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I am wondering about the wave/particle duality of an alpha particle in vacuum. Suppose a U-238 nucleus emits an alpha-particle in vacuum. Is the alpha particle initially a spherical wave propagating in all directions? Does the wave function of the alpha particle collapse only later, when it interacts with a remote object, like an atom or a detector? Does the recoil of the mother/daughter nucleus occur when the alpha particle is emitted, or later, when the wave function collapses?

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If you measure the recoil immediately after the emission, then you will know immediatly know which direction the particle is going in. If you wait until the first atom has been ionized then you will know which direction the nucleus is recoiling in without needing to measure it. The recoil and the ionization are correlated events. One implies and is implied by the other.

Actually this situation is one of the clearest example that shows that the notion of wavefunction collapse is unneccesary, but is a useful approximation. The best account I know of this is in Schiff's Quantum Mechanics and is based on earlier work by Nevill Mott. The idea is compute the amplitude for the ionization of two atoms by the emitted s-wave electron. He shows that the amplitude is zero unless the two atoms and the nucleus emitting the s-wave alpha particle all lie on a straight line, and that the amplitude is same for all such linear arrangements. Thus, once one atom is ionized you know that only other atoms on the nucleus-first-atom line will can be ionized i.e the ionization trail is a straight line. Showing that the two atoms and nucleus need to be a straight line requires second order perturbation theory in the Hilbert space of the alpha-particle's motion and internal states of the two atoms.

If you assume the alpha particle s-wave wavefunction "collapses" to a narrow beam at the moment that the first atom is ionized you can get away with first order perturbation in just the alpha particle wavefunction and the internal state of the second atom. The full "non-collapse" calculation shows that the first event is inevitably correlated with with the second and so on. The bottom line is that "wavefunction collapse" is unnecessary to get the straight line track as long as you use a big enough Hilbert space, but if you are lazy you can use the "collapse" notion to save your labour.

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