I’m teaching Physics 350: Quantum Mechanics this term, which is a junior/senior level elective course using Townsend’s book which deals with quantum mechanics in the state vector formalism. The room in which the class meets is the only one in the department that contains a whiteboard (using dry-erase markers) rather than a blackboard (using chalk).
In the first several weeks of the course, I have mostly been using blue markers, because that’s what’s been in the room. These fade into illegibility very quickly, so today I went into the stockroom to get more, and discovered a box of black markers of a different brand. The black markers write very clearly on the board, but when I attempted to erase the board at the end of class, the erasers just sort of smeared the ink around leaving a greyish smudge on the board.
From this, we can deduce that the operator W, which describes wiping the board clean, and the operator M, which describes making marks on the board, are non-commuting operators. Formally, the commutator [M,W] is:
[M,W] = MW – WM = 1
Furthermore, there is an uncertainty relationship between M and W such that:
ΔM2 ΔW2 ≥ (1/4) |< [M,W]>|2
We can represent an empty classroom by the vacuum state |vac>, which is a minimum uncertainty state satisfying ΔM2 ΔW2=1/4, so:
ΔM2 = 1/2 = ΔW2
Any coherent lecture state can be produced by successive applications of the knowledge-raising operator K+, which can be viewed as translating the state |vac> outward from the initial state of zero expectation value, preserving the uncertainty area.
And thus, we see that the chance of successfully erasing a whiteboard is:
P(W) ≈ Δ W = (1/2)1/2
or approximately 70%.
(Please note that this result only holds for coherent lecture states; incoherent lectures consisting of a classical mixture of many different states have uncertainty much larger than the minimum, and thus can be erased from both the boards and the students’ minds with higher probability.)
Belated attribution: The author wishes to thank B. Bartell for helpful discussion.
I took QM a few years ago using that book. I understood very little of this post. I’m debating on whether to blame the book or the professor. (I’m going to go with the Professor. He hadn’t taught QM in many years (if ever?) and it showed. Badly. We had a different professor for the second half of the course and things went much better, though it’s hard to come back after failing to grasp the first half.)
You know about the hidden variable: spray that liquid and you’ll achieve a classical state of perfectly clean whiteboard.
Try a different solvent (alcohol? Cheap scotch?) I think prof. needs to get out more 🙂 (it was funny though, and the end result seemed probable)
This reminded me how much I’ve forgotten from college 🙁
black markers suck. dry out faster and erase harder, probably a correlation. black markers suck.
And what about a generalized uncertainty principle that takes into account a minimum length scale from quantum gravity?
boy, you must be bored to come up with this one. Or writing an exam? Did your class get to the minimum uncertainty state as a topic of the lectures? I don’t remember that from undergrad.
I can’t quite wrap my head around the uncertainty relationship between M and W. Normally I associate uncertainty relationships with outcomes of measurements of observables, but I can’t figure out what a “measurement of M” or a “measurement of W” would be.
But aside from those pedantic difficulties:
Given the difficulties in getting whiteboards clean, I like the idea of the “vacuum” state of the board having equal noise/uncertainty apportioned between ink and whitespace. With a bunch of vigorous wiping with the ineffective erasers you could create a “squeezed” state where the center is much whiter than before, but only at the expense of a lot of marker gunk around the corners of the board. So in that sense I kinda like the model.
Perhaps you tried it already, but try painting over the drawing/text with the pens used earlier. The solvent in them usually makes both “inks” erasable.