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In the nuclear magnetic resonance imaging (MRI), the spatial resolution and temporal resolution can not be arbitrarily reduce simultaneously. What would be a simple inequality of the Heisenberg's uncertainty principle to elucidate that trade off?

I just thought of this question from a casual conversation with a radiologist. I am not familiar with the detailed working mechanism but only the vague notion of MRI. So I am not able to work out the exact inequality.

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  • $\begingroup$ Sounds like a homework question. A hint: the limiting constant is the maximum gradient strength. $\endgroup$
    – tesch1
    Apr 29, 2018 at 12:43
  • $\begingroup$ @tesch1: Thank you for the hint. But it is not a homework problem. I just added the explanation of the context of the question above. $\endgroup$
    – Hans
    Apr 29, 2018 at 19:40

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MRI should usually be thought of classically rather than quantum mechanically.

The limit that your radiologist friend was discussing is not a physical limitation, but a SNR limitation. All other things being equal, the SNR is proportional to the voxel size times the square root of the sampling time. So in principle you can arbitrarily increase both at the cost of arbitrarily poor SNR.

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  • $\begingroup$ "MRI should usually be thought of classically rather than quantum mechanically." Why? Can you please elaborate? Thanks! $\endgroup$ Aug 26, 2018 at 20:06
  • $\begingroup$ In even the smallest of voxels there are millions of spins, and they are all interacting with each other and the environment. The classical limit works well, and any time it works well it is preferable to use. $\endgroup$
    – Dale
    Aug 26, 2018 at 20:18
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    $\begingroup$ You have misunderstood my question. Heisenberg's uncertainty principle is not limited to quantum mechanics. It is a general mathematical property resulting from the non-commutativity of any two self-adjoint operators. Your statement of SNR limitation presumably refers to the Gabor limit in the Fourier analysis, but it is but one example of Heisenberg's uncertainty principle. See en.wikipedia.org/wiki/Uncertainty_principle#Harmonic_analysis. $\endgroup$
    – Hans
    Aug 26, 2018 at 20:47
  • $\begingroup$ What I am asking for is an explicit identification of the two non-commuting operators associated with MRI spatial and temporal resolutions, and express the limit in an explicit inequality. $\endgroup$
    – Hans
    Aug 26, 2018 at 20:48
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    $\begingroup$ Because I have been doing R&D in MRI pulse sequence design for the past 18 years and I have never seen such a relationship written down in any of the standard textbooks nor used in any of the research papers I have read. I know the relationship referred to by your radiologist friend and I am “familiar with the detailed working mechanism”. Maybe such a relationship could be devised, but I don’t think that it has been yet or I would likely be aware. $\endgroup$
    – Dale
    Sep 3, 2018 at 8:19

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