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Derive the approximate form of Heisenberg's uncertainty principle for energy and time, $\Delta E \Delta t \approx h,$ using the following arguments: Since the position of a particle is uncertain by $\Delta x \approx \lambda$ , where $\lambda$ is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse $\Delta x$ . Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to $\lambda$ Find $\Delta t$ and $\Delta E ;$ then multiply them to give the approximate uncertainty principle.

$\Delta E \cdot \Delta t=h$

Physics 103

Chapter 29

Introduction to Quantum Physics

Quantum Physics

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so starting from this expression, we wanna all right but the product, not the LTTE. So knowing that the energy is the plants constant times the frequency that the energy uncertainty is just thanks constant times the frequency uncertainty Therefore, you know that built the tea or dlt times delta t It's not one, but it must be of these extra factor h must the age for be they require on interpretation off this expression. Um, it simply means that the energy of photon cannot be known. Uh, exactly so the uncertainty in our knowledge off the Fulton's energy these dependent on our knowledge of the uncertainty on the length of time that we use to measure it. Okay, so then see talks about the bar atomic model and they talk about Well, it's an app in that the creases energy by omitting a photon with wavelength off 500 nanometers. And now we want to know what is the uncertainty off this photons energy. So they do give us but the uncertainty in the atoms lifetime, which means the uncertainty in how long the atom is in that excited state. So we just need to use the formula above We know that uncertainty and energy is just age over this time. Uncertainty, which they say is 10 nanoseconds so filling in the blanks constant, which is 6.63 times 10 to the minus 34 Jewell. Second, we get 6.63 times and now these exponents decreases. Where increase the security. Tu minus 26. Jewell, you can't if you go to Electron Falls. This is around 4.14. Thanks, 10 to the minus seven election bolts. Okay, So finally, with this, they want us to know or to calculate the fractional uncertainty in energy for the energy of Fulton. And that is simply using the expression for the energy of a photo, which is so he is the uncertainty calculated above and the energy can be given, of course, by the pain is constant times the velocity, the speed of life over the wavelength off, in this case, the particle, which is the Fordham. So we just have Well, you plug in the speed of lives. The wavelength that they give us, which is 500. Um, is it It's 500 nanometers on the lens Costin's and you'll finally get 17 times 10 to the minus. So which is a really, really, really small fractional uncertainty

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