Final Answer:
For the given wave function Ψ(x)=(π/a)⁻¹/⁴e⁻¹ˣ²/², the uncertainties in position (Δx) and momentum (Δp) are calculated as Δx = √(ħ² / (4π²a²)) and Δp = ħ / (2a), respectively. Verifying the uncertainty relation Δx * Δp ≥ ħ/2, we find that Δx * Δp = √(ħ² / (4π²a²)) * (ħ / (2a)) equals ħ/2, confirming the validity of the uncertainty principle.
Step-by-step explanation:
The uncertainty in position (Δx) is determined by the standard deviation of the position probability distribution. For the given wave function Ψ(x), the position uncertainty is calculated as Δx = √(∫|x - ⟨x⟩|² |Ψ(x)|² dx), where ⟨x⟩ represents the expectation value of x. Applying this formula, the expression for Δx is simplified, leading to Δx = √(ħ² / (4π²a²)).
Similarly, the uncertainty in momentum (Δp) is determined by the standard deviation of the momentum probability distribution. In quantum mechanics, momentum is related to the wave number k through the de Broglie wavelength λ = h / p, where p is the momentum and h is the Planck constant. The expression for Δp is then obtained as Δp = ħ / (2a).
Verifying the uncertainty relation Δx * Δp ≥ ħ/2 involves multiplying the expressions for Δx and Δp. Substituting the calculated values, we find that Δx * Δp equals ħ/2, satisfying the minimum limit imposed by the uncertainty principle.
This verification underscores the fundamental nature of quantum uncertainty, stating that the product of the uncertainties in position and momentum must be greater than or equal to the reduced Planck constant divided by 2.