I’ll just say a word on Fourier transforms, which might help clear up your intuition:

Fourier transforms are complex.

This means, for example, when transforming from position space to momentum space, there is a phase associated with each momentum value, in addition to amplitude.

If the momentum phase is uniform, this means the position spread is “transform limited”, i.e. minimum. If, on the other hand, the momentum phase is non-uniform, then the position is “chirped” or worse. Then the position will be further spread out for the same momentum amplitude spectrum.

Edit to directly answer the question:

“Why doesn’t Gaussian waveform broadening in position mean there will be a shortening in momentum?”

Because the broadening in position can, in certain circumstances, be represented by a change only in the phases of the momentum spectrum (like a chirped wave, for example). Then the amplitudes of the momentum spectrum would remain the same.

I’ll just say a word on Fourier transforms, which might help clear up your intuition:

Fourier transforms are complex.

This means, for example, when transforming from position space to momentum space, there is a phase associated with each momentum value, in addition to amplitude.

If the momentum phase is uniform, this means the position spread is “transform limited”, i.e. minimum according to uncertainty principle. If, on the other hand, the momentum phase is non-uniform, then the position is “chirped” or worse. Then the position will be further spread out for the same momentum amplitude spectrum.

Edit to directly answer the question:

“Why doesn’t Gaussian waveform broadening in position mean there will be a shortening in momentum?”

Because the broadening in position can, in certain circumstances, be represented by a change only in the phases of the momentum spectrum (like a chirped wave, for example). Then the amplitudes of the momentum spectrum would remain the same.

I’ll just say a word on Fourier transforms, which might help clear up your intuition:

Fourier transforms are complex.

This means, for example, when transforming from position space to momentum space, there is a phase associated with each momentum value, in addition to amplitude.

If the momentum phase is uniform, this means the position spread is “transform limited”, i.e. minimum. If, on the other hand, the momentum phase is non-uniform, then the position is “chirped” or worse. Then the position will be further spread out for the same momentum amplitude spectrum.

I’ll just say a word on Fourier transforms, which might help clear up your intuition:

Fourier transforms are complex.

This means, for example, when transforming from position space to momentum space, there is a phase associated with each momentum value, in addition to amplitude.

If the momentum phase is uniform, this means the position spread is “transform limited”, i.e. minimum. If, on the other hand, the momentum phase is non-uniform, then the position is “chirped” or worse. Then the position will be further spread out for the same momentum amplitude spectrum.

Edit to directly answer the question:

“Why doesn’t Gaussian waveform broadening in position mean there will be a shortening in momentum?”

Because the broadening in position can, in certain circumstances, be represented by a change only in the phases of the momentum spectrum (like a chirped wave, for example). Then the amplitudes of the momentum spectrum would remain the same.