DxDp = h/4p
Where Dx is uncertainty of position, Dp is uncertainty in momentum, h is Planck's constant 5.26e-33 Joule seconds, and p is linear momentum, P = mv.
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Where Dx is uncertainty of position, Dp is uncertainty in momentum, h is Planck's constant 5.26e-33 Joule seconds, and p is linear momentum, P = mv.
Δx * Δp >= ħ/2
To find the uncertainty in position, we need to calculate the uncertainty in momentum first. The momentum (p) of an object can be calculated using the formula:
p = m * v
where m represents the mass and v represents the velocity.
Given:
Speed of the electron (v) = (6.5 ± 0.1) * 10^5 m/s
Mass of the electron (m) = 9.109 * 10^-31 kg
Calculating the momentum of the electron:
p = m * v
p = (9.109 * 10^-31 kg) * (6.5 * 10^5 m/s)
p = 5.92085 * 10^-25 kg.m/s
Now we can calculate the uncertainty in momentum (Δp). Since the uncertainty in velocity is given as ±0.1 * 10^5 m/s, we can use this value to represent the uncertainty in momentum as well.
Therefore, the uncertainty in momentum is:
Δp = ±0.1 * 10^5 kg.m/s
Finally, we can use Heisenberg's uncertainty principle to find the uncertainty in position (Δx). Rearranging the formula, we have:
Δx >= ħ/(2 * Δp)
Plugging in the values:
ħ = 1.0545718 × 10^-34 J.s (Planck's constant divided by 2π)
Δp = ±0.1 * 10^5 kg.m/s
Calculating the uncertainty in position:
Δx >= (1.0545718 × 10^-34 J.s) / (2 * (±0.1 * 10^5 kg.m/s))
Note: The uncertainty in position will be a positive value since we are dealing with magnitudes.
Therefore, the uncertainty in position (in meters) is given by:
Δx >= 5.272859 × 10^-40 m
Hence, the uncertainty in the position of the electron is approximately 5.27 × 10^-40 meters.