What is the uncertainty of the position of the bacterium?

A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of m = 2.00 \rm fg (where a femtogram, \rm fg, is \rm 10^{-15}\; g) and is swimming at a velocity of v = 8.00 \mu m/s, with an uncertainty in the velocity of 5.00 \%. E. coli bacterial cells are around 1 \mu \rm m ( 10^{-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. 

Answer: The uncertainty in the position of the bacterium can be calculated using the Heisenberg Uncertainty Principle.

First convert 2 fg to kilograms: 

Mass = 2/1018 = 2*10-18 

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

Velocity = 8/106= 8*10-6 

Since there is an uncertainty of 5%, multiply the velocity of the bacterium by .05: 

Uncertainty in velocity = ( 8*10-6)(0.05) = (4*10-7) 

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: 

Uncertainty in momentum = (4*10-7)*(2*10-18)= 8*1025 

By Heisenberg’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.

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

delta x= 6.626*10-34 / (4pi * (8*1025) 

delta x= 6.626*10-34 / 1.005309 

delta x= 6.591004 *10-11 

So the uncertainty in the position of the bacterium is 6.591*10-11

Therefore, it is evident that the uncertainty in the position of the bacterium is more than the microscope’s viewing field and the bacteria cannot be located.

The Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is an important law in the quantum mechanics that states that the position and velocity of a particle cannot be measured exactly at the same time.

According to the principle, the more precise the position of a particle becomes, the more uncertain is its velocity and vice versa; the more accurately we know one of the values, the less accurately we know the other.

The principle explains why multiple variables cannot be measured simultaneously. The Heisenberg Uncertainty Principle applies to many conjugate pairs like momentum/position and energy/time.

Just like momentum and position, the exact energy of a system cannot be measured in a finite amount of time. Mathematically, The Heisenberg Uncertainty Principle states that:

ΔpΔx > h/4pi

where Δ refers to the uncertainty of that variable and h is the Planck’s constant.

This can also be observed practically by observing that the more precisely we want to measure the position of a particle, like an electron or a bacterium, the more disruptions it causes to the system.

The disruptions can be due to the methods used for measurement like the photon particles. The photon particles also have momentum of their own and when they collide with the test particles of electrons they pass on some of their momentum to them and cause changes in the momentum of the electrons.

Therefore, the more precisely we want to measure the position of the particle, the more changes it brings to the momentum and increases the variability and uncertainty.

Now this does not mean that we cannot find the exact position or the momentum of a particle; it simply means that both the variables cannot be exactly found at the same time.

Explanation of The Heisenberg Uncertainty Principle

The explanation to The Heisenberg Uncertainty Principle can most possibly be found in the wave nature of the particles. According to Louis de Broglie, all the microscopic particles like the electrons, protons, atoms, molecules, bacteria etc. have a dual character. They behave both as particles and as waves.

Therefore, the particles like bacteria also show the wave nature and they are most likely to be found at the places where the crests and troughs of the wave are at the maximum.

However, when the crests and troughs or the undulations in the waves are at the maximum, the wavelength gets disrupted and so does the momentum.

So, a particle that has a definite position will not be able have a definite velocity and the one with a definite and precise velocity will not have a defined position and can be found almost anywhere.

It is to be noted that the dual nature is applicable to all objects and particles, but it becomes more pronounced in the particles with subatomic masses as the effect of the change in velocity is more due to the less mass.

Therefore, we can conclude that since the mass of the bacterium is extremely small and according to the Heisenberg Uncertainty Principle, the exact velocity and position of the bacterium cannot be measured at the same time.

After the calculations, we observe that the uncertainty in the position of the bacterium is very high and it is more than the microscope’s viewing field, so the position of the bacterium cannot be traced and therefore, the bacterium cannot be located and drawn by the student.

References

  1. The Editors of Encyclopaedia Britannica, Uncertainty Principle: Encyclopaedia Britannica.
  2. Raymond Chang, Physical Chemistry for the Biosciences: University Science Books, 2005.
  3. Robert G. Mortimer, Physical Chemistry: San Diego Hardcourt Academic Press, 1993.
  4. Howard Wiseman, Explainer: Heisenberg’s Uncertainty Principle: The Conversation, January 2012.
  5. Alok Jha, What is Heisenberg Uncertainty Principle: The Guardian, November 2013.

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