theories:quantum_mechanics
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theories:quantum_mechanics [2018/05/11 17:11] jakobadmin [Roadmaps] |
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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)). | ||
theories/quantum_mechanics.txt · Last modified: 2018/06/08 13:57 by jakobadmin
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