Uncertainty Principle Formula

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

Where:

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?

Answer:

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?

Answer:

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