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?
To find the uncertainty in position (Δx), we can use the Heisenberg Uncertainty Principle equation:
Δx * Δp >= h/4π
Where Δp is the uncertainty in momentum and h is the Planck's constant (approximately 6.626 x 10^-34 J*s).
In this case, we are given the uncertainty in velocity rather than momentum. So we need to calculate the uncertainty in momentum first:
Δp = m * Δv
Where m is the mass of the bacterium and Δv is the uncertainty in velocity.
Given:
Mass of the bacterium (m) = 0.100 fg = 0.100 x 10^-15 g
Uncertainty in velocity (Δv) = 5.00% of 7.00 μm/s
To convert the units properly, we need to convert the mass to kilograms and the velocity to meters per second:
1 fg = 10^-18 g
1 μm = 10^-6 m
Mass of the bacterium (m) = 0.100 x 10^-15 g * (10^-18 kg / 1 g)
= 0.100 x 10^-33 kg
Uncertainty in velocity (Δv) = 5.00% * 7.00 μm/s * (10^-6 m / 1 μm)
= 5.00% * 7.00 x 10^-6 m/s
= 0.050 * 7.00 x 10^-6 m/s
= 0.350 x 10^-6 m/s
Now we can calculate the uncertainty in momentum:
Δp = (0.100 x 10^-33 kg) * (0.350 x 10^-6 m/s)
= 0.035 x 10^-39 kg·m/s
Finally, we can calculate the uncertainty in position (Δx):
Δx >= h / (4π * Δp)
Δx >= (6.626 x 10^-34 J*s) / (4π * 0.035 x 10^-39 kg·m/s)
Simplifying the equation, we get:
Δx >= 4.745 x 10^-25 m
Therefore, the uncertainty in the position of the bacterium is at least 4.745 x 10^-25 meters.