Fermilab

Dear J. Oldendick,
Your questions were forwarded to me. I hope you find these answers helpful. You were right that the answers to the questions about resonances and atomic levels are related, and that the uncertainty principle is involved. Let me get back to them. First you ask,

"Why is the cross section for compound nucleus formation not zero between resonances?"

In case you are asking this because you've seen a plot of nuclear cross sections, let me point out that the experiment cannot distinguish the multi-particle state, into which the resonance decays, from non-resonant production of that state. Let me sketch a picture of what I mean:

Here the slashes are meant to show incoming and outgoing particles (e.g. original nucleus and neutron coming in, and a smaller nucleus and an alpha particle going out). The key is that the non-resonant mechanism can occur at any energy.

Often, the coupling strength of the non-resonant production is so small that it can be neglected. This happens in the interaction of the E-M field with atoms. I don't think you were really asking about non-resonant production, so let's neglect it for the rest of this e-mail.

"But doesn't [the shape of the resonance curve] violate fundamental quantum theory? ... How can an atom absorb energy that does not correspond to an allowed excited state? Moreover, most texts treat atoms in an E-M field as a dipole. But doesn't changing the position of the electron cloud relative to the nucleus also change the energy of the atom? How can that be reconciled with quantum mechanics?"

These questions, as you suspected, all boil down to the uncertainty principle. Indeed, the uncertainty relations

delta x times delta p >= hbar delta E times delta t >= hbar

are more fundamental to quantum mechanics than the notion that a nuclear or atomic level has a single energy. (I will address why textbooks say they do below.) Look at the second uncertainty relation, between energy and time. It says that the precision of the energy of a state can only go to zero if you can, and do, look at it infinitely long. With a resonance, or an excited state, you cannot. The compound nucleus breaks up and the excited state drops down to a lower level. So, there should be a relation between the width in the resonance formula, Gamma = sqrt(f), and the halflife of the state. Indeed,

halflife = n hbar/Gamma,

where n is an uninteresting numerical factor (with ln(2) and so on). So, rather than contradicting fundamental quantum mechanics, the finite width is _required_ by fundamental quantum mechanics, namely the uncertainty principle. In many cases the width is much much smaller than the resolution of any apparatus used to measure the energy splittings. Then it is easier, even if a bit sloppy, to neglect the width and speak of only one allowed energy. That's what textbooks do when they say there's only a handful of allowed energies. In more precise language the difference between the quantum and a classical theory of an atom is this:

classical: physical states are arbitrarily close together in energy; any energy is thus, a priori, equally probable.

quantum: physical states have isolated (central) energies with well-defined energy spreads (widths); energies far (compared to widths) from central energies are highly improbable (though not impossible).

Saying the atom "never" experiences a transition to one of these intermediate energies is just short for so improbable that it won't happen during the course of the measurement (or the lifetime of the measurer, or the lifetime of the universe...).

Thank-you for your interest in physics and Fermilab,

Andreas Kronfeld Theoretical Physics Department Fermilab

Dear Andreas Kronfeld,
Thank you for your response to my questions. However, my questions concerned non-resonant absorption of energy (atoms) and non-resonant formation of a compound nucleus. I understand completely that the shape of a resonance curve is due to the uncertainty principle. What I am asking is simply:

How can there be non-resonant absorption of E-M radiation by atoms ? I understand that many times it can be neglected but nevertheless the 1/(E-Eo)^2+f factor shows that some absorption still occurs far away from resonance. How can that be? This is exactly what I cannot reconcile with quantum mechanics. Even if the photon energy is far away from the atomic energy levels, absorption still occurs. Similarly, I cannot understand how a compound nucleus can be formed even when the excitation energy is far away from an energy level in the compound nucleus.

Again thanks,
J. Oldendick
jimo@eurekanet.com

Dear J. Oldendick,
Let's clarify the language: when I say non-resonant, I mean the system was never in the state corresponding to the resonance. It seems to me that when you say non-resonant, you actually mean off the center (energy Eo) of the resonance, but scattering (i.e. absorption and, later, reemission) of through the resonant state. It is important to realize that the resonance is a physical state of finite lifetime, with an excitation spectrum 1/[(E-Eo)^2 + f]. Don't confuse it with the center, or peak, of the curve, at E = Eo. OK, back to your question, ``How can it be?'' In fact, detailed measurements of resonances reveal that the formula is correct: atoms do absorb energy far away from the (center of the) levels. It's just improbable, not impossible. The probability is proportional to 1/[(E-Eo)^2 + f]; the probability at E = Eo + 100 sqrt(f) is 10001 times smaller than at E = Eo. So the real question should be, ``How does quantum mechanics remain consistent with Nature?'' Well, it does, through the uncertainty principle and the connection of the uncertainty principle to the resonance shape, which, you say, you understand completely. Since Nature and quantum mechanics agree, I'm not sure what your problem is. What do you take as the "fundamentals" of quantum mechanics? The uncertainty principle (if you want to be quantitative, the uncertainty relations) is truly fundamental, more fundamental than even the Schroedinger equation. The notion that excitations can occur only at the (centers of) certain levels is an idealization. In the mathematical framework of quantum mechanics, one can cope with the idealization, by solving the Schroedinger equation while neglecting the quantum nature of the E-M field. But a unified treatment, treating both electrons and photons as quantum objects, would end up showing that the levels are not infinitely sharp, but resonances. Fundamentals are rarely the first thing one reads on a subject. Assuming the subject is logical, as physics is supposed to be, the fundamentals are the basic ideas that allow one to reconcile seemingly contradictory end results.

Andreas Kronfeld

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