there is an error in their argument; when properly corrected, it also leads to P (t) ∼ t−3 asymptotically. See R. M. Cavalcanti, “Comment on ‘Resonant Spectra and the Time Evolution of the Survival and Nonescape Prob- abilities’,” preprint quant-ph/9704023, submitted to Phys. Rev. Lett. 19 I. Affleck, “Quantum-Statistical Metastability,” Phys. Rev. Lett. 46, 388-391 (1981). 20 To find the eigenvectors one must, in fact, solve (55) to −βE first order in Γ: λ1 ≈ Γ1 + e Γ, λ2 ≈ Γ2 +Γ. 21 D. Boyanovsky, R. Willey and R. Holman, “Quantum me- chanical metastability: When and why?” Nucl. Phys. B 376, 599-634 (1992). 22 J. S. Langer, “Statistical Theory of the Decay of Metastable States,” Ann. Phys. (N.Y.) 54, 258-275 (1969). 23 J. S. Langer, “Theory of the Condensation Point,” Ann. Phys. (N.Y.) 41, 108-156 (1967). 24 D. Boyanovsky, R. Holman and A. Singh, “A Note on Ther- mal Activation,” Nucl. Phys. B 441, 595-608 (1995). 25 For a discussion of such issues, see U. Weiss, Quantum Dis- sipative Systems (World Scientific, Singapore, 1993), and 8 9