Question

what is the value of the product △x△p? ? use p=ℏk to find the uncertainty in the momentum of the particle.Express your answer in terms of quantities given in Part A and fundamental constants._____________

Answer 1

The value of the product ΔxΔp is governed by the Heisenberg Uncertainty Principle. To find the uncertainty in momentum (Δp), use Δp = ℏ/(2Δx) and plug in the provided Δx.

Step-by-step explanation:

The value of the product ΔxΔp is determined by the Heisenberg Uncertainty Principle, which states that ΔxΔp must be greater than or equal to ℏ/2. Using the provided equation p = ℏk, we can find the uncertainty in the momentum of a particle given its uncertainty in position (Δx).

Given that Δx is known, the uncertainty in momentum (Δp) can be expressed as ℏ/Δx based on the de Broglie's relations, which also use the reduced Planck constant (ℏ). The smaller the value of Δx, the larger the uncertainty in momentum (Δp) as they are inversely proportional.

Therefore to find the uncertainty in the momentum of the particle, we can rearrange the Heisenberg Uncertainty Principle to Δp = ℏ/(2Δx) and plug in the known value of Δx to obtain Δp in terms of ℏ and the given uncertainty in position.

Answer 2

The value of the product of uncertainties in position and momentum, according to Heisenberg's Uncertainty Principle, is at least ℏ/2. To find the uncertainty in momentum using p = ℏk, we express the uncertainty principle as ΔxΔp ≥ ℏ/2 and solve for Δp.

Step-by-step explanation:

The question pertains to Heisenberg's Uncertainty Principle in Quantum Mechanics, which states that one cannot simultaneously know the exact position (Δx) and the exact momentum (Δp) of a particle. The principle is typically represented by the inequality ΔxΔp ≥ ℏ/2, where ℏ is the reduced Planck constant (ℏ = h/2π). To find the uncertainty in the momentum of the particle using p = ℏk, where k is the wave number, we first write down the uncertainty principle as ΔxΔp ≥ ℏ/2 and substitute the given relation for momentum. The uncertainty in the momentum Δp can be rewritten using de Broglie's hypothesis as Δp ≥ ℏ/(2Δx).