According to the Heisenberg uncertainty principle, if the uncertainty in the speed of an electron is 3.5 x 10(3) m/s, the uncertainty in its position is at least
a)66 m
b)17 m
c)6.6 x 10-8 m
d)1.7 x 10-8 m
e)None of the above
6.6 x 10-8
The Heisenberg uncertainty principle states that the product of the uncertainties in the position and momentum of a particle must be greater than or equal to a constant value, specifically, Planck's constant divided by 4π.
Mathematically, it can be written as ∆x * ∆p ≥ h / (4π), where ∆x represents the uncertainty in position, ∆p represents the uncertainty in momentum, and h is Planck's constant.
In this case, we are given the uncertainty in the speed of an electron (∆v = 3.5 x 10^3 m/s). Remember that momentum (p) is equal to mass (m) times velocity (v), so we can rewrite the uncertainty in momentum as ∆p = m * ∆v.
Since the mass of an electron is constant, we can focus on the uncertainty in velocity (∆v) to calculate the uncertainty in momentum (∆p).
Now, to find the uncertainty in position (∆x), we need to rearrange the uncertainty principle equation:
∆x * ∆p ≥ h / (4π)
∆x ≥ h / (4π * ∆p)
Plugging in the values, we have:
∆x ≥ (6.63 x 10^-34 J s) / (4π * m * ∆v)
Since we are only interested in the minimum uncertainty, we want to find the maximum value for ∆x. To do this, we need to find the minimum value for ∆v, which is given as 3.5 x 10^3 m/s.
Substituting the values, we get:
∆x ≥ (6.63 x 10^-34 J s) / (4π * (9.11 x 10^-31 kg) * (3.5 x 10^3 m/s))
Calculating this expression, we find:
∆x ≥ 1.7 x 10^-8 m
Therefore, the correct answer is option d) 1.7 x 10^-8 m.