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| probability_and_uncertainty_in_quantum_mechanics [2021/02/08 22:07] – [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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| Since the particle will always be found somewhere if measured, this quantity needs to equal $1$. | Since the particle will always be found somewhere if measured, this quantity needs to equal $1$. |
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| You will not always be lucky enough to be give neatly normalized wavefunctions, so what should you do if you are given a wavefunction $\phi(x,t)$ such that $\int_{-\infty}^{+\infty} |\phi(x,t)|^2\,\mathrm{d}x \neq 1$? What I recommend doing is to always normalize the wavefunction as the first step in any calculation. To do this, define $\psi(x,t) = A\phi(x,t)$, where $A$ is a constant that you determine by imposing $\int_{-\infty}^{+\infty} |\psi(x,t)|^2\,\mathrm{d}x = 1$, and then work with $\psi(x,t)$ for the rest of the calculation. | You will not always be lucky enough to be given neatly normalized wavefunctions, so what should you do if you are given a wavefunction $\phi(x,t)$ such that $\int_{-\infty}^{+\infty} |\phi(x,t)|^2\,\mathrm{d}x \neq 1$? What I recommend doing is to always normalize the wavefunction as the first step in any calculation. To do this, define $\psi(x,t) = A\phi(x,t)$, where $A$ is a constant that you determine by imposing $\int_{-\infty}^{+\infty} |\psi(x,t)|^2\,\mathrm{d}x = 1$, and then work with $\psi(x,t)$ for the rest of the calculation. |
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| Equivalently, the Born rule can be generalized to unnormalized wavefunctions as follows. | Equivalently, the Born rule can be generalized to unnormalized wavefunctions as follows. |
| - //Joint measurement accuracy//: It is not possible to measure both $x$ and $p$ precisely at the same time. If we attempt to make a measurement that tells us both $x$ and $p$ then the accuracy of those measurements is limited by $\Delta x \Delta p \geq \hbar/2.$ | - //Joint measurement accuracy//: It is not possible to measure both $x$ and $p$ precisely at the same time. If we attempt to make a measurement that tells us both $x$ and $p$ then the accuracy of those measurements is limited by $\Delta x \Delta p \geq \hbar/2.$ |
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| All three of these are true in quantum theory. 2 and 3 have to be handled carefully because a quantum particle does not necessarily //have// a momentum or position that can be disturbed and how can we tell the accuracy of a measurement of a quantity that does not necessarily exist before the measurement. 2 and 3 are the subject of ongoing research and controversy, but it is possible to come up with definitions of "disturbance" and "accuracy" such that they are true. | All three of these are true in quantum theory. 2 and 3 have to be handled carefully because a quantum particle does not necessarily //have// a momentum or position that can be disturbed, so how can we tell the accuracy of a measurement of a quantity that does not necessarily exist before the measurement. 2 and 3 are the subject of ongoing research and controversy, but it is possible to come up with definitions of "disturbance" and "accuracy" such that they are true. |
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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 $p_{\text{light}} = h/\lambda \sim h/\Delta x$. |