formulas:canonical_commutation_relations
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formulas:canonical_commutation_relations [2018/05/03 14:04] jakobadmin [Abstract] |
formulas:canonical_commutation_relations [2018/05/13 09:18] (current) jakobadmin ↷ Page moved from equations:canonical_commutation_relations to formulas:canonical_commutation_relations |
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| However, can measure the <color firebrick>location</color> on different axes - e.g. the location on the x-axis and the location on the y-axis - with <color darksalmon>arbitrary precision</color>. Equally, we can measure the <color royalblue>momentum</color> in the direction of different axes with <color darksalmon>arbitrary precision</color>. | However, can measure the <color firebrick>location</color> on different axes - e.g. the location on the x-axis and the location on the y-axis - with <color darksalmon>arbitrary precision</color>. Equally, we can measure the <color royalblue>momentum</color> in the direction of different axes with <color darksalmon>arbitrary precision</color>. | ||
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| + | <blockquote>The action, in the form h/2π, also occurs in the Heisenberg Uncertainty Relationship, and here it sets the limit of precision to which the conjugate coordinates (momentum and position; or energy and time) can be determined simultaneously for one particle. It also occurs in the Dirac commutator relations (discovered by the British theoretical physicist, Paul Dirac (1902–84)) which show that the order in which measurements of these conjugate coordinates are carried out does make a difference - a tiny difference of magnitude h/2π . | ||
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| + | <cite>The Lazy Universe by Coopersmith</cite></blockquote> | ||
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| **Quantum Mechanics** | **Quantum Mechanics** | ||
| - | \begin{equation} \label{eq:commquantummech} [\hat{p}_i,\hat{x}_j] = -i \delta_{ij} .\end{equation} | + | \begin{equation} \label{eq:commquantummech} [\hat{p}_i,\hat{x}_j] = -i \hbar \delta_{ij} .\end{equation} |
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| \begin{equation} \label{qftcomm} [\Phi(x), \pi(y)]=\Phi(x) \pi(y) - \pi(y) \Phi(x) = i \delta(x-y) \end{equation} where $\delta(x-y)$ is the Dirac delta distribution and $\pi(y) = \frac{\partial \mathscr{L}}{\partial(\partial_0\Phi)}$ is the conjugate momentum. | \begin{equation} \label{qftcomm} [\Phi(x), \pi(y)]=\Phi(x) \pi(y) - \pi(y) \Phi(x) = i \delta(x-y) \end{equation} where $\delta(x-y)$ is the Dirac delta distribution and $\pi(y) = \frac{\partial \mathscr{L}}{\partial(\partial_0\Phi)}$ is the conjugate momentum. | ||
| - | It tells us that the fields in [[theories:quantum_field_theory|quantum field theory]] $\Phi(x)$ can‘t be simply a function, but must be __operators__. In contrast, ordinary functions and numbers commute: | + | It tells us that the fields in [[theories:quantum_field_theory:canonical|quantum field theory]] $\Phi(x)$ can‘t be simply a function, but must be __operators__. In contrast, ordinary functions and numbers commute: |
| For example $f(x)=3x$ and $g(y)= 7y^2 +3$ clearly commute $$ [f(x) , g(x)]= f(x)g(x) - g(x) f(x) = 3x (7y^2 +3) -(7y^2 +3) 3x =0. $$ | For example $f(x)=3x$ and $g(y)= 7y^2 +3$ clearly commute $$ [f(x) , g(x)]= f(x)g(x) - g(x) f(x) = 3x (7y^2 +3) -(7y^2 +3) 3x =0. $$ | ||
formulas/canonical_commutation_relations.1525349066.txt.gz · Last modified: 2018/05/03 12:04 (external edit)
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