Take a guitar string and pluck it. The oscillations on the string have the shape of sine or cosine functions. One can use these sine and cosine functions to construct virtually any function. This follows from a mathematical technique called Fourier theory. One would use a Fourier transform of a function to determine what the are that one needs for all the different sine and cosine functions to reconstruct said function. The coefficients also form a function that is called the spectrum and which is a function of frequency.
In physics sine and cosine functions are often extended to multiple dimensions to describe waves in space and time called plane waves. Fourier theory still works for these plane waves and can be used to reconstruct any field such as electromagnetic fields that exist in space-time.
The spectrum of a function usually have a specific width associated with it. The same is true for the function itself. There is an interesting relationship that exists between the width of the function and the width of its spectrum. If one stretches the function so that it becomes wider the effect on its spectrum is to make the spectrum narrower. One can actually see that the product of the width of a function and the width of its spectrum is a constant. For every function there is such a constant called the space-bandwidth product.
Now it turns out that one can not have space-bandwidth products that are arbitrarily small. The smallest value for the space-bandwidth product is found with a function called the Gaussian function. There exists no other function with a smaller space-bandwidth product.
When Max Planck discovered that there is a relationship between the frequency and the energy in black body radiation, he implicitly used Fourier theory to represent the radiation in terms of its frequency spectrum. As a result quantum mechanics was formulated as a Fourier expansion of particles and fields in terms of plane waves. Fourier theory therefore lies at the foundation of .
Since quantum mechanics is founded on Fourier theory the properties of Fourier transforms would also be built into quantum mechanics. The restriction that exists on space-bandwidth products would therefore also exist in quantum mechanics. This manifests as the so called Heisenberg uncertainty principle.
One can, for instance, express the Heisenberg uncertainty principle as a restriction that exists on the product of the uncertainty in position and the uncertainty in momentum. While the uncertainty in position is given by the width of the position space function that represents an object, the uncertainty in momentum is then given by the width of its spectrum. This is because the spatial frequencies (of which this spectrum is a function) are related to the momentum of the object via Planck’s constant. We see therefore that the is nothing else but the mathematical restriction in space-bandwidth product that exists for all functions.
Our conclusions is that the Heisenberg uncertainty principle is not a fundamental principle of nature. It is a direct result of the fact that quantum mechanics is formulated in terms of Fourier theory.