The speed of an electron is known to be between 3.0×10^6 m/s and 3.3×10^6 m/s . Estimate the uncertainty in its position.
To estimate the uncertainty in the position of an electron, we'll use Heisenberg's uncertainty principle, which states that it is impossible to simultaneously determine the precise position and momentum of a particle with absolute certainty.
The uncertainty principle is mathematically expressed as follows:
Δx * Δp >= h/4π
Where:
Δx = uncertainty in position
Δp = uncertainty in momentum
h = Planck's constant (approximately 6.626 x 10^-34 J.s)
π = Pi (approximately 3.14159)
We are given the speed (v) of the electron, but to find the uncertainty in momentum (Δp), we need to calculate its mass (m).
The formula for momentum is:
p = m * v
Rearranging the equation, we get:
m = p / v
From the given speed range, we can determine the range of momentum values:
p_min = m * v_min
p_max = m * v_max
Next, we can calculate the uncertainty in momentum by subtracting the minimum momentum from the maximum momentum:
Δp = p_max - p_min
Once we have the uncertainty in momentum (Δp), we can substitute the values into the uncertainty principle equation to find the uncertainty in position (Δx).
Let's proceed with the calculations:
Given:
v_min = 3.0 x 10^6 m/s
v_max = 3.3 x 10^6 m/s
First, calculate m using the minimum velocity:
m_min = p_min / v_min
Next, calculate m using the maximum velocity:
m_max = p_max / v_max
Then, calculate the minimum momentum:
p_min = m_min * v_min
And calculate the maximum momentum:
p_max = m_max * v_max
Finally, calculate the uncertainty in momentum:
Δp = p_max - p_min
Now substitute the uncertainty in momentum into the uncertainty principle equation:
Δx * Δp >= h / (4π)
Rearrange the equation to solve for Δx:
Δx >= (h / (4π)) / Δp
Evaluate the expression to find the final estimate for the uncertainty in position (Δx).