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hamiltonian [2016/09/06 18:10] nikolaj |
hamiltonian [2016/09/08 22:46] nikolaj |
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| === Discussion === | === Discussion === | ||
| + | == Eigenstate == | ||
| + | Given one system in isolation and a corresponding Hamiltonian with a gap (positive non-zero eigenstate), the (ratios of the) differences between eigenvalues is more relevant than the value of the eigenvalues themselves. This is because we may be able to rescale our units so that $w$, e.g. the lowest eigen-frequency, equals $2\pi$. | ||
| + | |||
| + | Now given eigenstates | ||
| + | |||
| $W\,|w\rangle = w\,|w\rangle$ | $W\,|w\rangle = w\,|w\rangle$ | ||
| Line 13: | Line 18: | ||
| $u = w + \Delta_{wu}$ | $u = w + \Delta_{wu}$ | ||
| - | The point of the following is the physical interpretation of the role of $\Delta_{wu}$, capturing the difference of eigenvalue of state $|u\rangle$ to a reference eigenstate $|w\rangle$. | + | the evolution is described by individually non-measurable oscillations ${\mathrm e}^{-iw}$ resp. ${\mathrm e}^{-iv}$ and, given we can rescale $w$ e.g. to $2\pi$, what's interesting is ${\mathrm e}^{-i\Delta_{wu}}$, capturing how much faster the second state oscillates. |
| + | == Superposition state == | ||
| For | For | ||
| Line 31: | Line 37: | ||
| 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. | ||
| - | |||
| - | We may be able to rescale our units so that $w$, e.g. the lowest eigen-frequency, equals $2\pi$. | ||
| === Reference === | === Reference === | ||
