Question

State Heisenberg Uncertainty principle. Give its mathematical expression.

Answer 1

Heisenberg uncertainty principle states that it is impossible to measure or calculate exactly, both the position and the momentum of an object. This principle is based on the wave-particle duality of matter. Another implication of this principle is that it is impossible to accurately measure the energy of a system in some finite amount of time.

Complete step by step answer:
In the field of quantum mechanics, Heisenberg’s uncertainty principle is a fundamental theory that explains why it is impossible to measure more than one quantum variable simultaneously.
This principle states that it is impossible to measure, both the position and momentum of an object. Although Heisenberg’s uncertainty principle can be ignored in the macroscopic world but it holds significant value in the quantum world.
Now, let us see the Heisenberg’s uncertainty principle formula.
If, $\Delta x$ is the error in the position measurement and $\Delta p$ is the error in the measurement of momentum, then,
$\Delta x\, \times \Delta p \geqslant \dfrac{h}{{4\pi }}$
Since, momentum $p = mv$
Therefore, this principle can be alternatively written as,
$\Delta x\, \times \Delta mv \geqslant \dfrac{h}{{4\pi }}$
Or, $\Delta x\, \times \Delta m\, \times \Delta v \geqslant \dfrac{h}{{4\pi }}$ where, $\Delta v$ is the error in the measurement of velocity
Therefore, $\Delta x\, \times \Delta v \geqslant \dfrac{h}{{4\pi m}}$ where, h$=$ Planck’s constant
However, the more precise our measurement of position is, the less accurate will be our momentum measurement and vice-versa.

Note:
The electromagnetic radiations and microscopic matter waves exhibit a dual nature of mass/momentum and wave character. Position and velocity/momentum of macroscopic matter waves can be determined accurately simultaneously. For e.g. the location and speed of a moving can be determined at the same time with minimum error. But in case of microscopic particles, it will not be possible to fix the position and measure the velocity of the particle simultaneously.