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advanced_tools:expectation_values [2017/11/05 16:12]
jakobadmin ↷ Page moved from advanced_notion:expectation_values to advanced_notions:expectation_values
advanced_tools:expectation_values [2017/12/04 08:01]
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-====== Expectation Values ====== 
- 
-<tabbox Why is it interesting?> ​ 
- 
-<tabbox Layman> ​ 
- 
-<note tip> 
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. 
-</​note>​ 
-  ​ 
-<tabbox Student> ​ 
- 
-<​blockquote>​ 
-To that end, consider a locally defined microscopic variable which I will denote 
-$\psi(r)$. In a ferromagnet this might well be the local magnetization,​ M(r), or spin 
-vector, S(r), at point r in ordinary d-dimensional (Euclidean) space; in a fluid it 
-might be the deviation $\delta p(r)$, of the fluctuating density at r from the mean density. 
-In QFT the local variables $\psi(r)$ are the basic quantum fields which are '​operator 
-valued.'​ For a magnetic system in which quantum mechanics was important, M(r) 
-and S(x) would, likewise, be operators. However, the distinction is of relatively 
-minor importance so that we may, for ease, suppose $\psi(r)$ is a simple classical variable. 
-It will be most interesting when $\psi$ is closely related to the order parameter for the 
-phase transition and critical behavior of concern. 
- 
-By means of a scattering experiment (using light, x rays, neutrons, electrons, etc.) 
-one can often observe the corresponding //pair correlation function// (or basic '​two-point 
-function'​) 
-$$ G(r)=\langle \psi(0) \psi(r)\rangle $$ 
-where the angular brackets $ \langle \cdot \rangle$ denote a statistical average over the thermal fluctuations that characterize all equilibrium systems at nonzero temperature. (Also understood, when $\psi(r)$ is an operator, are the corresponding quantum-mechanical 
-expectation values.) 
-Physically, $G(r)$ is important since it provides a direct measure of the influence of 
-the leading microscopic fluctuations at the origin $0$ on the behavior at a point distance 
-$r = |r|$ away. But, almost by definition, in the vicinity of an appropriate critical point 
-- for example the Curie point of a ferromagnet when $\psi = \vec M$ or the gas-liquid critical 
-point when $\psi =  \delta p$ - a strong "​ordering"​ influence or correlation spreads out over, 
-essentially,​ macroscopic distances. As a consequence,​ precisely at criticality one 
-rather generally finds a power-law decay, namely, 
-$$ G(r) \approx D/​d^{d-2+\eta} $$ 
-as $r \to \infty$ 
-which is characterized by the [[advanced_notions:​critical_exponent]] (or critical index) $d - 2 + \eta$. 
- 
-<​cite>​Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</​cite>​ 
-</​blockquote>​ 
- 
- 
-<note tip> 
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas. 
-</​note>​ 
-  
-<tabbox Researcher> ​ 
- 
-<note tip> 
-The motto in this section is: //the higher the level of abstraction,​ the better//. 
-</​note>​ 
- 
---> Common Question 1# 
- 
-  
-<-- 
- 
---> Common Question 2# 
- 
-  
-<-- 
-  ​ 
-<tabbox Examples> ​ 
- 
---> Example1# 
- 
-  
-<-- 
- 
---> Example2:# 
- 
-  
-<-- 
-  ​ 
-<tabbox History> ​ 
- 
-</​tabbox>​ 
- 
  
advanced_tools/expectation_values.txt · Last modified: 2017/11/05 16:13 by jakobadmin