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?
According to the Heisenberg Uncertainty Principle, there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. The principle states that the product of the uncertainties in the position (Δx) and the momentum (Δp) of a particle must be greater than or equal to a certain value. In mathematical terms, it is expressed as:
Δx * Δp >= h / (4π)
where h is the reduced Planck's constant (h/2π ≈ 1.05457 × 10^-34 J·s).
In this case, we will consider the uncertainty in the velocity (Δv) of the bacterium as the uncertainty in momentum (Δp), assuming the bacterium's mass remains constant.
Given the mass of the bacterium (mm = 0.100 fg = 0.100 × 10^-15 g) and the uncertainty in velocity (Δv = 5.00% of 7.00 μm/s), we first need to calculate the uncertainty in momentum (Δp).
Δv = (5.00/100) * 7.00 μm/s = 0.350 μm/s
The uncertainty in momentum (Δp) can be calculated using the mass (m) and velocity uncertainty (Δv) as:
Δp = m * Δv
Converting the mass to kilograms (1 g = 10^-3 kg), we have:
m = 0.100 × 10^-15 g = 0.100 × 10^-18 kg
Δp = (0.100 × 10^-18 kg) * (0.350 × 10^-6 m/s) = 0.035 × 10^-24 kg·m/s
Now, we can use the Heisenberg Uncertainty Principle to find the uncertainty in position (Δx):
Δx * Δp >= h / (4π)
Δx >= h / (4π * Δp)
Δx >= (1.05457 × 10^-34 J·s) / (4π * 0.035 × 10^-24 kg·m/s)
Δx >= 2.981 × 10^-11 m
Hence, the uncertainty in the position (Δx) of the bacterium is approximately 2.981 × 10^-11 meters.