Question

A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of m = 0.900 fg (where a femtogram, fg, is 10^−15g) and is swimming at a velocity of v = 2.00 μm/s , with an uncertainty in the velocity of 3.00 % . E. coli bacterial cells are around 1 μm ( 10^−6 m) in length. The student is supposed to observe the bacterium and make a drawing. However, the student, having just learned about the Heisenberg uncertainty principle in physics class, complains that she cannot make the drawing. She claims that the uncertainty of the bacterium's position is greater than the microscope's viewing field, and the bacterium is thus impossible to locate.
Part A

What is the uncertainty of the position of the bacterium? Express your answer with the appropriate units.

Answer 1

Answer:

9.73 x 10⁻¹⁰ m

Explanation:

According to Heisenberg uncertainty principle

Uncertainty in position x uncertainty in momentum ≥ h / 4π

Δ X x Δp ≥ h / 4π

Δp = mΔV

ΔV = Uncertainty in velocity

= 2 x 10⁻⁶ x 3 / 100

= 6 x 10⁻⁸

mass m = 0.9 x 10⁻¹⁵ x 10⁻³ kg

m = 9 x 10⁻¹⁹

Δp = mΔV

= 9 x 10⁻¹⁹ x 6 x 10⁻⁸

= 54 x 10⁻²⁷

Δ X x Δp ≥ h / 4π

Δ X x  54 x 10⁻²⁷ ≥ h / 4π

Δ X = h / 4π x 1 /  54 x 10⁻²⁷

= $\frac{6.6\times10^{-34}}{4\times3.14\times54\times10^{-27}}$

= 9.73 x 10⁻¹⁰ m