theories:quantum_mechanics
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theories:quantum_mechanics [2018/05/11 16:34] jakobadmin [Interpretations] |
theories:quantum_mechanics [2018/05/13 09:18] jakobadmin ↷ Links adapted because of a move operation |
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| - | Both Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulations both belong to the description known as **[[theories:quantum_mechanics:canonical|canonical quantum mechanics]]**. The relevant mathematical stage for both formulations is [[basic_tools:hilbert_space|Hilbert space]]. The connection between them lies in the identification of Heisenberg's infinite matrices $p_j$ and $q^i$ ($i,j=1,2,3$), representing the momentum and position of a particle moving in $\mathbb{R}^3$, with Schrödinger's operators $-i\hbar\partial/\partial x^j$ and $x^i$ (seen as a multiplication operator) on the Hilbert space $\mathcal H=L^2(\mathbb{R}^3)$, respectively. The key to this identification lies in the [[equations:canonical_commutation_relations|canonical commutation relations]] | + | Both Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulations both belong to the description known as **[[theories:quantum_mechanics:canonical|canonical quantum mechanics]]**. The relevant mathematical stage for both formulations is [[basic_tools:hilbert_space|Hilbert space]]. The connection between them lies in the identification of Heisenberg's infinite matrices $p_j$ and $q^i$ ($i,j=1,2,3$), representing the momentum and position of a particle moving in $\mathbb{R}^3$, with Schrödinger's operators $-i\hbar\partial/\partial x^j$ and $x^i$ (seen as a multiplication operator) on the Hilbert space $\mathcal H=L^2(\mathbb{R}^3)$, respectively. The key to this identification lies in the [[formulas:canonical_commutation_relations|canonical commutation relations]] |
| $$ [p_i,q^j]=-i\hbar \delta^j_i. $$ | $$ [p_i,q^j]=-i\hbar \delta^j_i. $$ | ||
| We usually call these two formulations the "**Heisenberg picture**" and the "**Schrödinger picture**", since, both descriptions are actually equivalent. In some sense, the transformation between them is "just a basis change in Hilbert space"((https://en.wikipedia.org/wiki/Heisenberg_picture)). | We usually call these two formulations the "**Heisenberg picture**" and the "**Schrödinger picture**", since, both descriptions are actually equivalent. In some sense, the transformation between them is "just a basis change in Hilbert space"((https://en.wikipedia.org/wiki/Heisenberg_picture)). | ||
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| - | The standard interpretation of quantum mechanics is presented in almost every textbook and known as the Copenhagen interpretation. | + | The standard (orthodox) interpretation of quantum mechanics is presented in almost every textbook and known as the Copenhagen interpretation. |
| According to this interpretation, particles do not possess specific dynamical properties (momentum, position, angular momentum, energy, etc.) until we perform a measurement. | According to this interpretation, particles do not possess specific dynamical properties (momentum, position, angular momentum, energy, etc.) until we perform a measurement. | ||
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| The wave function is interpreted statistically and it collapses once we measure it. Therefore, if we immediately repeat a measurement, we will get the same result again. | The wave function is interpreted statistically and it collapses once we measure it. Therefore, if we immediately repeat a measurement, we will get the same result again. | ||
| - | Regarding the question whether a particle already has a definite momentum etc. before we measure it, the Copenhagen interpretation states that | + | Regarding the question, whether a particle already has a definite momentum etc. before we measure it, the Copenhagen interpretation states that |
| >"observations not only disturb what has to be measured, they produce it!" - Pascual Jordan. | >"observations not only disturb what has to be measured, they produce it!" - Pascual Jordan. | ||
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| + | In contrast, hidden variable interpretations which are also called realist interpretations, state that | ||
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| + | >“the position of the particle was never indeterminate, but was merely unknown to the experimenter.” - Bernard d'Espagnat. | ||
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| + | A third popular interpretation is called the agnostic interpretation states that it makes no sense to ask such a question since how can we discuss anything that we can never measure. By definition, a property like momentum is undetermined until we measure it and a discussion about its value before measurement makes no sense: | ||
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| + | >"One should no more rack one’s brain about the problem of whether something one cannot know anything about exists all the same, than about the ancient question of how many angels are able to sit on the point of a needle." - Wolfgang Pauli | ||
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| * A good book on the topic is Roland Omnes, Interpretation of Quantum Mechanics, Princeton U. Press, Princeton, 1994. | * A good book on the topic is Roland Omnes, Interpretation of Quantum Mechanics, Princeton U. Press, Princeton, 1994. | ||
| * See also Elegance and Enigma - The Quantum Interviews by Schlosshauer | * See also Elegance and Enigma - The Quantum Interviews by Schlosshauer | ||
| + | * Making Sense of Quantum Mechanics by Bricmont | ||
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| + | ----- | ||
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| + | <blockquote>“If you are not confused by quantum mechanics, then you haven’t really understood it.” <cite>Niels Bohr</cite></blockquote> | ||
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| + | <blockquote>“I think I can safely say that nobody understands quantum mechanics.” <cite>Richard Feynman</cite></blockquote> | ||
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| - | __The Traditional Roadmap__ | + | -->The Traditional Roadmap# |
| **Basics** | **Basics** | ||
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| </WRAP> | </WRAP> | ||
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| + | </WRAP> | ||
| + | |||
| + | <-- | ||
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| + | -->Applications# | ||
| Quantum mechanics is technically difficult. Only a few extremely artificial textbook examples can be solved exactly. For everything else, we need to use approximation techniques to tackle realistic systems. | Quantum mechanics is technically difficult. Only a few extremely artificial textbook examples can be solved exactly. For everything else, we need to use approximation techniques to tackle realistic systems. | ||
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| * the Born approximation, | * the Born approximation, | ||
| * Fermi's golden rule. | * Fermi's golden rule. | ||
| - | </WRAP> | + | <-- |
| - | + | ||
theories/quantum_mechanics.txt · Last modified: 2018/06/08 13:57 by jakobadmin
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