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In a parallel universe the uncertainty in position and the uncertainty in momentum are directly related to each other.

What kinds of effects would this have on the quantum mechanics of this universe? Could this type of parallel universe have life or any complex structures?

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    $\begingroup$ When I don't know where something is I usually don't know its momentum either... They seem pretty proportional to me ;) $\endgroup$ – Samuel Dec 4 '15 at 21:14
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    $\begingroup$ The dimensionality of h-bar will have to change. That, in and of itself would invalidate dozens of equations, including the original formulation of h-bar, so basically the answer to this question will be "any answer you please, because you are so far away from actual physics that their equations have no power over the universe you are describing." $\endgroup$ – Cort Ammon Dec 4 '15 at 22:04
  • $\begingroup$ Being that the uncertainty principle is a known inequality, I disagree that it would be "any answer you please," since making it an equality by definition would preclude certain variations of the universe. Since h-bar is included in so many other formulae, one can surely change those as well and derive specific conclusions about how the hypothetical universe can be and formulate an answer. $\endgroup$ – The Anathema Dec 4 '15 at 22:25
  • $\begingroup$ Just to be clear, mathematically, you're saying that position and momentum satisfy $\Delta x\Delta p=\hbar/2$, rather than $\Delta x\Delta p\geq \hbar/2$ - and the same for any pair of conjugate variables? $\endgroup$ – HDE 226868 May 19 at 14:52
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Such a restriction requires a change in the foundations of quantum mechanics or its interpretation.

The Heisenberg's uncertainty principle is an inequality: $\sigma_p\cdot \sigma_x \geqslant \hbar/2$; as such, it doesn't forbid a system to be in the state where the uncertainty of the momentum and coordinate are directly related by, say, being inversely proportional to each other in a way that makes it an equality:

$$\sigma_x = \frac{\hbar}{2\sigma_p}.$$

However, if one keeps only states which satisfy this relation, one arrives at a contradiction with other principles of quantum mechanics. The condition above severely restricts the variety of quantum mechanical states a system can be in; for instance, the harmonic oscillator could only exist in what is called coherent states and cannot exist in states with definite energy; for a hydrogen atom, no stationary state (with a fixed energy level) is possible either. But the measurement process of, say, energy should put the system it into a state with definite energy; however these states are now forbidden. Thus, to stay consistent, one has to review at least the part of the foundations of quantum mechanics which tells what a measurement does to the particle.

Any other direct functional connection between uncertainties should lead to similar problems, because it leads to too few allowed states to be compatible with the existing interpretations of measurement in quantum mechanics.

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The uncertainty principle in general isn't an arbitrary rule that could be anything. It occurs as a direct consequence of the Universe having a smallest scale; a base resolution in a sense.

Now consider why some values are conjugate, famously including position v momentum and energy v time.

The wave v particle expression of a phenomena directly brings up this effect. If quantization happens, the inverse relation between particle and waves will happen, and the familar relationship of momentum v position will just show up without you having put it in explicitly.

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