Heisenberg’s uncertainty principle | Quantum mechanics | Physical basis of quantum mechanics

Heisenberg’s uncertainty principle

• According to Heisenberg’s uncertainty principle it is impossible to determine precisely and simultaneously two canonically conjugate variables.
• If we measure one physical quantity precisely then the other will have a great uncertainty.
• This principle is observed only in microscopic particles, not in macroscopic bodies.
• If Δp = uncertainty in momentum, and Δx = uncertainty in position, then according to Heisenberg's principle

                Δx ✕ Δp ≥ ħ / 2

• Here ħ = h / 2 and it is known as reduced Planck's constant

• If ΔE and Δt are the uncertainties in the energy and time then the uncertainty principle will be

                ΔE ✕ Δt ≥ ħ / 2

• and if ΔJ and Δϕ are the uncertainties in the angular momentum and angular position then the uncertainty principle will be

                ΔJ ✕ Δϕ ≥ ħ / 2

• In solving the problem related to Heisenberg uncertainty principle we use

                Δx ✕ Δp ≈ ħ

Physical significance of uncertainty principle

• If the position of a particle is determined with great accuracy, Δx = 0, then Δp = ∞, the uncertainty in momentum become infinite.
• If the momentum of a particle is determined with great accuracy, Δp = 0, then Δx = ∞, the uncertainty in position become infinite.

• If a particle of mass m moves with velocity v, then uncertainty relation.

                Δx ✕ Δv ≈ ħ / m

• Since for a very heavy particle m is very large, and therefore Δx ✕ Δv is very small.
• Therefore for very heavy particle we can measure the values of position and velocity both with accuracy.

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