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ito_integral [2016/07/06 13:00] nikolaj |
ito_integral [2016/07/06 13:02] nikolaj |
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| I.e. the Itō integral is defined with most left of the grid cells as in the explicit Euler-method numerical approximation scheme. The implicit Euler-method simply corresponds to a different notion of integral here and would give a different result. | I.e. the Itō integral is defined with most left of the grid cells as in the explicit Euler-method numerical approximation scheme. The implicit Euler-method simply corresponds to a different notion of integral here and would give a different result. | ||
| - | In quantum mechanics, the difference of the product above | + | In quantum mechanics, the difference of the products |
| $[x,p]_{\Delta t}:=x(t+\delta^2{\Delta t})\,p_{\Delta t}(t)-x(t)\,p_{\Delta t}(t)$ | $[x,p]_{\Delta t}:=x(t+\delta^2{\Delta t})\,p_{\Delta t}(t)-x(t)\,p_{\Delta t}(t)$ | ||
| - | is $m\,\kappa^2=m\frac{i\hbar}{2m}=\frac{i\hbar}{2}$. | + | equals $m\,\kappa^2=m\frac{i\hbar}{2m}=\frac{i\hbar}{2}$. |
| == Fractional quantum mechanics == | == Fractional quantum mechanics == | ||
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| $P(\Delta x) \propto \exp\left(c\frac{(\Delta x)^2}{\Delta t}\right)$ | $P(\Delta x) \propto \exp\left(c\frac{(\Delta x)^2}{\Delta t}\right)$ | ||
| - | as next-step distribution, and then | + | as next-step distribution, and then $\langle |x|\rangle\propto t^{1/2}$ gives the non-smooth curve. |
| - | + | ||
| - | $\langle |x|\rangle\propto t^{1/2}$ | + | |
| - | + | ||
| - | gives the non-smooth curve. | + | |
| You may want to look at other next-step distributions, | You may want to look at other next-step distributions, | ||
