I am currently exploring the mathematical structure of Quantum Mechanics on an introductory level. A couple of books and online sources (including this website) stated how the Uncertainty Principle is a consequence of two non-commuting operators like position-momentum or the spin operators. But in some books including Stephen Hawking's - "A Brief History of Time", the Uncertainty Principle is interpreted as a limitation imposed due to an inevitable tradeoff between wavelength and energy. For e.g. in the book "A Brief History of Time" on Page 60 the author writes -
In order to predict the future position and velocity of a particle, one has to be able to measure its present position and velocity accurately. The obvious way to do this is to shine light on the particle. Some of the waves of light will be scattered by the particle and this will indicate its position. However, one will not be able to determine the position of the particle more accurately than the distance between the wave crests of light, one needs to use light of a short wavelength in order to measure the position of the particle precisely. Now, by Planck's quantum hypothesis, one cannot use an arbitrarily small amount of light; one has to use at least one quantum. This quantum will disturb the particle and change its velocity in a way that cannot be predicted. Moreover, the more accurately one measures the position, the shorter the wavelength of the light that one needs and hence the higher the energy of a single quantum. So the velocity of the particle will be disturbed by a larger amount. In other words, the more accurately you try to measure the position of the particle, the less accurately you can measure its speed, and vice versa.
Hawking clearly stated it as a physical consequence, which is more intuitive than the mathematical one. Finally, my question is which one of the interpretation is correct? Or is it possible that both interpretations are equally plausible and that the mathematical derivation just happens to explain the physical cause through a different approach.