Locality/nonlocality for bound states - a question

[pawelsobko]

A recent preprint on Time in Quantum Theory
( http://www.rzuser.uni-heidelberg.de/~as3/TimeInQT.pdf ) by Dieter H Zeh has brought my attention to the question of the `speed of quantum changes'. While the classical discussions of nonlocality in Quantum Mechanics (QM) and consequences of Bell's Theorem are widely published, there are some other situations where nonlocality is rather hard to grok.

Consider a hydrogen atom in excited state. The electron wavefunction has some specific form, extending with exponentially vanishing factor, to infinity. Now, when the atom emits a photon (preferably for this analysis in spontaneous emission) the wavefunction changes.


The appropriate passage by Zeh is as follows:
A wave function(al) obeying a relativistic Schrödinger equation never
propagates faster than light with respect to the underlying presumed absolute spacetime. Recent reports of apparently observed superluminal phenomena were either based on inappropriate 'clocks', or on questionable interpretations of the wave function. For example, the exact energy eigenstate of a particle, bound to an attractive potential in a state of negative energy E = -|E|, would extend to spatial infinity according to exp(-√|E|r) outside the range of the potential. It has therefore been claimed to be able in principle to cause effects at an arbitrary distance within any finite time [10]. However, if the wave function of the bound system forms dynamically (according to the Schrödinger equation rather than by quantum jumps), it can only subluminally approach the exact eigenstate with its infinite exponential tail. This time-dependence requires a minimum energy spread that is in accord with the timefrequency
Fourier theorem.

When one looks up treatments of emission a little more detailed than simple Bohr pictures (in which, yes, the wavefunction changes instantaneously throughout the whole space!) one may get some sort of time dependence, especially for forced emission (as there is external element to the Hamiltonian - the external EM field. But the real question is for spontaneous emission in zero field?

Question: does the wavefunction change at the same moment in the whole space? Or, as Zeh suggests, is there a `wave' of changing wavefunction, spreading our from the atom?

 

[beautiful1]

pawelsobko,

I do not know if there is a definite answer, as the solution may depend on whether you think of the wavefunction as a statistical tool or a 'real' thing.

Either way, I have read a report about how Gisin's group tried to measure the 'speed' of quantum teleportation, with the hope of seeing when the state of a system changes. I beelive that the result was that the state changed faster than everything else. See:
Sundays in a Quantum Engineer's Life
http://arxiv.org/abs/quant-ph/0104140

Hope that helps.
b-

 

[mgelfan]

Question: does the wavefunction change at the same moment in the whole space? Or, as Zeh suggests, is there a `wave' of changing wavefunction, spreading our from the atom?

Isn't this one of those questions that the principles of quantum theory assure can never be answered ... no matter how one interprets the reality of the wavefuntion?

 

[pawelsobko]

pawelsobko,

I do not know if there is a definite answer, as the solution may depend on whether you think of the wavefunction as a statistical tool or a 'real' thing.

Either way, I have read a report about how Gisin's group tried to measure the 'speed' of quantum teleportation, with the hope of seeing when the state of a system changes. I beelive that the result was that the state changed faster than everything else. See:
Sundays in a Quantum Engineer's Life
http://arxiv.org/abs/quant-ph/0104140

Hope that helps.
b-

Thanks. I'll go through Gisin's paper. I agree that the state of the matter seems to be somewhat undecided. This is why I asked...
If you are interested there are more papers that touch the subject, for example:
Hegerfeldt, G. C. Instantaneous Spreading and Einstein Causality in Quantum Theory, Annalen Phys., 1998, 7, 716-725 http://lanl.arxiv.org/pdf/quant-ph/9809030,
Hegerfeldt, G. C. Horzela, A. & Kapuscik, E. (ed.) Particle localization and the Notion of Einstein Causality Extensions of Quantum Theory, Apeiron, Montreal, 2001 http://lanl.arxiv.org/pdf/quant-ph/0109044,
Schulman, L. S. Muga, J. G.; Sala Mayato, R. & Egusquiza, I. L. (ed.) Jump Time and Passage Time: The Duration of a Quantum Transition Time in Quantum Mechanics, Springer-Verlag, Berlin, 2002, 99-+ http://arxiv.org/pdf/quant-ph/0103151,
Schulman, L. S. Observational line broadening and the duration of a quantum jump J. Phys. A: Math. Gen., 1997, 30, L293-L299 http://www.iop.org/EJ/S/UNREG/q54YpQjj7HhELDA.xUqjiw/article/0305-4470/30/9/006/a709l6.pdf,
Uffink, J. The rate of evolution of a quantum state American Journal of Physics, 1993, 61, 935-936 http://www.fys.ruu.nl/~wwwgrnsl/jos/publications/evolution/evol3.pdf ,

 

[mgelfan]

... I agree that the state of the matter seems to be somewhat undecided. This is why I asked ...

