Question

Write Heisenberg’s uncertainty principle.

Answer 1

For a macroscopic system, it is possible to measure or to know about both position and momentum of the system, simultaneously, at any instant. But, for microscopic particle, which is moving it is not the case, we can either know its momentum or position with some degree of accuracy, not both at a time. This understanding is encoded in the Heisenberg uncertainty principle.

Complete step by step answer:
Definition: Heisenberg’s uncertainty principle states that it is impossible to measure both the position and momentum of a particle accurately at a given instance.
In mathematical form, $\Delta x \times \Delta p \geqslant \dfrac{h}{{4\pi }}$ where, x and p are position and momentum respectively.
Let $\Delta x$ be the error in position measurement and $\Delta p$ be the error in momentum measurement. If $\Delta x = 0$ or $\Delta p = 0$, then we can say we measurement is accurately done for x and p respectively. But, error in measurement follow the mathematical form of Heisenberg’s uncertainty principle is:
$\Delta x \times \Delta p \geqslant \dfrac{h}{{4\pi }}$ ………………….. (1)
This puts a lower bound on the error in position and momentum measured together for any particle. If we measure the position of a particle accurately, we will lose the information about its velocity, on the other hand, if we are able to measure the momentum of a particle accurately, we will lose the information about its position.

Additional Information:
Here we can see that in equation (1), We have introduced term error in definition. It implies that mean distance can be found mean velocity can be found but exact velocity and momentum together can’t be found.
$\Rightarrow {x_{exact}} = {x_{mean}} \pm \Delta x$
Similarly,
${v_{exact}} = {v_{mean}} \pm \Delta v$
A particle, moving with high speed and has very infinitesimal mass, exists predominantly in a dual state of particle and wave. Due to this duality, they can produce both types of phenomena which in general is reserved for one of kind. However, Heavier bodies also have a dual wave with them. But they are dominated with matter properties more due to heavy mass and slow speed.

Note:
In short, more accurately you know the momentum or velocity of a moving microscopic particle; less accurately you know about its position and vice versa.