Summary: We get ourselves a toy in j-space based on the ideas from Quantum Mechanics. The particle in a box and the extreme shallow-well problems are discussed.
We have a dilemma. We had set the origin in discrete measurement space as a reference described by an observer of capability equal to the capability of the observer making measurements. The measurements themselves are set to be PE1j measurements. The observers making measurements have limited capability defined as [0j, ∞j ]. Therefore the events outside the capability of the observers can be regarded as indeterminate. What does it mean by an indeterminate state? Let us try to answer that. First we describe a methodology.
Consider dropping a small pebble into the pond in Chicago. The action will cause a ripple we can measure. Now if the pond is large enough, which actually it is, the ripple will not change anything really. We call it an infinitesimal perturbation. We also know that the contents within pond can be considered uniform, therefore the results obtained from our infinitesimal action and subsequent measurements will be applicable everywhere in the pond. And we will be able to predict the behavior of its contents or a similar system to a much higher accuracy.
In physics we have similar methodologies to analyze a given system. For example the principle of virtual work allows the study of a physical system in equilibrium. We displace a system in equilibrium by an infinitesimal amount. Since the displacement is infinitesimal, the total work sums up to zero hence the name virtual work. We can thus obtain the relationship between the different components of the internal forces maintaining the equilibrium state. In Quantum Mechanics we apply a small perturbation to a system in its lowest energy state and study its behavior in its excited states.
Let us consider the indeterminate state and introduce a action δA in it by the application of an infinitesimally small measurement force. The shape of the indeterminate state is deliberately drawn not to be symmetric. We note the application of the measurement force by an observer immediately puts the observer in discrete j-space.
The action δA will introduce the complimentary simultaneous states, one of them will allow a measurement to take place other will not allow a measurement to take place in the discrete measurement space. Combined together they will result in an indeterminate state. We just separate them by an infinitesimal amount below the observer's measurement threshold.
Both of these states will behave as sources for the observer in the discrete measurement space. We have introduced a perturbation which has restricted the space within discrete measurement space into regions where measurement can be made and measurements can not be made. Or in other words the regions where a particle's existence is measured or not measured.
Further for an observer with low efficiency the required measurement force will be much larger and resolution between the both types of regions will be much coarser. This situation is analogous to the collider schemes where to access information about fundamental particles, a very large measurement force and hence a much higher energy input is required.
An observer who can measure states created by only one of either "continuum" or "nil" sources but not both, is considered a zero-efficiency observer.
We note the concept of the region where the particles can not be measured is different than conventional vacuum state which is based on source and sink concept. In this case the particles are physically removed to create vacuum which is essentially a classical thermodynamic system.
Particle in Box
" You should not be led along blindly."
- Shang-Keng Ma, in Statistical Mechanics, World Scientific.
"...we have to assume that there is a limit to the fineness of our powers of observation and the smallness of the accompanying disturbances..."
- Dirac in The Principles of Quantum Mechanics, Oxford Science Publications.
Sears and Zemansky, University Physics, Pearson's Education, Inc. is a good starting point for basic quantum mechanics concepts.