theories:quantum_mechanics
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theories:quantum_mechanics [2018/05/11 17:11] jakobadmin [Roadmaps] |
theories:quantum_mechanics [2018/06/08 13:57] jakobadmin [Interpretations] |
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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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| There is no general consensus as to what the fundamental principles of quantum mechanics are and what it really "means". While almost any physicist can do calculations((At least in the standard, Hilbert space formulation)) in quantum mechanics, the stories that are told about what we really do when we perform these calculations vary wildly. For example, a common question is whether a particle in quantum mechanics already has well-defined properties before we measure it or if they only take on definite values as soon as we measure them. | There is no general consensus as to what the fundamental principles of quantum mechanics are and what it really "means". While almost any physicist can do calculations((At least in the standard, Hilbert space formulation)) in quantum mechanics, the stories that are told about what we really do when we perform these calculations vary wildly. For example, a common question is whether a particle in quantum mechanics already has well-defined properties before we measure it or if they only take on definite values as soon as we measure them. | ||
| - | The thing is that experimentally outcomes stay the same no matter which interpretation we believe in((This is similar to the statement that it doesn't matter which formulation we use. But here it makes at least some difference since some scenarios can be calculated more easily in a specific formulation.)). In this sense, discussions about the intereprtation of quantum mechanics are mostly a matter of taste. | + | The thing is that experimentally outcomes stay the same no matter which interpretation we believe in((This is similar to the statement that it doesn't matter which formulation we use. But here it makes at least some difference since some scenarios can be calculated more easily in a specific formulation.)). In this sense, discussions about the interpretation of quantum mechanics are mostly a matter of taste. |
| Important notions regarding the interpretation of quantum mechanics are | Important notions regarding the interpretation of quantum mechanics are | ||
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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 | >"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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| + | An amazing discussion of the Copenhagen interpreation and how it came about can be found in Quantum Dialogue by Mara Beller. | ||
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theories/quantum_mechanics.txt · Last modified: 2018/06/08 13:57 by jakobadmin
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