I don't think that the state of the matter (regarding your question) is undecided. Or, if it is, it doesn't need to be.

Wavefunction collapse is, by definition, instantaneous and nonphysical. It occurs in the unitary space where quantum states evolve, not in the 3D space where empirical data is gathered.

 

[pawelsobko]

I don't think that the state of the matter (regarding your question) is undecided. Or, if it is, it doesn't need to be.

Wavefunction collapse is, by definition, instantaneous and nonphysical. It occurs in the unitary space where quantum states evolve, not in the 3D space where empirical data is gathered.

Whether the transition is in wavefunction picture or in abstract Hilbert space, the consequences should be the same. Or we would get different QM predictions depending on what representation we use. This would spell trouble for QM...

But even in Hilbert space, abstracting from any physical spacetime representation, we can ask the question whether the transition is really instantaneous. In my search I have not found any convincing argument that it must be so.

Whether using GRW or Jadczyk's EEQM one may consider the time between "localisation" events and the time of such events.

 

[mgelfan]

Whether the transition is in wavefunction picture or in abstract Hilbert space, the consequences should be the same. Or we would get different QM predictions depending on what representation we use. This would spell trouble for QM...

But even in Hilbert space, abstracting from any physical spacetime representation, we can ask the question whether the transition is really instantaneous. In my search I have not found any convincing argument that it must be so.

Whether using GRW or Jadczyk's EEQM one may consider the time between "localisation" events and the time of such events.

The transition from one physical state to another, different, physical state can't be instantaneous. Such transitions involve changes in the positions of the objects comprising the physical state(s). There is some time interval associated with any measurement operation.

However, if we're talking about changes in an imaginary space, then instantaneous transitions are no problem at all. The evolution of quantum states occurs in an imaginary space. Quantum states are not real physical states. This is precisely why quantum nonlocality, say, is no threat to relativity.

I think your original question is a matter of interpretation (taste) and semantics, notwithstanding that answering it one way or the other might give some sort of satisfaction.

I don't see any interpretation or semantic adjustment which gives the question any physical meaning. One can take quantum states to be real physical states, one can take wavefunctions to be real physical waves in some real physical medium (of unknown and unknowable structure), but that would be a matter of faith.

 

[pawelsobko]

The transition from one physical state to another, different, physical state can't be instantaneous. Such transitions involve changes in the positions of the objects comprising the physical state(s). There is some time interval associated with any measurement operation.

However, if we're talking about changes in an imaginary space, then instantaneous transitions are no problem at all. The evolution of quantum states occurs in an imaginary space. Quantum states are not real physical states.

This is strict Copenhagen interpretation. A lot of people accept it (I don't). But my initial question remains valid even in this interpretation. Let me rephrase it as follows: is it possible to measure, in any way, the difference in the state of the atom after emitting the photon, close to the atom and, say, at Betelguese? Would such measurements give the same time, regardless of the place/distance from the center of the atom? Or would there be measurable differences, which in turn might be interpreted by "spreading of change in nonphysical entity such as wavefunction"?

I think your original question is a matter of interpretation (taste) and semantics, notwithstanding that answering it one way or the other might give some sort of satisfaction.

I don't see any interpretation or semantic adjustment which gives the question any physical meaning. One can take quantum states to be real physical states, one can take wavefunctions to be real physical waves in some real physical medium (of unknown and unknowable structure), but that would be a matter of faith.

Remember, for forty years after EPR paper the topic seemed purely philosophical discussion between Einstein and Bohr. Then in late 1970's it became, reluctantly, a topic of experiments, then triumphant Aspect et al QM victory and now we have industry of quantum cryptography and computing.
So the border of philosophy and experiment is shifting. Faith may be decided, maybe five years from now, maybe today, by some smart experimentalist...