Timeline for Quantum mechanics formalism and C*-algebras
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 30, 2013 at 8:42 | comment | added | jjcale | If you know the expectation values of all projection operators then you can also calculate the expectation values of the unbounded operators. | |
| Aug 13, 2011 at 19:20 | answer | added | Jonathan Gleason | timeline score: 4 | |
| Feb 20, 2011 at 20:44 | vote | accept | Naz Miheisi | ||
| Feb 20, 2011 at 20:44 | vote | accept | Naz Miheisi | ||
| Feb 20, 2011 at 20:44 | |||||
| Feb 20, 2011 at 10:29 | answer | added | Stefan Waldmann | timeline score: 28 | |
| Feb 19, 2011 at 19:12 | comment | added | Paul Siegel | It's also worth mentioning that von Neumann in fact invented much of what we now consider to be basic C*-algebra and von Neumann algebra theory in order to handle operators coming from quantum mechanics that he knew not to be bounded | |
| Feb 19, 2011 at 19:08 | comment | added | Paul Siegel | I don't have anything substantially new to add to Pieter Naaijkens' comment and Tim van Beek's answer. I just want to comment that the use of C*-algebraic techniques in this context should be considered natural and even expected. The real point is that an essentially self adjoint unbounded operator $T$ can be characterized (in a certain sense) by its functional calculus, i.e. by the \textit{bounded} operators $f(T)$ where $f$ is a bounded continuous function on the spectrum of $T$. This is also the reason why C*-algebraic techniques are useful for understanding differential operators. | |
| Feb 19, 2011 at 18:28 | answer | added | Tim van Beek | timeline score: 19 | |
| Feb 19, 2011 at 16:50 | comment | added | Pieter Naaijkens | One thing you can do is to consider $U(s) = e^{i s P}$ and $V(t) = e^{i t Q}$ for example, if $P,Q$ are the position and momentum operators and $s,t$ real numbers. $U(s)$ and $V(t)$ are unitaries.You then have the Weyl form of the canonical commutation relations, $U(s) V(t) = e^{ist} V(t) U(s)$. | |
| Feb 19, 2011 at 16:39 | comment | added | user5831 | If I remember correctly then your question is explained very nicely in Strocchi's lecture notes "An Introduction to the Mathematical Structure of Quantum Mechanics - A Short Course for Mathematicians". | |
| Feb 19, 2011 at 16:30 | history | asked | Naz Miheisi | CC BY-SA 2.5 |