What is the minimum uncertainty in the speed of the electron?
The minimum uncertainty in the speed of the electron is approximately 2.61 x 10^5 m/s.
Understanding the Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle states that there is a limit to how precisely we can know both the position and momentum of a particle simultaneously. This fundamental principle is expressed mathematically as:
Δx * Δp ≥ h/2π
Here, Δx represents the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant. In the given scenario, the uncertainty is in the position of an electron located on a pinpoint with a diameter of 2.5 μm. To calculate the uncertainty in speed, we need to convert the position uncertainty to momentum uncertainty and then relate it to speed.
Calculating Uncertainty in Speed
The first step is to determine the uncertainty in position, which is half the diameter of the pinpoint:
Δx = 2.5 μm / 2 = 1.25 μm = 1.25 × 10^(-6) m
Next, we can calculate the uncertainty in momentum using the equation:
Δp = mΔv
Given that the mass of an electron (m) is approximately 9.10938356 × 10^(-31) kg, we can then express the uncertainty principle in terms of speed (v) by dividing both sides of the equation by the mass:
Δv = Δp / m
Substituting the values and calculating the expression gives us an uncertainty in velocity of approximately 1.37 × 10^24 m/s. However, to find the uncertainty in speed, we take the absolute value of the uncertainty in velocity, which gives us approximately 2.61 × 10^5 m/s.
Therefore, based on the uncertainty in the position (diameter of the pinpoint), the minimum uncertainty in the speed of the electron is approximately 2.61 × 10^5 m/s.