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## Summary:

- I have read that the uncertainty relation for momentum and position is unrelated to the uncertainty relation for energy and time. But I am confused because when I think about it, the two uncertainty principles actually seem like two sides of the same coin. The "uncertainty in position" can instead be thought of as the spatial delocalization of the wave function, in analogy with the "lifetime" in the energy-time uncertainty relation, which is the temporal delocalization of the wave function.

## Main Question or Discussion Point

In a few textbooks in introductory quantum mechanics which I have looked through (e.g. Griffiths), it is heavily emphasized that the momentum-position uncertainty relation has a completely different meaning from the energy-time uncertainty relation, and that they are quite unrelated and only superficially look similar. But I don't see why this is the case. Please let me explain.

From what I know, there are two ways to think about the momentum-position uncertainty principle. One way (which I believe is only valid in non-relativistic quantum mechanics) is that position should be treated as a dynamical variable that is an

But there is another way to think about the momentum-position uncertainty principle. We can treat the position as a

As far as I can tell, these two forms of the momentum-position uncertainty principle are equivalent (within the context of non-relativistic quantum mechanics) since if the probability distribution in the outcome of a position measurement has a standard deviation Δx, that precisely means that if one places a particle detector in space, then the probability density that the detector will detect the particle will vary with the position of the detector according to a distribution with standard deviation Δx.

But the second of the above two interpretations of the momentum-position uncertainty principle (which treats position as a parameter) seems strikingly similar in meaning to the energy-time uncertainty principle. The energy-time uncertainty principle states that if the probability density of detecting a particle at a certain spatiotemporal location has a

It seems that the momentum-position uncertainty principle is indeed totally analogous to the energy-time uncertainty principle since the former has two equivalent formulations (one with position as a dynamical variable, and the other with position as a parameter). Even though time can

But then, what

From what I know, there are two ways to think about the momentum-position uncertainty principle. One way (which I believe is only valid in non-relativistic quantum mechanics) is that position should be treated as a dynamical variable that is an

*attribute*of the particle. Then, the momentum-position uncertainty principle says that if the probability distribution in the outcome of a position measurement has a standard deviation Δx, then the probability distribution in the outcome of a momentum measurement must have a standard deviation no less than ħ/(2Δx).But there is another way to think about the momentum-position uncertainty principle. We can treat the position as a

*parameter*similar to time, rather than as an*attribute*of the particle. Then, the momentum-position uncertainty principle states that if the probability density of detecting a particle at a certain spatiotemporal location has a*spatial dependence*following a distribution with a standard deviation Δx, then the probability distribution in the outcome of a momentum measurement at that time must have a standard deviation no less than ħ/(2Δx).As far as I can tell, these two forms of the momentum-position uncertainty principle are equivalent (within the context of non-relativistic quantum mechanics) since if the probability distribution in the outcome of a position measurement has a standard deviation Δx, that precisely means that if one places a particle detector in space, then the probability density that the detector will detect the particle will vary with the position of the detector according to a distribution with standard deviation Δx.

But the second of the above two interpretations of the momentum-position uncertainty principle (which treats position as a parameter) seems strikingly similar in meaning to the energy-time uncertainty principle. The energy-time uncertainty principle states that if the probability density of detecting a particle at a certain spatiotemporal location has a

*temporal dependence*following a distribution with a standard deviation Δt, then the probability distribution in the outcome of an energy measurement must have a standard deviation no less than ħ/(2Δt). Compare this with the momentum-position uncertainty principle, which (as already mentioned) states that if the probability density of detecting a particle at a certain spatiotemporal location has a*spatial dependence*following a distribution with a standard deviation Δx, then the probability distribution in the outcome of a momentum measurement must have a standard deviation no less than ħ/(2Δx).It seems that the momentum-position uncertainty principle is indeed totally analogous to the energy-time uncertainty principle since the former has two equivalent formulations (one with position as a dynamical variable, and the other with position as a parameter). Even though time can

*only*be treated as a parameter, the fact that we can always change from treating position as a dynamical variable to treating position as a parameter seems to mean that the two uncertainty principles are actually intimately related, one focusing on the*spatial*dependence of the wave function and the other focusing on the*temporal*dependence of the wave function.But then, what

*is*the point that these textbooks (such as Griffiths) seem to make? Is there any*mistake*in my argument above? Any clarification is appreciated.