It states that the position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. There exists a minimum value for the product of the uncertainties of these two measurements. There is also a minimum for the product of the uncertainties of the energy and time. It arises from the wave properties inherent in the quantum mechanical description of nature. The uncertainty is inherent in nature.
(Position uncertainty) * (momentum uncertainty) ≥ (Planck’s constant) /2
(Energy uncertainty) * (time uncertainty) ≥ (Planck’s constant) /2
The equations are:
Δx Δp ≤ ℏ/2
ΔE Δt ≤ ℏ/2
ℏ: Planck’s constant
Δx: Position uncertainty
Δp: Momentum uncertainty
ΔE: Energy uncertainty
Δt: Time uncertainty
Uncertainty Principle Formula Questions:
1) Assume an electron is confined to a atom of size 0.4 nm, what is the energy average of the particle in the atom?
From the equation above we find Δp, which is the average momentum of the particle in the atom
Δp ≤ ℏ/2(0.4 nm) = 1.66*10(-24) Kg*m/s
The energy is given by Δp2/2m, where 9.10938356 × 10-31 kg is the mass of the electron
ΔE ≤ 9.4 eV
2) Consider the same particle above, what is the time average of the electron in the atom?
The formula for the time comes from the second equation of the uncertainty principle
Δt ≤ ℏ/ 2ΔE
Then Δt ≤ 4.13*10(15) eV*s/2(1.66*10(-24) Kg*m/s)
Δt ≤ 1.24*10(39) s ≈ 3.94*10(31) years