Why no Bound-Free superposition states in QM ?

[Widdekind]

Can QM admit the possibility of particle states, which are super-positions, of bound-states (E<0) and free (plane-wave, E>0) states ??

 

I understand, that for a free particle (E>0), incident on a potential well, to become bound into that well (E<0), would require a discontinuous, von Neumann Type 1 Process, "quantum jump" -- with an associated photon emission.

 

Now, imagine that the Overlap Integral (< F | B >), between the free state (F) and bound-state (B), was (say) 10%, so that the Transition Probability was 10%² = 1%. Imagine, further, that that 10% portion of the Wave Function were to "quantum jump", down into the bound state, "on its own" (emitting a 1% strength photon). Would not the application, of a discontinuous "jump" process, to a previously continuous Wave Function, necessarily introduce discontinuities, into said Wave Function ("at the edges", where the Localized bound-state "ended") ? And, aren't all Wave Functions required to be smooth & continuous, so as to furnish well defined second derivatives ?

 

It seems, qualitatively and w/o mathematical rigor, that, insofar as a discontinuous "quantum jump" transition is required, to "arrest" a free-state (E>0, spatially varying phase) wave packet, into a bound state (E<0, spatially uniform phase), then the only way to apply that discontinuous process, to the free 'particle', and still wind up with a smooth, continuous, & computable wave packet, would be to apply that said discontinuous "jump" procedure, to the whole wave function, completely "relocating & repackaging" the same, down into the potential well.

 

Any application, of a discontinuous procedure, in a 'pieces parts' manner, might necessarily introduce mathematically impossible discontinuities into the wave function.

Edited August 26, 2010 by Widdekind

[Bob_for_short]

If you scatter a projectile from an atom (a bound relative motion state), then the final atomic state is a superposition of all excited atomic states allowed by the energy conservation law. These excited states may include the ionized states (free relative motion of atomic components).

[Widdekind]

If you scatter a projectile from an atom (a bound relative motion state), then the final atomic state is a superposition of all excited atomic states allowed by the energy conservation law. These excited states may include the ionized states (free relative motion of atomic components).

 

Wow, does that involve "entanglement", of the scattering projectile, with the atomic wave function(s) ? If, then, the scatterer is later "observed", does such "measurement" cause the entangled atomic wave functions to collapse, into one particular state (ionized or otherwise) ?

[Bob_for_short]

Wow, does that involve "entanglement", of the scattering projectile, with the atomic wave function(s) ? If, then, the scatterer is later "observed", does such "measurement" cause the entangled atomic wave functions to collapse, into one particular state (ionized or otherwise) ?

 

Yes in principle and no in reality. To obtain entanglement one has to use the conservation laws. For that the initial atomic state and that of a projectile have to be well determined (by the preparation devices). After scattering the projectile energy loss should be measured very accurately to judge in what energetic state is the target atom in the final state. Resolving the projectile energy is a very hard problem. So in practice it is not done. In practice it may be easier to observe the target atom state directly (traces of ion and electron) or indirectly (atomic radiation).

[Widdekind]

Yes in principle and no in reality. To obtain entanglement one has to use the conservation laws. For that the initial atomic state and that of a projectile have to be well determined (by the preparation devices). After scattering the projectile energy loss should be measured very accurately to judge in what energetic state is the target atom in the final state. Resolving the projectile energy is a very hard problem. So in practice it is not done. In practice it may be easier to observe the target atom state directly (traces of ion and electron) or indirectly (atomic radiation).

 

Atomic radiation would constitute an "optical micro-signal" (s), which could, in principle, be amplified into a "macro-signal" (S), which human scientists could observe. That would also constitute an "irreversible act of amplification", which would "register" the phenomena, and coincide with wave function collapse, to quote physicist John Wheeler. Would, then, if only in theory, the (optical) micro-signal also collapse the entangled scatterer into a specific state of its own ?

[Bob_for_short]

Atomic radiation would constitute an "optical micro-signal" (s), which could, in principle, be amplified into a "macro-signal" (S), which human scientists could observe. That would also constitute an "irreversible act of amplification", which would "register" the phenomena, and coincide with wave function collapse, to quote physicist John Wheeler. Would, then, if only in theory, the (optical) micro-signal also collapse the entangled scatterer into a specific state of its own ?

