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| classes:2009:fall:phys4101.001:lec_notes_0911 [2009/09/14 08:15] – 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 === |
| The main points of today's lecture: | The main points of today's lecture: |
| * discussion and interpretation in quantum mechanics (i.e. <p>, <math>\Psi^*\Psi dx=P(x)dx</math>, etc...) | * discussion and interpretation in quantum mechanics (i.e. <p>, <math>\Psi^*\Psi dx=P(x)dx</math>, etc...) |
| * Bohm's postulate and Yuichi's discussion on how Bohm's may have used classical ideas to come up with <math>\Psi^*\Psi dx=P(x)dx</math> | * Born's postulate and Yuichi's discussion on how Born's may have used classical ideas to come up with <math>\Psi^*\Psi dx=P(x)dx</math> |
| * Why we use <math>|\Psi|^2</math>, and not just <math>\Psi</math>, or some linear combination of <math>\Psi</math> for the probability | * Why we use <math>|\Psi|^2</math>, and not just <math>\Psi</math>, or some linear combination of <math>\Psi</math> for the probability |
| * Normalization for t=0 and how the normalization works for time t | * Normalization for t=0 and how the normalization works for time t |
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| ==How Born may have come up with the idea that Psi squared is the Probability Density== | ==How Born may have come up with the idea that Psi squared is the Probability Density== |
| We then discussed how Born may have come up with the interpretation that Psi squared is the Probability Density. Yuichi told us that Born probably looked to classical physics, where he came up with this idea that <math>\Psi^*\Psi dx=P(x)dx</math>. Yuichi explained that when we looked at electromagnetic wave E, and look at <math> |E|^2 \prop</math> Intensity <math>\prop</math> Energy density <math>\prop</math> number of particle <math>\prop probability </math>. From this, we can see that the square of a classical wave can be interpreted as a probability, which Yuichi speculates Born used to interpret <math> |\Psi|^2 </math> to be the probability density. | We then discussed how Born may have come up with the interpretation that Psi squared is the Probability Density. Yuichi told us that Born probably looked to classical physics, where he came up with this idea that <math>\Psi^*\Psi dx=P(x)dx</math>. Yuichi explained that when we looked at electromagnetic wave E, and look at <math> |E|^2 \prop</math> Intensity <math>\prop</math> Energy density <math>\prop</math> number of particle <math>\prop</math> probability. From this, we can see that the square of a classical wave can be interpreted as a probability, which Yuichi speculates Born used to interpret <math> |\Psi|^2 </math> to be the probability density. |
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| This concluded the lecture. | This concluded the lecture. |
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| | //Yuichi// |
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| | 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>.\\ |
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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> 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>. |
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| | 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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