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| classes:2009:fall:phys4101.001:lec_notes_0911 [2009/09/14 14:38] – yk | classes:2009:fall:phys4101.001:lec_notes_0911 [2009/09/23 22:43] (current) – yk |
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| ===== Sept 11 (Fri) ===== | ===== Sept 11 (Fri) Probability interpretation ===== |
| ** Responsible party: Schrödinger's Dog, Devlin** | ** Responsible party: Schrödinger's Dog** |
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| **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]]**\\ |
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| | **[[Q_A]]** |
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| === Main points === | === Main points === |
| 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>.\\ | 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>.\\ |
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| 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> to <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>. | 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>. |
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| *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>. | *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. | 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. |
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| | ==Schrodinger's Dog== |
| | Thanks Yuichi, this makes a lot more sense! |
| ==End of lecture== | ==End of lecture== |
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| 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! | 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! |
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| | **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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