{"id":4643,"date":"2010-04-28T11:15:12","date_gmt":"2010-04-28T11:15:12","guid":{"rendered":"http:\/\/scienceblogs.com\/principles\/2010\/04\/28\/on-the-quantum-physics-of-whit\/"},"modified":"2010-04-28T11:15:12","modified_gmt":"2010-04-28T11:15:12","slug":"on-the-quantum-physics-of-whit","status":"publish","type":"post","link":"http:\/\/chadorzel.com\/principles\/2010\/04\/28\/on-the-quantum-physics-of-whit\/","title":{"rendered":"On the Quantum Physics of Whiteboards"},"content":{"rendered":"

I’m teaching Physics 350: Quantum Mechanics this term, which is a junior\/senior level elective course using Townsend’s book<\/a> 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).<\/p>\n

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.<\/p>\n

From this, we can deduce that the operator W<\/b>, which describes wiping the board clean, and the operator M<\/b>, which describes making marks on the board, are non-commuting operators. Formally, the commutator [M<\/b>,W<\/b>] is:<\/p>\n

\n

[M<\/b>,W<\/b>] = MW<\/b> – WM<\/b> = 1<\/p>\n<\/blockquote>\n

Furthermore, there is an uncertainty relationship between M<\/b> and W<\/b> such that:<\/p>\n

\n

ΔM<\/b>2<\/sup> ΔW<\/b>2<\/sup> ≥ (1\/4) |< [M<\/b>,W<\/b>]>|2<\/sup><\/p>\n<\/blockquote>\n

We can represent an empty classroom by the vacuum state |vac<\/i>>, which is a minimum uncertainty state satisfying ΔM<\/b>2<\/sup> ΔW<\/b>2<\/sup>=1\/4, so:<\/p>\n

\n

ΔM<\/b>2<\/sup> = 1\/2 = ΔW<\/b>2<\/sup><\/p>\n<\/blockquote>\n

Any coherent lecture state can be produced by successive applications of the knowledge-raising operator K<\/b>+<\/sub>, which can be viewed as translating the state |vac<\/i>> outward from the initial state of zero expectation value, preserving the uncertainty area.<\/p>\n

And thus, we see that the chance of successfully erasing a whiteboard is:<\/p>\n

\n

P(W) ≈ Δ W<\/b> = (1\/2)1\/2<\/sup><\/p>\n<\/blockquote>\n

or approximately 70%.<\/p>\n

(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.)<\/p>\n

<\/p>\n

Belated attribution:<\/strong> The author wishes to thank B. Bartell for helpful discussion.<\/p>\n","protected":false},"excerpt":{"rendered":"

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).… Continue reading On the Quantum Physics of Whiteboards<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"1","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[134,7,23,11,24],"tags":[],"class_list":["post-4643","post","type-post","status-publish","format-standard","hentry","category-course_reports","category-physics","category-quantum_optics","category-science","category-silliness","entry"],"_links":{"self":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/4643","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/comments?post=4643"}],"version-history":[{"count":0,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/4643\/revisions"}],"wp:attachment":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/media?parent=4643"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/categories?post=4643"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/tags?post=4643"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}