In a quantum system, with time--independent interactions, and  a finite number of degrees of freedom, the answer is No. The reason is that, due to tunneling, no phase transition can take place, which would lead to distinct state spaces. For such state spaces to appear, the system must, not only, have more than one phase, but tunneling between these must be suppressed. This can happen, if one has an infinite number of degrees of freedom-but this is a necessary condition, not a sufficient one.

Of course finite-dimensional spaces of different dimension, aren't equivalent. Infinite-dimensional spaces, on the other hand, may or may not be equivalent. 

For the potential in question the space of states consists of the bound state sector and the scattering state sector. For a rectangular well of finite depth and width, there will be a finite number of bound states, so, just by varying the depth of the well, one can change the space of states in ways that aren't unitarily equivalent. On the other hand, by varying the width of the well, by a finite amount, one doesn't change the number of bound states, therefore the spaces of states of all wells of the same, finite depth, differing finite widths and, of course, same boundary conditions, are equivalent. 

The space of scattering states is infinite dimensional and  varying width or depth by finite amounts doesn't affect it: the spaces labeled by these parameters are unitarily equivalent, so there is just one such space. This is what is meant by the absence of a phase transition. 

Quantum mechanics is not logically inconsistent and it is  possible to understand tunneling in a perfectly consistent way. Tunneling doesn't violate special relativity, for a number of reasons, the first being that quantum mechanics, as discussed here, isn't consistent with special relativity, to start with. This is, however, irrelevant. A major source of confusion is that quantum mechanics is formulated in phase space, whereas particles and wave packets move in spacetime. It isn't easy to describe the map between the two. One way is through the path integral formalism, that can be given a rigorous formulation using a lattice. Quantum mechanics then becomes a one-dimensional quantum field theory, whose properties can be studied mathematically. In particular, the fact that tunneling through finite barriers by quantum fluctuations can describe the absence of phase transitions in low dimensional systems is well known, from the progress made in constructive field theory in the 1970s and 1980s and beyond. 

Many of the difficulties, indeed, can be traced back to confusion regarding unitary equivalence of the space of states of the physical system. 

When the classical potential has multiple minima, the quantum potential has only one minimum, since tunneling occurs-there isn't a barrier at all, it's a semi-classical approximation.  That's why it doesn't make sense to ask about the velocity of a particle tunneling below the barrier, when taking only the leading correction into account. The neglected corrections change the potential completely. The *only* feature of the quantum potential that is captured correctly by the semi-classical approximation, in this case,  is that the energy levels of the full theory display a non-analytic dependence on the parameters of the potential. 

Newtonian mechanics is invariant under Gallilean transformations, relativistic mechanics is invariant under Poincaré transformations. In both cases, all effects, where Planck's constant appears, are neglected, therefore both describe classical mechanics, where the physical configurations are defined as solutions of the (classical) equations of motion.  The former is the limiting case of the latter, when the typical speeds are much smaller than the speed of light. 

When Planck's constant isn't neglected but typical speeds are so small with respect to the speed of light that reativistic effects can be neglected, it turns out that the potential energy, for a system of a finite number of particles (one particle is a special case) can have only one minimum. Since particles can't be created in this, non-relativistic, context, the number of particles is well-defined. If the energy spectrum has spacing that is a non-analytic function of the parameters, therefore of Planck's constant, then the potential, in the semi-classical or classical approximation, has multiple minima and tunneling is said to occur. If the spacing is analytic in Planck's constant, then the potential in the semi-classical and classical approximations, also, has a unique minimum and tunneling doesn't occur. In the  classical approximations the velocity can be computed from the equation of motion, when quantum effects are relevant its distribution can be computed. Tunneling doesn't allow any particle to travel faster than the speed of light, because there isn't any case where such a statement can be shown to make sense. Tunneling's  just a term for describing situations, where the energy spectrum's spacing is non-analytic in the parameters. Whenever velocities can be defined, the non-relativistic approximation is found to be consistent. 

Dear Juan D. Bulnes,

Despite some of the bizarre responses that this post appears to have elicited - your question is an interesting and rather straight forward one to answer.  The simple answer is: yes. there are many systems where a non-trivial modification of the Hilbert space accessible to quantum particles is possible.  And perhaps more interestingly, this can be a very useful action.

