Chemistry Heisenberg Uncertainty Principle

Question

A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of mmm = 0.100 fg (where a femtogram, fg, is 10−15g) and is swimming at a velocity of v = 6.00 μm/s, with an uncertainty in the velocity of 9.00 % . 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.

What is the uncertainty of the position of the bacterium?



Answer 1

Uncertainty as it relates to the Heisenberg uncertainty principle in practice only applies to particles with mass on the order of electrons and other similar mass particles, since the uncertainty in position of a particle is proportional to its de Broglie wavelength:

λ = h/mv = Δx

where λ is the de Broglie wavelength in meters, h is the Planck constant, m is mass in kilograms, v is velocity in meters per second, and Δx is the uncertainty in position.

In this case, the de Broglie wavelength of an E. Coli bacterial cell is:

h/mv = (6.62607 × 10^-34 J s) / ((1.0 × 10^-19 kg) × (6.00 × 10^-6 m/s))

= 1.10 × 10^-9 m or 0.0011 micrometers.

Therefore, the uncertainty in the position is 0.0011 micrometers.

This is three orders of magnitude smaller than the size of the cell, making it too small to detect with an optical microscope.