A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of m = 2.00 fg (where a femtogram, fg, is 10−15g) and is swimming at a velocity of v = 7.00 μm/s , with an uncertainty in the velocity of 2.00 % . 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?

Heisenberg's uncertainty principle states that it is not possible to know with infinite precision the position and the momentum of an object at the same time. Mathematically, this is written as:

[tex]\Delta x \Delta p \geq \frac{h}{4\pi}[/tex]

where:

[tex]\Delta x[/tex] is the uncertainty on the position

[tex]\Delta p[/tex] is the uncertainty on the momentum

[tex]h=6.63\cdot 10^{-34} Js[/tex] is the Planck's constant

Since the momentum can be written as product of mass (m) and velocity:

[tex]p=mv[/tex]

The uncertainty on the momentum can be written as (assuming the mass is known with infinite precision):

[tex]\Delta p = m\Delta v[/tex]

Therefore, the previous equation can be rewritten as:

[tex]m\Delta x\Delta v\geq \frac{h}{4\pi}[/tex]

In this problem, we have:

[tex]m=2.00 fg = 2.00\cdot 10^{-15} g = 2.00\cdot 10^{-18} kg[/tex] is the mass of the E.Coli

[tex]v=7.00\mu m/s = 7.00\cdot 10^{-6} m/s[/tex] is the E.Coli velocity

The uncertainty on the velocity is 2.00% of this value, so:

[tex]\Delta v = \frac{2}{100}v=0.02\cdot 7.00\cdot 10^{-6}=1.4\cdot 10^{-7} m/s[/tex]

Therefore, if we now re-arrange the equation, we can find [tex]\Delta x[/tex], the minimum uncertainty on the position of the bacterium:

[tex]\Delta x \geq \frac{h}{4\pi m\Delta v}=\frac{6.63\cdot 10^{-34}}{4\pi (2.0\cdot 10^{-18})(1.4\cdot 10^{-7})}=1.88\cdot 10^{-10} m[/tex]