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retarded_propagator_._time-independent_hamiltonian [2014/08/20 13:56] nikolaj |
retarded_propagator_._time-independent_hamiltonian [2016/08/30 20:38] nikolaj |
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| | @#55CCEE: context | @#55CCEE: $\mathcal H$ ... Hilbert space | | | @#55CCEE: context | @#55CCEE: $\mathcal H$ ... Hilbert space | | ||
| | @#55CCEE: context | @#55CCEE: $H\in\mathrm{Observable}(\mathcal H)$ | | | @#55CCEE: context | @#55CCEE: $H\in\mathrm{Observable}(\mathcal H)$ | | ||
| - | |||
| | @#FFBB00: definiendum | @#FFBB00: $P:\mathbb{R}\times\mathbb{R}\to L(\mathcal H,\mathcal H)$ | | | @#FFBB00: definiendum | @#FFBB00: $P:\mathbb{R}\times\mathbb{R}\to L(\mathcal H,\mathcal H)$ | | ||
| | @#FFBB00: definiendum | @#FFBB00: $P(t,s):=\exp\left(i\,(t-s)\frac{1}{\hbar}H\right)$ | | | @#FFBB00: definiendum | @#FFBB00: $P(t,s):=\exp\left(i\,(t-s)\frac{1}{\hbar}H\right)$ | | ||
| - | ==== Discussion ==== | + | ----- |
| + | === Discussion === | ||
| Let $N\equiv \mathrm{dim}(V)$ and let $i,j,k,n$ range from $1$ to $N$. | Let $N\equiv \mathrm{dim}(V)$ and let $i,j,k,n$ range from $1$ to $N$. | ||
| Let $\{|E_n\rangle\}$ as well as $\{|Q_n\rangle\}$ be two orthonormal sets of basis vectors where in particular $H|E_n\rangle=E_n|E_n\rangle$. | Let $\{|E_n\rangle\}$ as well as $\{|Q_n\rangle\}$ be two orthonormal sets of basis vectors where in particular $H|E_n\rangle=E_n|E_n\rangle$. | ||
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| To define the continuum limit of this expression is a difficult task. In quantum field theory, an explicit expression for the $\lim_{\Delta t\to 0}\sum_{k=1}^{m_\Delta}\Delta S_{i_{k-1}i_{k}}$ is taken as a starting point and identified with the [[action]] corresponding to the Hamiltonian $H$. The measure, in essence given by the product $\prod_{k=1}^m\Delta\mu_{i_{k-1}i_{k}}$, is generally ill-defined. | To define the continuum limit of this expression is a difficult task. In quantum field theory, an explicit expression for the $\lim_{\Delta t\to 0}\sum_{k=1}^{m_\Delta}\Delta S_{i_{k-1}i_{k}}$ is taken as a starting point and identified with the [[action]] corresponding to the Hamiltonian $H$. The measure, in essence given by the product $\prod_{k=1}^m\Delta\mu_{i_{k-1}i_{k}}$, is generally ill-defined. | ||
| - | ==== Parents ==== | + | ----- |
| === Context === | === Context === | ||
| [[Hilbert space]] | [[Hilbert space]] | ||
| === Requirements === | === Requirements === | ||
| - | [[Observable]], [[Linear operator algebra]] | + | [[Observable]], |
| + | [[Linear operator algebra]] | ||
| === Related === | === Related === | ||
| + | [[Hamiltonian]], | ||
| [[Linear first-order ODE system]] | [[Linear first-order ODE system]] | ||
