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on_physical_units_._note [2016/08/30 18:57] nikolaj |
on_physical_units_._note [2016/09/28 17:17] nikolaj |
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| * $h_F\,\left(\tau_t\dfrac{{\mathrm d}}{{\mathrm d}t}\right)^2x=l_x\,F(x,\frac{{\mathrm d}}{{\mathrm d}t}x,t)$ | * $h_F\,\left(\tau_t\dfrac{{\mathrm d}}{{\mathrm d}t}\right)^2x=l_x\,F(x,\frac{{\mathrm d}}{{\mathrm d}t}x,t)$ | ||
| - | $ \tau_t^2 \Delta_{t_1}^{t_2} x' = l_x \int A(x(t), x'(t), t)\,{\mathrm d}t $ | + | $ h_A\,\tau_t^2\,\Delta_{t_1}^{t_2} x' = l_x\,\int A(x(t), x'(t), t)\,{\mathrm d}t $ |
| where $\tau_t$ is a time constant characteristic for the system and $h$'s other parameters, possibly to be fixed in experiment. The usage of all the constants isn't standard. | where $\tau_t$ is a time constant characteristic for the system and $h$'s other parameters, possibly to be fixed in experiment. The usage of all the constants isn't standard. | ||
| - | The indices like in $h_A$ mostly denotes that they share unit with that index expression. $f=1/\tau_t$ is a characteristic frequency (with $f=\tau_\pi\omega$ where $\tau_\pi\equiv{2\pi}$ and with a characteristic circle frequency $\omega$) such as the first gap $f_1-f_0$. | + | The indices like in $h_A$ mostly denotes that they share unit with that index expression. $f=1/\tau_t$ is a characteristic rate or frequency (with $f=\tau_\pi\omega$ where $\tau_\pi\equiv{2\pi}$ and with a characteristic circle frequency $\omega$) such as the first gap $f_1-f_0$. It makes sense to call it "frequency" as soon as we deal with an expression or system that soon enough returns to its previous state, like ${\mathrm e}^{-2\pi\cdot{\mathrm i}ft}$ does. |
| $A$ is a linear Hermitean operator and $F$ some function and all constants may be absorbed into them. We may call $A$ the arousal of a state (it's often $\propto$ the occuptation and we may naturally choose it to have units of action). | $A$ is a linear Hermitean operator and $F$ some function and all constants may be absorbed into them. We may call $A$ the arousal of a state (it's often $\propto$ the occuptation and we may naturally choose it to have units of action). | ||
| In quantum mechanics in general, but in particular if we pull out the frequency of the Hamiltonian like that, we want to think of the state occupation numbers as the main extensive quantity. The one that can be traded between subsystem. | In quantum mechanics in general, but in particular if we pull out the frequency of the Hamiltonian like that, we want to think of the state occupation numbers as the main extensive quantity. The one that can be traded between subsystem. | ||
