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hamiltonian [2016/09/08 22:47] nikolaj |
hamiltonian [2016/09/08 23:00] (current) nikolaj |
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| $u = w + \Delta_{wu}$ | $u = w + \Delta_{wu}$ | ||
| - | the evolution is described by individually non-measurable oscillations ${\mathrm e}^{-iw}$ resp. ${\mathrm e}^{-iv}={\mathrm e}^{-iu}{\mathrm e}^{-i\Delta_{wu}}$ and what's interesting is the second factor, capturing how much faster the second state oscillates. | + | the evolution is described by individually non-measurable oscillations ${\mathrm e}^{-iw}$ resp. ${\mathrm e}^{-iu}={\mathrm e}^{-iw}{\mathrm e}^{-i\Delta_{wu}}$ and what's interesting is the second factor, capturing how much faster the second state oscillates. |
| == Superposition states == | == Superposition states == | ||
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| Up to a phase, $(e^{it\Delta_{wu}}-1)$ generates oscillations about $|\psi\rangle$. | Up to a phase, $(e^{it\Delta_{wu}}-1)$ generates oscillations about $|\psi\rangle$. | ||
| This is somewhat related to Rabi-oscillations. | This is somewhat related to Rabi-oscillations. | ||
| + | |||
| + | == On interaction terms == | ||
| + | Say we start with a model $W$ and eigenstates $|w\rangle$, $|u\rangle$ and then decide to make it "more realistic" and add interaction terms, making for a new Hamiltonian $W'$. Usually $W'=W+qI$, where $q\in{\mathbb R}$ is a scalar called couple constant. One now said this describes an interacting system, but that's a relative notion: All Hamiltonians like $W$ or $W'$ that one considers are hermitean and thus diagonalizable, i.e. there are states $|w'\rangle$, $|u'\rangle$ for $W'$ which are not interacting. The nomenclature basically just comes from sticking to the old states and this is mostly done because those are the ones one can compare. | ||
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| + | So given a system with $W'$ is e.g. in an eigenstate $|u'\rangle$ with oscillation frequency $u'$ ("total energy of the system"), then expressed as superposition of $|w\rangle$ and $|u\rangle$ (making for a state that undergoes an back and forth according to $\Delta_{wu}$) we have a notion of total energy flowing between those non-eigenstates of the "interacting system". | ||
| === Reference === | === Reference === | ||