In general, Hilbert space dimensionality does not change in elementary proofs and implementations, but this is sometimes a simplification, sometimes a statement of fundamental resources (i.e. a finite number of qubits), and sometimes a lack of creativity.  Let me explain.

The simplest element I can think of to illustrate changing Hilbert space in a discrete representation is that of an optical beamsplitter.  A single photon travelling through space can be described as a single mode (all right - I'm assuming that this is a transform limited single photon pulse of well-defined polarisation).  This is a one-dimensional Hilbert space, and without any other degrees of freedom to modify, is probably fairly trivial.  Let's call this mode |0>, and we'll furthermore ignore the translation of the photon, so anywhere along the path of mode |0> will also be |0>

If the photon impinges on a beamsplitter, it can either be reflected or transmitted.  Assume that the transmitted beam is essentially unaltered, so we'll still call this |0> (we could define this as |0'> to indicate that we've moved to a qualitatively different Hilbert space, but this just adds complexity without giving us new insight).

The reflected part is a new mode, which we can call |1>.  This has doubled the number of states that are accessible to the photon, and not coincidentally has given the opportunity to observe new physics (interference etc.).

In many qubit protocols, the photon subspace is implicitly assumed to have access to both modes, |0> and |1>, but to only occupy mode |0>.  This is an entirely valid way to view the problem, except that it can miss the fact that mode |1> really should contain the vacuum mode in the example I provided, and it also blinds one to the possibility that I can expand the Hilbert space arbitrarily by adding more beamsplitters.

Now we do these actions all the time, and understanding, and even optimising accessible Hilbert spaces is quite important.  

Apart from the issue of introducing elements such as beamsplitters, continuous modifications of potentials give rise to variations in the number of accessible states - for example changing the width of a finite potential well alters the number of bound modes within that well.  One can argue, of course, that the total number of states available is not altered (it is still infinite up to the bounds of causality, but of course this means counting both the bound and the unbound states).

There is a rather beautiful paper - Quantum computing using shortcuts through higher dimensions, Lanyon et al., Nature Physics 5, 134 (2009) - which shows that changing the Hilbert space dimension can provide more efficient quantum gate operations: so controlling Hilbert space dimension is pretty important, and is a widely overlooked resource IMO.

Of course, messing around with Hilbert space dimension (or to be more precise - accessible Hilbert space dimension) can be fraught with difficulties.  Adding extra dimensions is not so tricky, as shown above, but subtracting dimensions requires doing work on the system, and can lead to decoherence and (if not careful) apparently non-unitary evolution.  The reason for saying 'apparently' here is that the evolution won't be non-unitary (it can't be), but it can appear so in the specific subspace that you actually care about.  So this is where we have to stop talking in terms of generalities, and try to zoom in on a particular implementation that you care about.

I hope that this helps.

Dear Akira Kanda,

I am sure that you are trying to do some good here, but instead of railing against physicists and attacking the field, you would be far better served by actually learning some physics and attempting to apply it to problems to test consistency.

Given your posts here, and on other questions, I doubt that you actually want to learn, but in case you do, you may be interested to learn that the cloud chamber (or Wilson chamber if you will) was one of the early, troubling experiments for the new quantum physicists.  Heisenberg in particular was troubled by it because he saw the cloud chamber as producing realistic, classical trajectories, which were incompatible with the other experimental observations (e.g. Stern-Gerlach effect), and seemingly incompatible with succesful quantum mechanical results (e.g. Bohr atom).  

Such a clear, apparent inconsistency troubled him, and indeed caused much concern as to the validity of the new quantum theory.

Inconsistencies like this are what propel science forward, and Heisenberg's resolution based on sequential measurements forms the basis for how we understand evolution and motion in the presence of measurement.  If you want an even better example for understanding how we go from particle-like localisation (and hence classical-like behaviour) to wave-like propagation (and hence non-measured quantum-like behaviour*), then you should visit Aephraim Steinberg in your university (University of Toronto) and ask him to show you one of the most beautiful demonstrations of such effects, which he and colleagues published in Science [Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer, S. Kocsis, B. Braverman, S. Ravets, M.J. Stevens, R.P. Mirin, L.K. Shalm, and A.M. Steinberg, Science 332, 1170 (2011)].

