Calculate the uncertainty in speed of an electron if its position in space is known within the range of ±2pm. Assume me = 9×(10^-31) kg
To calculate the uncertainty in speed of an electron, we can make use of Heisenberg's uncertainty principle. According to the principle, the product of the uncertainties in position and momentum of a particle should be greater than or equal to a certain value.
In this case, the uncertainty in position is given as ±2 pm (picometers), which is equivalent to ±2 × 10^(-12) meters.
Given that the electron's mass is me = 9 × 10^(-31) kg, the uncertainty principle can be expressed as:
Δx * Δp ≥ h/4π
Where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant (h = 6.626 × 10^(-34) J·s).
We can rearrange the equation to solve for the uncertainty in momentum:
Δp ≥ h / (4π * Δx)
Substituting the given values:
Δp ≥ (6.626 × 10^(-34) J·s) / (4π * 2 × 10^(-12) m)
Calculating:
Δp ≥ 1.05 × 10^(-22) kg·m/s
Now, since momentum (p) is defined as the product of mass (m) and velocity (v), we can substitute the values into the formula:
Δp = m * Δv
Δv = Δp / m
Substituting the values:
Δv = (1.05 × 10^(-22) kg·m/s) / (9 × 10^(-31) kg)
Calculating:
Δv ≈ 1.17 × 10^8 m/s
Hence, the uncertainty in speed of the electron is approximately 1.17 × 10^8 m/s.