what is the relationship between these two things
Perhaps more resolution in a spectrogram is equivalent to knowing more the position of the electron and less resolution is knowing the velocity of the electron.
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source: (older thread: Which time-frequency coefficients does the Wavelet transform compute?)
In Quantum Mechanics, state of system specified by a vector (wave function which is a vector in function space), and you could use different basis to represent this vector (imagine one vector in 2 different coordinate system which lead to different components in those coordinate system but they both represent the same vector). For systems composed from moving particles, one basis is the position basis and the second one is momentum basis. The transform between these two basis is the Fourier transform. Considering the wave function give us a probabilistic interpretation, to find the momentum or position of a particle you have to find the average and after that to have a sense about error, you have to find the standard deviation.
The Heisenberg Uncertainty principle, tell us the multiplication of position error and momentum error could not be smaller than some value, for any possible state of system.
Considering this is a property of Fourier transform you could extend that to signals where the signal amplitude and it's spectrum are different representation of same thing in different basis, and say multiplication of effective bandwidth of pulse (as standard deviation of power spectrum around central frequency) and the effective pulse width (as standard deviation of signal's power around it's center in time) could not be smaller than some value for every possible signal.
The standard spectrogram (in the complex and redundant form) is a linear transformation that unfolds a 1D time signal onto a 2D time-frequency space. There, each coefficient represent a "time-times-frequency" square whose dimensions correspond to the standard-deviations or second-order moments of the Weyl-Heisenberg inequality for Fourier analysis.
Higher dimensional versions exist.
The spectogram is a specific member of a class of tools used for time-freq analysis of signals.
The time-freq analysis of a signal may be performed by a number of tools, but most common is to use a time dependent Fourier transform (or a wavelet transform) which is a Fourier transform applied on a sliding window accross the whole signal length.
Such a time-freq analysis provides not only a spectral view of the signal (inside the window) but also its time information of the events in it.
An event is something a like a pulse, a transient, a time-varying frequency of sinusoid (a modulated signal). These signals carry information both in their frequencies and in their timings.
The time-freq analysis has a frequency-resolution determined by the window length & shape, and a time-resolution determined by the window jumps.
A shorter window will provide a better time localisation (resolution) of a particular event, whereas a longer window will provide a better frequency resolution. The dilemma between the short and long windows for better time and frequency resolutions is reminiscent of a well known dilemma from quantum physics which was stated as Heisenberg's uncertainty principle.
Therefore, it's analogously named as the uncertainty principle of signal processing. But there's no physical link in between the two. It says that you cannot increase time and frequency resolutions simultaneously. If one is increasing, then the other must decrease. And their product, which is constant for a given observation time, is called the capacity of analysis.
Unless you increase the total observation interval (per sampling frequency) and thus increase the capacity of analysis, your frequency & time resolutions will be inversely dependent on the other.