*When I say 'classical-like' and 'quantum-like', what I really mean is 'evolution that we understand as being essentially following classical laws', or 'the evolution that we understand as being quantum' respectively.  ALL evolution is quantum and to be rigorous must be solved using quantum-mechanical laws of motion, although sometimes (often) classical approximations serve us well.

Dear Juan D. Bulnes

I would like you not to believe QM too much. 

I agree QM is absolutely right in mixed states. But I don't think quantum application such as quantum computer or transmission utilizing entanglement or wave-packet reduction i.e., concept of pure state, will be fulfillment.

QM has some paradoxes in dispute so far. In addition, some famous physicists insisted QM is incomplete.

For example, why do we think QM can be described in definite metric Hilbert space ? Real  time-space is Minkowski space that is indefinite metric space.

So, I believe we should study QM with indefinite metric Hilbert space. When we study QM with indefinite metric Hilbert space, we may get right answers about the paradoxes include your question.

Sorry for my intrusive behavior, but why don't you read following articles?

sincerely

Article Unobservable Potentials to Explain a Quantum Eraser and a De...

Dear Juan,

Glad to help!

You are also quite correct to approach concepts like the quantification of entanglement in systems with non-conserved Hilbert space dimensionality, indeed in most cases beyond rather simple ones, entanglement is often a difficult concept to understand and rigorously quantify (which isn't to say that entanglement isn't real or indeed useful, it just points to the fact that it is something that we don't fully understand).  Nevertheless there is outstanding work going on in trying to understand entanglement as a fundamental aspect of nature and as resource for various quantum tasks.

At the risk of promoting further reading, I'd like to recommend a paper by one of my colleagues, Jared Cole: Understanding entanglement sudden death through multipartite entanglement and quantum correlations, JH Cole, Journal of Physics A: Mathematical and Theoretical 43 (13), 135301.  One of the main points of this work is to highlight how difficult it can be to apply entanglement metrics to high-dimensional systems, and how blind application can mask the underlying physics.

All the best

Dear Akira Kanda - your comments are not germane to this thread, however I will attempt to answer you.  Your comments are in bold

Node on what you said in your post constitute an answer to some very basic and fundamental questions asked.

Some examples I repeat:

1. Schrodinger combined de Broglie relation which is relativistic and Hamiltonian which is classic to obtain Schrodinger wave equation. I wonder why you have classical m in the wave equation? Anything wrong with your reasoning?

In the standard formulation of the Schroedinger equation, you are correct that this is a non-relativistic equation.  It is perfectly possible to write down the Dirac equation to take into account relativistic effects.  I don't understand your problem here, the Schroedinger equation is not sufficiently general to predict everything.

2. Moreover, Relativity Theory and Classical dynamics contradicts each other if you have capacity to understand what contradiction means.

I don't know what you mean by classical dynamics here.  Because you are contrasting it with relativity, I assume you mean pre-Einsteinian dynamics.  Of course there's a contradiction here (I do know what that means): in this context we have the issue that without taking into account relativistic effects, we cannot understand fundamental processes of magnetism and the constant speed of light (and other processes).  So the evolution is rather straight forward - we have Newtonian (classical) dynamics - de Broglie wave hypothesis, Schroedinger equation, Dirac equation (and of course other formalisms).

3. For your information, let me tell you the consequence of this contradiction at the deepest level of QM. QM assumes relativity and it proves tunneling which proves that any particle can move faster than c. Only the brain of physicists can digest this.

Now you do seem to be getting off topic.  We must be quite clear, there is at present no reconciliation between GR and QM.  This is despite both of these theories being outstanding in their utility and predictive power in their own domains.  QED, for example, allows the calculation of the anomalous magnetic dipole moment of the electron.  This is the most accurately verified prediction of physics.  We also have gravitational lensing, blackholes and the expansion of the universe.  GR works.

Tunneling is a very important phenomenon, and one that can be easily demonstrated in any undergraduate laboratory, especially in the context of photon tunneling, i.e. frustrated total internal reflection.  It is arguable what is meant by the time taken for a particle to traverse a classically forbidden region, and indeed this is still debated.  What is not under question is whether causality is broken in such experiments, and this is really what we care about.  We also worry about whether information can be sent faster than light (which is really closely related).  Again, I note that your University of Toronto colleague Aephraim Steinberg is also an expert on this, perhaps you should visit him.

