The Heisenberg Uncertainty Principle:

A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of mm = 0.100fg\rm fg (where a femtogram, fg\rm fg, is 10−15g\rm 10^{-15}\; g) and is swimming at a velocity of vv = 7.00ìm/s\mu m/s , with an uncertainty in the velocity of 5.00%\% . E. coli bacterial cells are around 1 ìm\mu \rm m ( 10−6 m10^{-6}~\rm 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? Delta X =?express the answer in meter?

To find the uncertainty in the position of the bacterium, we can use the Heisenberg uncertainty principle formula:

Δx * Δp ≥ h/4π

Where Δx represents the uncertainty in position, Δp represents the uncertainty in momentum, and h is the Planck constant.

In this case, we are given the uncertainty in velocity (Δv), but we need to convert it to uncertainty in momentum (Δp) since momentum is the product of mass and velocity.

The uncertainty in momentum can be calculated using the formula:

Δp = m * Δv

Where m is the mass of the bacterium.

Given:
Mass of bacterium (m) = 0.100 fg (10^-15 g)
Velocity of bacterium (v) = 7.00 μm/s (10^-6 m/s)
Uncertainty in velocity (Δv) = 5.00%

First, let's convert the uncertainty in velocity to meters per second:

Δv = (5.00/100) * v = (0.05) * 7.00 μm/s = 0.35 μm/s

Next, let's convert the mass of the bacterium to kilograms:

m = 0.100 fg = 0.100 * 10^-15 g = 0.100 * 10^-18 kg

Now we can calculate the uncertainty in momentum:

Δp = m * Δv = (0.100 * 10^-18 kg) * (0.35 * 10^-6 m/s) = 0.035 * 10^-24 kg*m/s

Finally, we can use the Heisenberg uncertainty principle to find the uncertainty in position:

Δx * Δp ≥ h/4π

Rearranging the formula to solve for Δx:

Δx ≥ (h/4π) / Δp

Since h/4π is a constant (approximately 5.27 * 10^-35 m), we can substitute the values:

Δx ≥ (5.27 * 10^-35 m) / (0.035 * 10^-24 kg*m/s)

Calculating this expression will give us the minimum value for the uncertainty in position. However, it is important to note that this calculation assumes a delta function wave function, which may not be accurate in the case of a bacterium.

Please perform the calculation to find the uncertainty in the position of the bacterium by substituting the values and solving for Δx.