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--- classes:2009:fall:phys4101.001:lec_notes_0911 [2009/09/14 08:18] – yk
+++ classes:2009:fall:phys4101.001:lec_notes_0911 [2009/09/23 22:43] (current) – yk
@@ Line 1: / Line 1: @@
-===== Sept 11 (Fri) =====
+===== Sept 11 (Fri) Probability interpretation =====
-** Responsible party: Schrödinger's Dog, Devlin**
+** Responsible party: Schrödinger's Dog**
-**To go back to the lecture note list, click [[lec_notes]]**
+**To go back to the lecture note list, click [[lec_notes]]**\\
+**previous lecture note: [[lec_notes_0909]]**\\
+**next lecture note: [[lec_notes_0914]]**\\
+**[[Q_A]]**
 === Main points ===
@@ Line 35: / Line 38: @@
  This concluded the lecture.
+//Yuichi//
+Clarification on this point.  What we want to show is that <math>\frac{d}{dt} \int_{-\infty}^{\infty} f(x,t) dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial t} f(x,t) dx</math>.  \\
+By the definition of differentiation, <math>\frac{d}{dt} \int_{-\infty}^{\infty} f(x,t) dx=\lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} f(x,t+\Delta t) dx - \int_{-\infty}^{\infty} f(x,t) dx}{\Delta t} </math>.\\
+In order to get the difference between the two terms in the numerator of the RHS, it may be convenient to express <math>f(x,t+\Delta t)</math> in terms of <math>f(x,t)</math>.  If we CHOOSE to compare these two terms for the same value of //x//* (note that this is up to me), we find that <math>f(x,t+\Delta t) = f(x,t)+\frac{\partial f(x,t)}{\partial t}\Delta t</math>.  By subbing this into the above expression with a limit, we find that <math>\lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} f(x,t+\Delta t) dx - \int_{-\infty}^{\infty} f(x,t) dx}{\Delta t} = \lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} \frac{\partial f(x,t)}{\partial t}\Delta t dx}{\Delta t} = \int_{-\infty}^{\infty} \frac{\partial f(x,t)}{\partial t} dx</math>.
+*Even though it's silly to choose different value of //x// by <math>\Delta x</math>, then <math>f(x+\Delta x,t+\Delta t) = f(x,t)+\frac{\partial f(x,t)}{\partial t}\Delta t+ \frac{\partial f(x,t)}{\partial x}\Delta x</math>.
+Then, <math>\lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} f(x+\Delta x,t+\Delta t) dx - \int_{-\infty}^{\infty} f(x,t) dx}{\Delta t} = \lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} \frac{\partial f(x,t)}{\partial t}\Delta t dx + \frac{\partial f(x,t)}{\partial x}\Delta x dx}{\Delta t} = \int_{-\infty}^{\infty} \frac{\partial f(x,t)}{\partial t} dx + \Delta x[f(x,t)]_{-\infty}^{\infty}</math>.  Provided that <math>f(x,t)</math> is a normalizable function, the last term should be zero.
+==Schrodinger's Dog==
+Thanks Yuichi, this makes a lot more sense!
 ==End of lecture==
 S.D.: I wasn't really clear on the last proof, but feel that it isn't a important detail. Besides that, the rest of the lecture was great!
+**To go back to the lecture note list, click [[lec_notes]]**\\
+**previous lecture note: [[lec_notes_0909]]**\\
+**next lecture note: [[lec_notes_0914]]**\\

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