advanced_notions:uncertainty_principle
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advanced_notions:uncertainty_principle [2018/05/11 16:04] jakobadmin [Why is it interesting?] |
advanced_notions:uncertainty_principle [2018/07/26 17:56] (current) 77.177.199.249 [FAQ] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | We can generate a wave in a long rope by shaking it rhythmically up and down. | ||
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| + | {{ :advanced_notions:wave1rope.png?nolink&600 |}} | ||
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| + | Now, if someone asks us: "Where precisely is the wave?" we wouldn't have a good answer since the wave is spread out. In contrast, if we get asked: "What's the wavelength of the wave?" we could easily answer this question: "It's around 6cm". | ||
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| + | We can also generate a different kind of wave in a rope by jerking it only once. | ||
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| + | {{ :advanced_notions:wave2rope.png?nolink&600 |}} | ||
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| + | This way we get a narrow bump that travels down the line. Now, we could easily answer the question: "Where precisely is the wave?" but we would have a hard time answer the question "What's the wavelength of the wave?" since the wave isn't periodic and it is completely unclear how we could assign a wavelength to it. | ||
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| + | Similarly, we can generate any type of wave between these two edge cases. However, there is always a tradeoff. The more precise the position of the wave is, the less precise its wavelength becomes and vice versa. | ||
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| + | This is true for any wave phenomena and since in quantum mechanics we describe particle using waves, it also applies here. In quantum mechanics, the wavelength is in a direct relationship to its momentum. The larger the momentum, the smaller the wavelength of the wave that describes the particle. A spread in wavelength, therefore, corresponds to a spread in momentum. As a result, we can derive an uncertainty relation that tells us: | ||
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| + | The more precisely we determine the location of a particle, the less precisely we can determine its momentum and vice versa. The thing is that a localized wave bump can be thought of as a superposition of dozens of other waves with well-defined wave-lengths((This is exactly the idea behind the [[basic_tools:fourier_transform|Fourier transform]].)): | ||
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| + | {{ :advanced_notions:fourieruncertainty.png?nolink&600 |}} | ||
| + | In this sense, such a localized bump does not have one specific wavelength but is a superposition of many. | ||
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| + | ---- | ||
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| <blockquote> | <blockquote> | ||
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| <cite>https://www.scientificamerican.com/article/common-interpretation-of-heisenbergs-uncertainty-principle-is-proven-false/</cite> | <cite>https://www.scientificamerican.com/article/common-interpretation-of-heisenbergs-uncertainty-principle-is-proven-false/</cite> | ||
| </blockquote> | </blockquote> | ||
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| + | <blockquote>Bohr, for his part, explained uncertainty by pointing out that answering certain questions necessitates not answering others. To measure position, we need a stationary measuring object, like a fixed photographic plate. This plate defines a fixed frame of reference. To measure velocity, by contrast, we need an apparatus that allows for some recoil, and hence moveable parts. This experiment requires a movable frame. Testing one therefore means not testing the other. <cite>https://opinionator.blogs.nytimes.com/2013/07/21/nothing-to-see-here-demoting-the-uncertainty-principle/</cite></blockquote> | ||
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| <tabbox Concrete> | <tabbox Concrete> | ||
| - | Whenever we measure the momentum or the location of a particle, we get a precise answer. However, if repeat our measurement on equally prepared systems, we do not always get exactly the same result. Instead, the results are spread. | + | Whenever we measure an observable in quantum mechanics, we get a precise answer. However, if repeat our measurement on equally prepared systems, we do not always get exactly the same result. Instead, the results are spread around some central value. |
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| + | While we can prepare our systems such that a repeated measurement always yields almost exactly the same value, there is a price we have to pay for that: the measurements of some other observable will be wildly scattered. | ||
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| + | The most famous example is the position and momentum uncertainty: $$ \sigma_x \sigma_p \geq \hbar/2,$$ where $\hbar$ denotes the reduced Planck constant and the $\sigma_x$ means the standard variation if we perform multiple measurements of the position $x$ for equally prepared particles. Analogously, $\sigma_p$ denotes the standard variation if we measure the momentum $p$. | ||
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| + | Hence, if we try to know the position $x$ very accurately, which means $\sigma_x \ll \hbar/2$, then our knowledge about the momentum becomes much worse: $\sigma_p \gg \hbar/2 $. This follows directly from the inequality $ \sigma_x \sigma_p \geq \hbar/2.$ | ||
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| \begin{equation} \sigma_A \sigma_B \geq \big | \frac{1}{2i} \langle [A,B] \rangle \big|^2 . \end{equation} | \begin{equation} \sigma_A \sigma_B \geq \big | \frac{1}{2i} \langle [A,B] \rangle \big|^2 . \end{equation} | ||
| + | See also | ||
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| + | * [[https://arxiv.org/abs/quant-ph/0608138|The certainty principle (review)]] by D. A. Arbatsky | ||
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| that can be consistently attributed to it. Indeed, an observable (such | that can be consistently attributed to it. Indeed, an observable (such | ||
| as q) that is not invariant under the automorphisms of a state (such | as q) that is not invariant under the automorphisms of a state (such | ||
| - | as |jp〉) cannot define an “objective” property of the latter. Hence, | + | as |jp〉) cannot define an “objective” property of the latter. Hence, |
| the expectation value function will have a non-zero dispersion. We | the expectation value function will have a non-zero dispersion. We | ||
| have argued that this dispersion measures the extent to which the | have argued that this dispersion measures the extent to which the | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| According to the uncertainty principle, it is impossible to know several pairs of variables at the same time with arbitrary accuracy. | According to the uncertainty principle, it is impossible to know several pairs of variables at the same time with arbitrary accuracy. | ||
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| + | In some sense, it completely encapsulates what is different about quantum mechanics compared to [[theories:classical_mechanics|classical mechanics]]. | ||
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| + | <blockquote> | ||
| + | A philosopher once said ‘It is necessary for the very existence of science that the same conditions always produce the same results’. Well, they don’t! | ||
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| + | - Richard Feynman | ||
| + | </blockquote> | ||
| <tabbox FAQ> | <tabbox FAQ> | ||
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| --> Is there a time-energy uncertainty relation?# | --> Is there a time-energy uncertainty relation?# | ||
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| <-- | <-- | ||
| - | --->What's the origin of the uncertainty? # | + | -->What's the origin of the uncertainty?# |
| <blockquote>Quantum mechanics uses the generators of the corresponding symmetry as measurement operators. For instance, this has the consequence that a measurement of momentum is equivalent to the action of the translation generator. (Recall that invariance under translations leads us to conservation of momentum.) The translation generator moves our system a little bit and therefore the location is changed.<cite>Physics from Symmetry by J. Schwichtenberg</cite></blockquote> | <blockquote>Quantum mechanics uses the generators of the corresponding symmetry as measurement operators. For instance, this has the consequence that a measurement of momentum is equivalent to the action of the translation generator. (Recall that invariance under translations leads us to conservation of momentum.) The translation generator moves our system a little bit and therefore the location is changed.<cite>Physics from Symmetry by J. Schwichtenberg</cite></blockquote> | ||
| <-- | <-- | ||
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| </tabbox> | </tabbox> | ||
| - | {{tag>theories:quantum_theory:quantum_mechanics theories:quantum_theory:quantum_field_theory}} | ||
advanced_notions/uncertainty_principle.1526047469.txt.gz · Last modified: 2018/05/11 14:04 (external edit)
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