In article <1992May22.195706.19
...@midway.uchicago.edu> j
...@midway.uchicago.edu writes:
>Exponential decay is in fact a theoretical necessity. It is a generic
>quantum mechanical feature of problems in which you have a discrete
>state (e.g. an excited atomic or nuclear state) coupled to a continuum
>of states (e.g. the atomic or nuclear system in the ground state and
>an emitted photon flitting around somewhere). There is nothing ad hoc
>about it. The original paper is Weisskopf & Wigner, 1930, Z. Physik,
>63, 54. If you can't get a translation from German, (or don't speak
>German), see Gasiorowicz, "Quantum Physics" 1974 (Wiley), pp 473-480,
>or Cohen-Tannoudji, Diu, & Laloe, "Quantum Mechanics" 1977 (Wiley),
>vol 2, pp 1343-1355.
>The essence of the result is the effective modification of the energy of
>the excited state by a small complex perturbation, E --> E + (dE - i*R/2)
>where dE is the small radiative energy correction (Lamb shift) and R
>is the decay rate. The time dependent phase factor is thus also modified:
>exp(-i*E*t) --> exp[-i*(E+dE)*t]exp[-R*t/2]. This is the source of the decay;
>probabilities, which go as the square of the amplitudes, will exhibit a time
>dependence exp[-R*t].
This is indeed the conventional wisdom. Let me begin by saying:
1) I agree that the exponential decay law is backed up by theory in this
sort of situation and is far from an ad hoc "curve fitting" sort of
thing.
2) The exponential law is apparently an excellent approximation, and as
far as I know no deviations from it have ever been observed. Here I
am not talking about the (necessary) deviations due to finite sample
size. I am talking about deviations present in the limit as the sample
size approaches infinity.
3) If you ever wanted someone to actually calculate a decay rate for
you, I'm sure Graziani would do a whole lot better job than I would.
What follows has nothing to do with the important job of getting an
answer that's good enough for all practical purposes. It is a matter of
principal (my specialty). There's no real conflict.
Okay. So, Graziani has offered the conventional wisdom, what everyone
knows about radioactive decay, that it is a "theoretical necessity".
It's precisely because this is so well-entrenched that I thought I
should point out that one can easily prove that quantum-mechanical decay
processes cannot be EXACTLY exponential. There are approximations in
all of the arguments Graziani cites.
Let me just repeat the proof that decay processes aren't exactly
exponential. It uses one mild assumption, but if the going gets rough I
imagine someone will raise questions about this assumption. It'd be
nice to get a proof with even weaker assumptions; I vaguely recall that
one could use the fact that the Hamiltonian is bounded below to do so.
This is just the proof that Robert Israel gave a while ago (an improved
version of mine).
Let psi be the wavefunction of a "new-born radioactive nucleus",
together with whatever fields that are involved in the decay. Let P be
the projection onto the space of states in which the nucleus has NOT
decayed. Let H be the Hamiltonian, a self-adjoint operator. The
probability that at time t the system will be observed to have NOT
decayed is
||P exp(-itH) psi||^2
The claim is that this function cannot be of the form exp(-kt) for all
t>0, where k is some positive constant.
Just differentiate this function with respect to t and set t = 0.
First, rewrite the function as
<exp(-itH) psi, P exp(-itH) psi>,
and then differentiate to get
<-iH exp(-itH) psi, P exp(-itH) psi> +
<exp(-itH) psi, -iPH exp(-itH) psi>
and set t = 0 to get
<-iH psi, psi> + <psi, -iH psi> = 0
Here we are using P psi = psi. Since we get zero, the function could
not have been equal to exp(-kt) for k nonzero.
That should satisfy any physicist. A mathematician will worry about why
we can differentiate the function. This is a simple issue if you know
about unbounded self-adjoint operators. (Try Reed and Simon's Methods
of Modern Mathematical Physics vol. I: Functional Analysis, and vol. II:
Fourier Analysis and Self-Adjointness.) For the function to be
differentiable it suffices that psi is in the domain of H. For
physicists, this condition means that ||H psi|| < infinity.
[Let me put in a digression only to be read by the most nitpicky of
nitpickers, e.g. myself. An excited state psi, while presumably an
eigenvector for some "free" Hamiltonian which neglects the interactions
causing the decay, is not an eigenvector for the true Hamiltonian H,
which of course is why it doesn't just sit there. One might worry,
then, that the eigenvector psi of the "free" Hamiltonian is not in the
domain of the true Hamiltonian H. This is a standard issue in
perturbation theory and the answer depends on how singular the
perturbation is. Certainly for perturbations that can be treated by
Kato-Rellich perturbation theory any eigenvector of the free Hamiltonian
is in the domain of the true Hamiltonian H, cf. Thm X.13 vol. II R&S.
But I claim that this issue is a red herring, the real point being
that any state we can *actually prepare* has ||H psi|| < infinity.
Instead of arguing about this, I would hope that any mathematical
physicists would just come up with a theorem with weaker hypotheses.]
As Israel pointed out, this argument shows what's going on: when you are
SURE the nucleus has not decayed yet (i.e., it's "new-born"), the decay
rate must be zero; the decay rate then can "ramp up" very rapidly to the
value obtained by the usual approximate calculations.
Physicists occaisionally mistrust mathematicians on matters such as
these. Arcane considerations about the domains of unbounded
self-adjoint operators probably only serve to enhance this mistrust,
which is ironic, of course, since the mathematicians are simply trying
to avoid sloppiness. In any event, just to show that this isn't
something only mathematicians believe in, let me cite the paper:
Time scale of short time deviations from exponential decay, Grotz and
Klapdor, Phys Rev. C 30 (1984), 2098-2100.
"In this Brief Report we discuss critically whether such quantum
mechanically rigorously demanded deviations from the usual decay
formulas may lead to observable effects and give estimates using the
Heisenberg uncertainty relation.
It is easily seen that the exponential decay law following from a
statistical ansatz is only an approximation in a quantum mechanical
description. [Gives essentially the above argument.] So for very small
times, the decay rate is not constant as characteristic for an
exponential decay law, but varies proportional to t. [....]
Equations (2) and (3) tell us that for sufficiently short times, the
decay rate is whatever [arbitrarily - these guys are German] small.
However, to make any quantitative estimate is extremely difficult.
Peres uses the threshold effect to get a quantitative estimate for the
onset of the exponential decay [...] Applying this estimate to double
beta decay yields approximately 10^{-21} sec, which is much too small to
give any measurable effect. [They then go on to argue with Peres.]"
This is all I want to say about this, unless someone has some nice theorems
about the allowed behavior of the function
||P exp(-itH) psi||^2
when H is bounded below and psi is not necessarily in the domain of H.
(This would probably involve extending t to a complex half-plane.)
