Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
|
its_about_time_._note [2016/01/05 15:36] nikolaj |
its_about_time_._note [2016/01/05 17:12] nikolaj |
||
|---|---|---|---|
| Line 15: | Line 15: | ||
| $h_\Psi\cdot\dfrac{{\mathrm d}}{{\mathrm d}t}\Psi = (-2\pi{\mathrm i})\,H\,\Psi$ | $h_\Psi\cdot\dfrac{{\mathrm d}}{{\mathrm d}t}\Psi = (-2\pi{\mathrm i})\,H\,\Psi$ | ||
| - | and note that if $h_\Psi$ is chosen unit less, the conserved quantity of the system (what is also called energy) is a frequency. | + | and note that if $h_\Psi$ is chosen unitless, the conserved quantity of the system (what is also called energy) is a frequency. |
| + | The cost associated with a process is basically inversely proportional to how quick it takes place, so that Reaching higher frequencies is more expensive. But for a more precise desciption in QM, one must look at the state spaces in the models. | ||
| + | A Hamiltonian $H(\langle q,p\rangle)$ is like an $\langle q,p\rangle$-indexed list of prices. A Hamiltonian operator $H$ acting on state vectors in QM is like a stamp with prices. | ||
| - | Reaching higher frequencies is "more expensive". This idea is realized if $\hbar$ is unitless and thus energy (the cost-unit par exellance) is actually a unit of frequency. Then, understand formulas, many units like e.g. mass ($kg=s/m^2$, after setting $\hbar$ unitless) can or should be multiplied by natural constants (e.g. $c^2$) so that they represent a frequency or time scale. | + | To understand many formulas, units like e.g. mass ($kg=s/m^2$, after setting $\hbar$ unitless) can or should be multiplied by natural constants (e.g. $c^2$) so that they represent a frequency or time scale. |
| - | + | ||
| - | A Hamiltonian $H(\langle q,p\rangle)$ is like an $\langle q,p\rangle$-indexed list of prices. A Hamiltonian operator $H$ acting on state vectors in QM is like a stamp with prices. | + | |
| == Canonical variables == | == Canonical variables == | ||
