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Subtleties in Quantum Mechanical Metastability

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Subtleties in Quantum Mechanical Metastability 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 <x<aand x>aby the in- dices 1 and 2, respectively, the corresponding wave func- Since the corresponding value of B/A is very small (∼ tions ϕj (x)(j=1,2) satisfy the free-particle Schr¨odinger n/λ), one may be tempted to say that the probability of equation: finding the particle outside the well is negligible in com- parison with the probability of finding the particle inside 2 1 d the well. Normalizing ψ in such a way that the latter − ϕ (x)=Eϕ (x). (8) n 2 dx2 j j equals one when t = 0, the probability of finding the particle inside the well at time t,ifitwereinthen-th Since the wall at the origin is impenetrable, ϕ (0) must 1 decaying state at t =0,wouldbe be zero; the solution of Eq. (8) which obeys this bound- a ary condition is 2 Pn(t)= |ψn(x, t)| dx =exp(−Γnt). (18) √ 0 ϕ1(x)=Asin kx (k = 2E ). (9) Z The trouble with this interpretation is that Im kn ≡ 1,6,13 To determine ϕ2 (x), we follow Gamow’s reasoning −Kn/2 < 0, and so ψn(x, t) diverges exponentially as and require ϕ2(x) to be an outgoing wave. Therefore, we x →∞, since, according to (10), select, from the admissible solutions of Eq. (8), 2 2 |ψn(x, t)| = |Bn| exp(−Γnt + Knx) (19) ikx ϕ2(x)=Be . (10) outside the well. Because of this “exponential catastro- phe”, the decaying states are nonnormalizible and, there- The wave function must be continuous at x = a,sothat fore, cannot be accepted as legitimate solutions of the ϕ(a)=ϕ (a), or 1 2 Schr¨odinger equation (although one can find in the liter- B ature the assertion that they are “rigorous” solutions of = e−ika sin ka. (11) 14 A the time-dependent Schr¨odinger equation ). 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.
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