60 lines
2.5 KiB
TeX
60 lines
2.5 KiB
TeX
\section{Numerical Simulations}\label{sec:numerical}
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\subsection{The Algorithm}
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We solve the two-particle Schr\"{o}dinger equation for the Hamiltonian
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\eqref{eq:main} via the explicit staggered method \cite{numerics}. We
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choose our units such that $\hbar = 1$ and the neutron mass $m_n =
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1$. The wave function is evaluated on a grid of discrete values for
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the independent variables
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\begin{equation}
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\Psi(x_1, x_2, t) = \Psi(x_1 = l\Delta x, x_2 = m\Delta x, t = n\Delta
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t) \equiv \Psi^n_{l, m},
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\end{equation}
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where $l$, $m$ and $n$ are integers. We separate it into real and
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imaginary parts,
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\begin{equation}
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\Psi^n_{l,m} = u^{n+1}_{l,m} + iv^{n+1}_{l,m},
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\end{equation}
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which allows us to solve the Schr\"{o}dinger equation via a finite
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difference method with a pair of coupled equations:
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\begin{equation}
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\begin{array} {lcl}
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u^{n+1}_{l,m} & = & u^{n-1}_{l,m} - \left\{ \alpha \left[ v^n_{l+1,m}
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+ v^n_{l-1,m} + v^n_{l,m+1} + v^n_{l,m-1} - 4v^n_{l,m} \right] - 2
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\Delta t V_{l,m}v^n_{l,m} \right\} ,
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\end{array}
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\end{equation}
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\begin{equation}
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\begin{array} {lcl}
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v^{n+1}_{l,m} & = & v^{n-1}_{l,m} + \left\{ \alpha \left[ u^n_{l+1,m}
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+ u^n_{l-1,m} + u^n_{l,m+1} + u^n_{l,m-1} - 4u^n_{l,m} \right] - 2
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\Delta t V_{l,m}u^n_{l,m} \right\},
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\end{array}
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\end{equation}
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where
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\begin{equation}
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\alpha = g^{xx}\frac{\Delta t}{\Delta x^2},
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\end{equation}
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and $V_{l,m}$ is the combined value of the potential due to the
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harmonic well and neutron-neutron interaction on the discretised
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grid given by
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\begin{equation}
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V_{l,m} = \frac{1}{2}\omega^2(l^2 + m^2)\Delta x^2 + \nu\delta_{l,m}.
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\end{equation}
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In order to evaluate the value of the real part at step $n+1$ we need
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to know the values of the real part at $n-1$ and the imaginary part at
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$n$. The same is true for evaluating the imaginary part. Therefore, we
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do not have to calculate both parts at every time step and we compute
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the real and imaginary parts at slightly different (staggered)
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times. The form of the discretised equations is identical to those
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presentied in \cite{numerics}. However, the value of $\alpha$ is now
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scaled by the oscillating metric tensor component $g^{xx}$. This novel
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modification does not lead to instabilities and simulations have been
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confirmed to conserve probability to the same precision as before.
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For the more general Laplacian \eqref{eq:general}, the equations have to
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be modified even further and new terms are introduced for the
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wavefunction's first derivative terms present. This too, does not
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cause instabilities and conserves probability very well.
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