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| probability_and_uncertainty_in_quantum_mechanics [2022/09/06 18:16] – [1.vii.3 The Uncertainty Principle] admin | probability_and_uncertainty_in_quantum_mechanics [2022/09/06 18:17] (current) – [1.vii.3 The Uncertainty Principle] admin |
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| Preparation uncertainty is the version that is usually proved in undergraduate quantum mechanics textbooks and we will do so in section 2. However, Heisenberg originally argued for 2. | Preparation uncertainty is the version that is usually proved in undergraduate quantum mechanics textbooks and we will do so in section 2. However, Heisenberg originally argued for 2. |
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| Heisenberg's argument is a semi-classical argument that does not pass muster in full-blown quantum mechanics, but it will give you an idea of where the principle came from. He imagines observing the position of an electron by scattering light off it. He treats the electron classically, as if it had a well defined position and momentum, and applied quantum principles to the light. This setup is often called the //**Heisenberg Microscope**//. I will give a toy version of the argument here that gets the order of magnitude right. Deriving precisely $\hbar/2$ as the limit requires a more detailed argument, which you can find at [[http://spiff.rit.edu/classes/phys314/lectures/heis/heis.html|this link]]. | Heisenberg's argument is a semi-classical argument that does not pass muster in full-blown quantum mechanics, but it will give you an idea of where the principle came from. He imagines observing the position of an electron by scattering light off it. He treats the electron classically, as if it had a well defined position and momentum, and applies quantum principles to the light. This setup is often called the //**Heisenberg Microscope**//. I will give a toy version of the argument here that gets the order of magnitude right. Deriving precisely $\hbar/2$ as the limit requires a more detailed argument, which you can find at [[http://spiff.rit.edu/classes/phys314/lectures/heis/heis.html|this link]]. |
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| To observe an electron's position with accuracy $\Delta x$, we need to use light of wavelength $\lambda \sim \Delta x$ or smaller. By the de Broglie relation, this corresponds to a photon of momentum $p_{\text{light}} = h/\lambda \sim h/\Delta x$. | To observe an electron's position with accuracy $\Delta x$, we need to use light of wavelength $\lambda \sim \Delta x$ or smaller. By the de Broglie relation, this corresponds to a photon of momentum $p_{\text{light}} = h/\lambda \sim h/\Delta x$. |