An electron traveling at 2.1×E5 m/s has an uncertainty in its velocity of 7.98×E4 m/s.
What is the uncertainty in its position?
To find the uncertainty in position, we can use the uncertainty principle which states that the product of the uncertainties in position and momentum (or velocity) of a particle is inversely proportional to the Planck's constant (h).
The uncertainty principle is expressed as:
Δx * Δp ≥ h/4π
Here, Δx is the uncertainty in position, Δp is the uncertainty in momentum (or velocity), and h is the Planck's constant.
Given values:
Velocity (Δv) = 2.1x10^5 m/s
Uncertainty in velocity (Δv) = 7.98x10^4 m/s
First, let's convert the velocity and uncertainty in velocity to momentum and uncertainty in momentum using the mass of an electron (me = 9.11x10^-31 kg):
Momentum (p) = mass * velocity
Δp = mass * Δv
Substituting the values:
p = (9.11x10^-31 kg) * (2.1x10^5 m/s)
Δp = (9.11x10^-31 kg) * (7.98x10^4 m/s)
Now, let's calculate the uncertainty in position (Δx) using the uncertainty principle formula:
Δx * Δp ≥ h/4π
Rearranging the formula to solve for Δx:
Δx ≥ h/(4π * Δp)
Substituting the values:
Δx ≥ (6.63x10^-34 J-s)/(4π * (9.11x10^-31 kg) * (7.98x10^4 m/s))
Calculating the uncertainty in position:
Δx ≥ 5.25x10^-10 meters
Therefore, the uncertainty in the position of the electron is approximately 5.25x10^-10 meters.
To determine the uncertainty in the electron's position, we can use Heisenberg's uncertainty principle, which states that the product of the uncertainties in position and velocity of a particle must be greater than or equal to a constant (h-bar) divided by 2.
Mathematically, it can be written as:
Δx * Δv ≥ h-bar / 2
In this case, we are given the uncertainty (Δv) in the electron's velocity as 7.98×E4 m/s, and we need to find the uncertainty (Δx) in its position. We can rearrange the formula to solve for Δx:
Δx ≥ h-bar / (2 * Δv)
Now, let's calculate the value using the given values.
The reduced Planck constant, h-bar, is approximately 1.0546 × 10^−34 Joule-seconds.
Let's substitute the values into the equation:
Δx ≥ (1.0546 × 10^−34 Js) / (2 * 7.98×E4 m/s)
Δx ≥ 6.623 × 10^−40 m²/s
Thus, the uncertainty in the electron's position is approximately 6.623 × 10^−40 m²/s.