An electron traveling at 4.3×105 m/s has an uncertainty in its velocity of 1.92×105 m/s.
What is the uncertainty in its position?
To determine the uncertainty in the position of an electron, we can use the Heisenberg uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is always greater than or equal to a constant value called Planck's constant (h). The equation for the Heisenberg uncertainty principle is given as:
Δx * Δp ≥ h/4π
Where:
Δx is the uncertainty in position,
Δp is the uncertainty in momentum, and
h is the Planck's constant (approximately 6.63 x 10^-34 J·s).
In this case, we are given the uncertainty in the electron's velocity, not momentum. However, momentum (p) is related to velocity (v) by the equation p = mv, where m is the mass of the electron.
To calculate the momentum uncertainty, we multiply the mass of the electron by the velocity uncertainty:
Δp = m * Δv
Given that the mass of an electron (m) is approximately 9.11 × 10^-31 kg and the uncertainty in velocity (Δv) is 1.92 × 10^5 m/s, we can substitute these values into the equation:
Δp = (9.11 × 10^-31 kg) * (1.92 × 10^5 m/s)
Calculating Δp will give us the uncertainty in momentum.
Finally, we can use the Heisenberg uncertainty principle equation to determine the uncertainty in position:
Δx * Δp ≥ h/4π
Rearranging the equation to isolate Δx:
Δx ≥ h/(4π * Δp)
Now we can substitute the calculated value of Δp into the equation and solve for Δx.