Question

State Heisenberg’s uncertainty principle.

Answer 1

Hint: Heisenberg’s uncertainty principle, in essence says that there is always an error while simultaneously measuring the momentum and position of a body in the quantum world. It can also be extended to other pairs of quantities.

Complete step-by-step answer:
Heisenberg’s uncertainty principle formed one of the cornerstones for the development of quantum mechanics. Heisenberg’s uncertainty principle, though devised extensively for quantum mechanics, works equally well for classical mechanics, thus enhancing the correctness of the principle.
Heisenberg’s uncertainty principle states that the momentum and position of a body in the quantum world cannot be measured simultaneously with an arbitrarily very high amount of precision. There will always be an error in the measurement of one of the quantities of the pair if the other is to be measured simultaneously with a high degree of precision. This principle can in fact be extended to other pairs of quantities like entanglement and coherence.
The mathematical form of the Heisenberg’s Uncertainty principle is as follows,
$\Delta x.\Delta p\ge \dfrac{h}{4\pi }$ --(mathematical form of Heisenberg’s uncertainty principle) $\therefore \Delta x.m\Delta v\ge \dfrac{h}{4\pi }$ --[$\left( \because \Delta p=\Delta mv+m\Delta v=m\Delta v \right)$ as$\Delta m=0$ as mass remains constant] $\therefore \Delta x.\Delta v\ge \dfrac{h}{4\pi m}$ ---(1) where $\Delta x$ and $\Delta p$ are the uncertainties in position and momentum of the body respectively, m is the mass of the body, v is the velocity of the body and $h$ is the Planck's constant equal to $6.636\times {{10}^{-34}}J.s$

Note: The Heisenberg’s uncertainty principle is equally valid for macroscopic objects like a car or a human but plugging in the comparatively huge value of mass in (1), will give a very small value for $\dfrac{h}{4\pi m}$ essentially meaning that the errors while simultaneously measuring the body’s momentum and position will be negligible. So, they can be measured accurately and this is where quantum mechanics converts into classical mechanics.
The Heisenberg uncertainty principle is the reason why the Bohr atomic model failed. Bohr postulated that the electron (a quantum particle) revolves around the nucleus in fixed paths called orbits with a fixed velocity. Thus, in essence he said that the position and momentum of the electron could be simultaneously accurately calculated, which is in strict disagreement with Heisenberg’s uncertainty principle.