Question

A biology student must use a microscope to track a single bacterium traveling along the length of a cell. The bacterium is `1.20` `f"g"` and travels at a speed of `1.00 xx 10^(-6) (pm 5.00%) "m/s"` [ . . . ] ?

A biology student must use a microscope to track a single bacterium traveling along the length of a cell. The bacterium is `1.20` `f"g"` and travels at a speed of `1.00 xx 10^(-6) (pm 5.00%) "m/s"`. The minimum viewing window of the microscope is `1` `mu"m"`.

Before starting, she complains that she will NOT be able to get a clear image of the bacterium through the microscope and demands better equipment. Based on what you know about the Heisenberg Uncertainty Principle, is the student correct? Why or why not?

Answer 1

The student is not correct. The uncertainty is at least `8.88 xx 10^(-4)` `mu"m"`, but that is over `1000` times smaller than the minimum viewing window needed to see the whole bacterium.

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Alright, well, we can tell that this is about the Heisenberg Uncertainty Principle, because obviously the question talks about it.

`barul(|" "stackrel(" ")DeltaxDeltap >= ℏ//2" "|)`

• `Deltax` is the uncertainty in the position.
• `Deltap = mDeltav` is the uncertainty in the momentum.
• `ℏ = h//2pi` is the reduced Planck's constant.
• `m` is the mass of the bacterium in `"kg"`.

We assume the bacterium is restricted to one dimension, and that that dimension is along the cell length.

Noting the conversions needed to get from `"g" -> "kg"`, `mu "m" -> "m"`, and `f"g" -> "g"`, the uncertainty in the position is then:

`color(blue)(Deltax) >= (ℏ//2)/(mDeltav)`

`= overbrace((6.626 xx 10^(-34) cancel"kg"cdot"m"^(cancel(2))"/"cancel"s" // 4 pi))^(ℏ//2)/(underbrace(1.20 xx 10^(-15) "g" xx cancel"1 kg"/(1000 cancel"g"))_(m) xx underbrace(0.0500 xx 1.00 xx 10^(-6) cancel"m/s")_(Deltav))`

`>= color(blue)(8.88 xx 10^(-10))` `color(blue)("m")` or `8.88 xx 10^(-4)` `mu"m"`.

So the student is not correct.

The uncertainty is at most `1126.12` times smaller than the minimum required viewing window to which the microscope should be adjusted, and the image should be clear... microscopes for bacterium are fairly standard fare...