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A good experiment is Raman spectroscopy, it's quite simple, all you really need is a laser optical Goniometer with diffraction grating and sensor. The sensor could be the human eye if the room is very dark. Arduino and computing could be used to automate the detection and plot results.

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First, a somewhat minor point is that $x = 0,0.01a,0.02a,...a,1.01a,....2a....100a$ actually gives a list of 10001 points, not 10000 points. I will assume that you actually meant to say $x = 0,0.01a,...a,1.01a,....2a....99.99a$. Second, you say that $$V(x)=\sum_{K}e^{iKx}V_{K}$$ where $K =\frac{2\pi n}a$ and $n=0,1,2,3$, but this gives a non-Hermitian ...

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There are a variety of methods for determining the spectrum of a Hamiltonian system in one or more dimensions. For systems with small numbers of degrees of freedom, a direct matrix approach can be taken, often using some variant of the Sinc Discrete Variable Representation of the Hamiltonian. Depending on the size of the Sinc-DVR matrix, either direct ...

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What you are proposing will work, it is essentially what is known as the shooting method for solving the eigenvalue problem. Note that the eigenfunction is defined up to a multiplicative constant so you can just set $\epsilon$=1 and there is only one parameter $\epsilon'$ to vary in order to achieve a properly decaying solution at infinity. The shooting ...

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As I commented, I would think that any 3D hydrodynamics code would work. The basics of hydrodynamics can be summed up in the following five equations: \begin{eqnarray} \frac{\partial \rho}{\partial t}+\nabla\cdot\rho\mathbf{v}=0 \tag{1} \\ \frac{\partial \rho\mathbf{v}}{\partial t}+\nabla\cdot\left[\rho\mathbf{v}\otimes\mathbf{v}+P\mathbb ...

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First a word of advise: don't ignore the air! Research in the past couple of years has shown that the air (in particular the air pressure) is crucial in determining whether the droplet makes a splash and if so, what the dynamics are. One of the landmark paper in this field is the one from Weitz Group in PRL in 2012. Then on to your question of modelling the ...

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