A student is examining a bacterium under the microscope. The E. coli bacterial c

Question

A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of m = 1.60 fg (where a femtogram, fg, is 1015g) and is swimming at a velocity of v = 1.00 m/s , with an uncertainty in the velocity of 1.00 % . E. coli bacterial cells are around 1 m ( 106 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.

Explanation / Answer

The E. coli bacterial cell has a mass of m = 1.60 fg (where a femtogram, fg, is 1015g) and is swimming at a velocity of v = 1.00 m/s , with an uncertainty in the velocity of 1.00 % . E. coli bacterial cells are around 1 m ( 106 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.

First convert 2fg to kilograms:

1.60 /10^18 = 1.6 *10^-18

Then convert micro meters per second (um/s) to m/s:

1/10^6= 1*10^-6

Since there is an uncertainty of 1%, multiply the velocity of the bacterium by .01:
( 1*10^-6)(0.01) = (1*10^-8)

By multiplying the uncertainty in the velocity by the mass you'll get an uncertainty in momentum which will help you find the uncertainty of the position:

(1*10^-8)*(1.6*10^-18)= 1.6*10^-26

By Heisenburg's uncertainty principle:

delta x * mass * delta y >/= (h/4pi) where delta x is the uncertainty in position, delta y is uncertainty in velocity, and h is planck's constant.

you can find that:

delta x= h / (4pi * m * (delta y)) where delta x is the uncertainty in position, delta y is uncertainty in velocity, and h is planck's constant.
so:

delta x= 6.626*10^-34 / (4pi * m * delta y)

delta x= 6.626*10^-34 / (4pi * (1.6*10^-26)

delta x= 3.29550204e-9

So the uncertainty of the position of the bacterium is 3.2955*10^-9

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