Yes, I think so, except I do not like the term collapse. Any observation is getting a specific information. Another matter that one-time measurement may not contain all information about the subject in question (wave function).

 

Even in classical statistics every single observation has only one issue whereas many observations give the issue distribution (probability density). We do not call a one-time issue of dice experiment a "probability collapse". The notion of probability belongs to an ensemble, it is its characteristic.

[Widdekind]

Yes, I think so, except I do not like the term collapse.

 

I would gather, that the principal objection to wave function "collapse", is the super-luminal instantaneity, of such "quantum jumps" ? Even Erwin Schrodinger, a founding father of QM, hated "these damned quantum jumps" [direct quote]. Never-the-less, even so, suspending disbelief on such super-luminality, for 10 seconds [9... 8... 7... ], are there any other (major) objections, to regarding wave functions as real, and their "collapses" as equally real (if disconcertingly discontinuous) ?

 

If you regard the [math]\Psi[/math] as real, then wave functions expand (Type 2 process, SWE) and contract (Type 1 process, "quantum jump"), a little like a squid splaying its tentacles, and then squirting away, as it swims (cf. particle tracks, thru cloud chambers). Were one want (desperately, perhaps) to view the [math]\Psi[/math] as real, one would require some kind of "two-component medium" (a little like the O₂ & N₂ in Earth air), thru which light & matter waves would propagate, and whose coupled oscillations were [math]\pi / 2[/math] out-of-phase (the complex-valued SWE can be recast, as two real-valued, if coupled, PDEs -- the "imaginary" number "i" merely amounts to a convenient accounting trick). The [math]\Psi[/math] (for a single particle) assigns to every point in space two real numbers, whose positive & negative values could correspond to "over-" & "under-" densities, in the "two-component aether" (at which point, I appeal to John Bell, who admitted the possibility of this position [brown&Davies. Ghost in the Atom.]).

 

If you scatter a projectile from an atom (a bound relative motion state), then the final atomic state is a superposition of all excited atomic states allowed by the energy conservation law. These excited states may include the ionized states (free relative motion of atomic components).

 

General Principle of QM -- collisions do not cause "collapses" (??)

 

Among all interactions between a macroscopic object & various other systems, very few are actually measurements. The window of a bubble chamber is bombarded continuously by air molecules, and none of these collisions ends in a measurement. One must therefore specify what is peculiar in the interactions giving rise to a measurement.

 

R.Omnes. Interpretation of QM, pg. 329.

And, double-slit experiments, performed with sliding slit-barriers, which are "kicked" & displaced some distance to the left & right, as the electron matter waves penetrate the slits, en route to the macro-detector screen (D) -- wherein they are absorbed into one of the micro-detector phosphor grains (d) -- still show interference effects, essentially similar to that seen with a fixed slit-barrier (zero "kick" displacement) (T.Bastin. Quantum Theory & Beyond, pp. 43-44.).

 

And, moreover, every "entanglement" interaction involves a direct collision, with wave functions overlapping. Thus, if 'particle' collision caused "collapse", then "entanglements" could never occur, in the first place.

[Widdekind]

If you scatter a projectile from an atom (a bound relative motion state), then the final atomic state is a superposition of all excited atomic states allowed by the energy conservation law. These excited states may include the ionized states (free relative motion of atomic components).

 

The projectile can, apparently, be a photon as well:

 

Though we've been speaking of "observation", we've not really said what constitutes an observation; it's ultimately a controversial issue. When a photon bounces off an isolated atom, does that photon observe the atom?

 

For a photon bouncing off an atom, there is a clear answer: The photon does not observe the atom. After the encounter, the photon is a wave of probability moving off in all directions. The photon and atom are together in a superposition state, that includes all possible positions of the atom before [??] their encounter. This can be confirmed with a complex two-body interference experiment. According to the Copenhagen interpretation, only when a macroscopic measuring instrument records the direction along which the photon came away from the atom does the existence of the atom in a particular position become a reality. Only then is the atom's position observed...

 

Bouncing a photon off an atom does not create the atom's position until the photon is detected.

 

Rosenblum & Kuttner. Quantum Enigma, pg. 103.

Apparently, too, were one able to detect the electron before the photon, the wave function collapse of the former would trigger that of the latter, so that any residual patterns, in the electron's wave function, would still be observable. (??)

Edited September 3, 2010 by Widdekind