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| classes:2009:fall:phys4101.001:q_a_0914 [2009/09/14 10:52] – x500_porte210 | classes:2009:fall:phys4101.001:q_a_0914 [2009/09/20 17:41] (current) – x500_vinc0053 |
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| If i understand the question, | If i understand the question, |
| i believe this correspondence is shown on page 19. Figure 1.7 shows a wave that has no clean position in time and 1.8 shows a pulse which has position in time, but a poorly definable wavelength. | i believe this correspondence is shown on page 19. Figure 1.7 shows a wave that has no clean position in time and 1.8 shows a pulse which has position in time, but a poorly definable wavelength. |
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| | ===The Doctor 21:50 9/14/09=== |
| | You could probably say that a single pulse is a poorly defined wave in that it's wavelength is ill-defined. But then you would be saying that a regular periodic wave is a poorly defined wave in that you don't know it's position. I actually don't remember seeing "poorly defined wave" used anywhere and it may be an unimportant definition. Mostly you want to just be looking at the poorly defined wavelength/position stuff. |
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| ==== Dark Helmet 12:33am 09/13 ==== | ==== Dark Helmet 12:33am 09/13 ==== |
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| ==== John Galt ==== | === John Galt 15:00 9/14/09=== |
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| From Wikipedia: | From Wikipedia: |
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| ***::<math>i\hbar {\partial \over \partial t}\Psi=-{\hbar^2 \over 2m}\nabla^2\Psi + V(x)\Psi</math> | ::<math>i\hbar {\partial \over \partial t}\Psi=-{\hbar^2 \over 2m}\nabla^2\Psi + V(x)\Psi</math> |
| This is the '''time dependent Schrödinger equation'''. It is the equation for the energy in classical mechanics, turned into a differential equation by substituting: | This is the '''time dependent Schrödinger equation'''. It is the equation for the energy in classical mechanics, turned into a differential equation by substituting: |
| ::<math>E\rightarrow i\hbar {\partial\over \partial t} \;\;\;\;\;\; p\rightarrow -i\hbar {\partial\over \partial x}</math> | ::<math>E\rightarrow i\hbar {\partial\over \partial t} \;\;\;\;\;\; p\rightarrow -i\hbar {\partial\over \partial x}</math> |
| </math> | </math> |
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| Which is the original '''time-independent Schrödinger equation'''.*** | Which is the original '''time-independent Schrödinger equation'''. |
| | ---------------------------------------------------------------------------- |
| | This doesn't entirely answer your question... also, it seems the majority of the chapter deals with the time dependent version... |
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| This doesn't entirely answer your question... but it seems to me that the substitution of energy for momentum makes time dependence less important, but that there is still some (probably negligible) time dependence for the "time independent" equation. Anyone else have a better idea on this one? | ====Andromeda 15:34 9/13/09==== |
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| ====Andromeda==== | |
| I dont understand the 1st paragraph of page 29...why doesnt the exponential part cancel in the general solution like in the separable solutions? | I dont understand the 1st paragraph of page 29...why doesnt the exponential part cancel in the general solution like in the separable solutions? |
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| === Zeno 9/13 10:00 === | === Zeno 9/13 10:00pm? === |
| Across pages 26 and 27 Griffiths shows that //Stationary States// have a definite total energy which is constant in time. The exponential function only has the variables time and energy, which are both constant for Stationary States and therefore cancel. The key is that this only works for Stationary States, which are the separable solutions. Remember that Separable Solutions represent a //very small// class of solutions to the Schrodinger equation. There are potentially infinitely many more classes of solutions that //aren't// Separable Solutions, and in these classes the energy is not required to be definite and constant in time; with varying energy the combined exponential terms will not exactly cancel. The exact solutions we obtain with the method of Separable Solutions are based on a highly specific and restrictive class of problems, representing a very small portion of number of potential cases there are. It's a lot like division problems in arithmetic that early mathematicians restricted to only working with whole numbers. They could easily solve problems exactly that were expressed in whole numbers, but fractal numbers posed conceptual and technical issues. They had greatly simplified mathematics compared to the number of potential arithmetical problems involving all Rational numbers or including all Real Numbers (or real and imaginary that we use today), but their analysis of potential problems was greatly restricted. Just like it took more analytical development to be able to