4. QM proves Uncertainty Principle which asserts that no particle can have trajectory. This is because once a particle is localized the probability of it being found in near by is almost zero. So, there is no way for a particle to leave trajectory in Wilson Chamber.

Unfortunately you don't understand the uncertainty principle.  Firstly, there is rather important result, Ehrenfest's theorem, which shows that expectation values of operators obey classical laws.  

I don't understand what you are getting at by saying 'once a particle is localised the probability of it being found in near by is almost zero' - I assume you believe that the measurement of position is precise - it isn't.  In the cloud chamber, the particle causes nucleation of a cloud in the vicinity of the ionized particle.  What is happening here from a QM measurement point of view is that the nucleating particles are effectively measuring the centre of charge, which they do to finite precision.  Typically one would expect that the measurement of position could be done at best to the standard quantum limit, where the uncertainty in position and momentum are equal.  The nucleation of the cloud provides a measurement record of where the particle was, so if we assume for a moment that the measurement is preformed stroboscopically (it isn't, it is performed effectively randomly, but that doesn't alter our argument) then the particle is localised to a finite region of space with finite uncertainty in momentum.  It then evolves for a short period of time according to quantum mechanical evolution before be re-measured and relocalised, again to finite precision.  These sequential measurements give a measurement record of the evolution of the particle, but note that we would not expect to ever see the particle simultaneously in two locations, because that would violate conservation of particle number and hence conservation of energy.

Again, the cloud chamber does not provide any practical or philosophical problems for quantum mechanics, and it is also important to realise the difference between a cloud chamber and an interferometer, where concepts like delocalisation ARE important.  Measurement theory is a well studied branch of QM and leads to ideas of decoherence, again of enormous practical significance.  And if you really want to see some beautiful results exploring the control of measurement and how this affects the evolution of a particle, I recommend (as I did before) that you talk to one of your colleagues at U Toronto, Steinberg, and ask to see his experiment (in the Science paper I mentioned previously).

5. The fact that the wrenched rain of wrenched theoretical physicists did not see that Schrodinger created a logically inconsistent theory is clearly proven by Klein0-Gordon-Dirac who tried to relativize already relativistic theory of QM by replacing energy and momentum variables of the relativistic energy momentum relation of Einstein by energy operator and momentum operator.

This seems more of a rant than a question to me.  Again, if you want to know where the important questions lie, then start looking at the differences between the treatment of time (a parameter in the Schroedinger equation) and space (operators in the SE), and the Dirac equation where we have to do away with position operators.

......

List goes on. You failed to respond to any of these simple questions. You guys built a Spanish castle on dung heap

I really can't understand why people feel the need to attack the most successful treatments and understanding that we have of the universe, especially from such an ignorant basis.  I don't think that the job of physics is complete, and there are inconsistencies.  Perhaps the resolution to quantum gravity will come from an entirely new perspective (likely).  Perhaps some of the conceptual problems we have in understanding QM are due to some failing in the formalism (highly unlikely). Irrespective of what the future for QM holds, this is an outrageously successful theory and if you are going to put up an alternative, you need to prove that your alternative can explain everything that QM explains and more.  You need to find testable inconsistencies and show that the results of experiment do not accord with rigorous application of QM.  Your examples of cloud chambers and tunnelling demonstrably do not contradict QM, as you would know if you had actually researched the topics rather than making wild and offensive assertions.

And finally I will point out that the greatest achievements in QM today are resolving philosophical arguments about the nature of reality.  To be explicit. before the pioneering work of Bell, Aspect and others, Einstein and Bohr could have their debates and fire thought experiments at each other.  Whilst diverting, without the ability to test the thought experiments, there was actually little progress.  Bell's genius (and CHSH) was to put the EPR experiment onto a testable footing, which was ultimately tested.  Aspect and his group showed us precisely what a measurement was.  We are in a golden age of understanding QM because we have reached the stage where experiments can rule out hypotheses about what is 'really going on'.  If you want to be a part of this golden age, learn some physics and talk with and listen to people.  There's always room for more people, especially smart people with unconventional thinking that they're prepared to test.  If you don't want to do that, and just want to rant, then please do so in a way that doesn't get in the way of real inquiry because it just wastes everyone's time.

I hope you have a good day.