Quantum Mechanics is perhaps the greatest discovery in the field of Physics since the days of Isaac Newton. Heisenberg’s Uncertainty principle is one of the most important tenets of this field. it is one of the most famous (and probably misunderstood) ideas in Physics. Let us try to understand more about the Heisenberg Uncertainty Formula.
Heisenberg Uncertainty Formula
The basic statement of the principle is that it is impossible to measure the position (x) and the momentum (p) of a particle with absolute accuracy or precision. The more accurately we know one of these values, the less accurately we know the other. Take for instance the scenario in which we try to view an electron. To view an electron we shine some photons(light) on it. These photons impart some energy to the electrons they are incident upon.
This leads to the electrons gaining some momentum and the calculations about the momentum are altered, alternatively, because the electrons move so fast, by the time the incident photons report back their positions the electrons would have already moved from there, thereby affecting the calculations about the position.
This principle was fundamental to understanding the structure of an atom which was not understandable using the Newtonian or classical mechanics. It helped to overcome the deficiencies of the classical models of the atoms like the Bohr’s model, Rutherford model etc.
It basically stated that the product of the errors in the measurement of the momentum and the position is equal to a constant. Thus making it easier to measure the values. This led to a significant boost for the nascent field of quantum mechanics.
This principle also applies to the macroparticles like a tennis ball thrown, a car moving on a road etc. But as the dimensions of an object increase the principles of quantum mechanics do not yield significant results. Thus we can accurately know the position of a car as well as its speed simultaneously.
This allows technologies like Google Maps etc. to function without any errors. So, Heisenberg’s principle cannot be applied to macroparticles. As the results would turn out to be meaningless.
Let us look at this concept mathematically,
Let the error in the measurement of the position and momentum be Δp and Δx respectively then :
Δp × Δx ≥ h ⁄ 4π
where h is the Planck’s constant with a value of 6.626 x 10-34 joule seconds.
Solved Examples for Heisenberg Uncertainty Formula
1) An electron in a molecule travels at a speed of 40m/s. The uncertainty in the momentum Δp of the electron is 10−6 of its momentum. Compute the uncertainty in position Δx if the mass of an electron is 9.1×10−31 kg using Heisenberg Uncertainty Formula.
Answer: Given measurements are,
v = 40m/s, m = 9.1×10−31 kg, h = 6.626×10−34 Js and Δp = P×10−6
We know that, P = m×v
P = 9.1×10−31×40 = 364×10−31 kgm/s
Δp = 364×10−37
Heisenberg Uncertainty principle formula is given as,
2) Position of a chloride ion on a material can be determined to a maximum error of 1μm. If the mass of the chloride ion is 5.86 × 10-26Kg, what will be the error in its velocity measurement?
Answer: ∆x = 10-6 m; ∆X × ∆mV ≥ h4π = 6.626×10−344×3.14 = 5.28×10-35Js
⸪ ∆V ≥ h4πmΔx≥6.626 × 10−344 × 3.14 × 5.86 × 10−26 × 10−6 = 9 × 10-4m s-1