A discussion on Heisenberg’s uncertainty relation based on the calculation of some typical systems

We calculate the uncertain quantities of particle position and momentum, x²-x² and p²-p², using a few typical systems from classical and quantum mechanics. The results show that the product of the uncertain quantities varies with different states. Taking the harmonic oscillator and infinitely deep potential well as examples, we show that a relation analogous to the uncertainty relation of quantum mechanics also exists in classical mechanics. This gives a new understanding of the Heisenberg uncertainty relation, and provides additional insight into the wave function of quantum mechanics. Finally, two related experimental phenomena are discussed.