Question

A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of m� = 1.20 fgfg (where a femtogram, fgfg, is 10−15g10−15g) and is swimming at a velocity of v� = 5.00 μm/s�m/s , with an uncertainty in the velocity of 7.00 %% . E. coli bacterial cells are around 1 μm�m ( 10−6 m10−6 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.

What is the uncertainty of the position of the bacterium?

Express your answer with the appropriate units.

Answer 1

Answer: the uncertainty of the position of the bacterium is approximately 1.255 x 10^(-10) meters (or 125.5 picometers, pm).

Step-by-step explanation:

The Heisenberg uncertainty principle states that there is a fundamental limit to how precisely we can simultaneously know the position (Δx) and momentum (Δp) of a particle, and this limit is given by the inequality:

Δx * Δp ≥ ħ / 2

where ħ (pronounced "h-bar") is the reduced Planck constant, approximately equal to 1.0545718 x 10^(-34) J·s.

In this case, we are dealing with a bacterium's mass and velocity, and we want to find the uncertainty in its position. The uncertainty in position (Δx) can be related to the uncertainty in momentum (Δp) using the following relationship:

Δp = m * Δv

where m is the mass of the bacterium, and Δv is the uncertainty in its velocity.

Given:

Mass of the bacterium (m) = 1.20 fg = 1.20 x 10^(-15) g

Velocity of the bacterium (v) = 5.00 μm/s = 5.00 x 10^(-6) m/s

Uncertainty in velocity (Δv) = 7.00% = 0.07 (as a decimal)

Now, calculate Δp:

Δp = (1.20 x 10^(-15) g) * (0.07 * 5.00 x 10^(-6) m/s)

Δp = (1.20 x 10^(-15) g) * (3.50 x 10^(-7) m/s)

Now, convert the mass from grams to kilograms (since 1 g = 10^(-3) kg):

Δp = (1.20 x 10^(-18) kg) * (3.50 x 10^(-7) m/s)

Δp = 4.20 x 10^(-25) kg·m/s

Now, we can use the Heisenberg uncertainty principle to find the uncertainty in position (Δx):

Δx * Δp ≥ ħ / 2

Δx * (4.20 x 10^(-25) kg·m/s) ≥ (1.0545718 x 10^(-34) J·s) / 2

Δx * (4.20 x 10^(-25) kg·m/s) ≥ 5.272859 x 10^(-35) J·s

Now, solve for Δx:

Δx ≥ (5.272859 x 10^(-35) J·s) / (4.20 x 10^(-25) kg·m/s)

Δx ≥ 1.255204 x 10^(-10) m

So, the uncertainty of the position of the bacterium is approximately 1.255 x 10^(-10) meters (or 125.5 picometers, pm).

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