work with all Real Numbers, right now we have exact solutions in QM only for Separable Solutions until we develop our mathematics to correctly handle more classes of solutions. | Across pages 26 and 27 Griffiths shows that //Stationary States// have a definite total energy which is constant in time. The exponential function only has the variables time and energy, which are both constant for Stationary States and therefore cancel. The key is that this only works for Stationary States, which are the separable solutions. Remember that Separable Solutions represent a //very small// class of solutions to the Schrodinger equation. There are potentially infinitely many more classes of solutions that //aren't// Separable Solutions, and in these classes the energy is not required to be definite and constant in time; with varying energy the combined exponential terms will not exactly cancel. The exact solutions we obtain with the method of Separable Solutions are based on a highly specific and restrictive class of problems, representing a very small portion of number of potential cases there are. It's a lot like division problems in arithmetic that early mathematicians restricted to only working with whole numbers. They could easily solve problems exactly that were expressed in whole numbers, but fractal numbers posed conceptual and technical issues. They had greatly simplified mathematics compared to the number of potential arithmetical problems involving all Rational numbers or including all Real Numbers (or real and imaginary that we use today), but their analysis of potential problems was greatly restricted. Just like it took more analytical development to be able to work with all Real Numbers, right now we have exact solutions in QM only for Separable Solutions until we develop our mathematics to correctly handle more classes of solutions. |
| (I hope that description and analogy helps) | (I hope that description and analogy helps) |
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| Relating it to the Fourier series should make it a bit more clear. A real problem that hopefully helps to visualize it is describing a specific wave function, say a square wave, in terms of the summation of many sinusoidal waves. This would be a real world example of sound waves that works for describing the quantum mechanical view of particles as well. | Relating it to the Fourier series should make it a bit more clear. A real problem that hopefully helps to visualize it is describing a specific wave function, say a square wave, in terms of the summation of many sinusoidal waves. This would be a real world example of sound waves that works for describing the quantum mechanical view of particles as well. |
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| | ===vinc0053 09/20 17:35=== |
| | I like to think of the simplest case where you know the wave is only in the ground state. Then you have <math>c_1</math> equal to 1 and all other constants equal to zero. Then you can add the next state by, for example, having <math>c_1, c_2</math> hold values reflecting their proportional make-up, with all other constants equal to zero. |
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| | ====John Galt 11:02 9/14/09==== |
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| | What causes the uncertainty in position of a photon? Since the wave function (edit: probability function) does not spread out over time, I would assume that it is the same as it was during its original emission. Does it reflect an inability to measure the time of emission properly, or just the lack of resolution of devices measuring the position of photons? I understand that position is uncertain due the velocity of the photon, but it seems to me that since the function does not spread out over time, an accurate enough recording of the time of emission would lessen the uncertainty in the position of a photon (narrowing the probability function) at any given point. Is this a reasonable assumption? |
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| | === Spherical Chicken 14:13 9/14/09=== |
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| | I think it would be fair to say that if photons had a set energy, and could not be energized to higher states, they would not be terribly uncertain as we would not change them (by adding energy) in observation. Viewing a photon by using a photon wouldn't add energy to it. |
| | However it is my understanding that photons can come in different energies -- even though they have a definite velocity. So although we don't per say change it's velocity, we can change it's energy, and hence it's momentum. Thus it's position and momentum are uncertain. as is it's energy and time. |
| | Unlike an electron, whose velocity changes with the photon we observe it with a photon doesn't necessarily change velocities (as C is constant) but it's energy does change. |
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| | is this correct? Or am I as well misunderstanding the concept? |
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| | ==== time to move on ==== |
| | It's time to move on to the next Q_A: [[Q_A_0916]] |
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