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What's the mathematical process and physical logic? The Fourier transform of position space ($\vec x$ domain) is wave number space ($\vec k$ domain). This is an unambiguous, well understood mathematical result. By the De Broglie hypothesis, the momentum is $\vec p = \hbar \vec k$. This is physical hypothesis with experimental confirmation. Although ...

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How accurate do you need to be? The problem with these calculations is that massless, frictionless pulleys are usually out of stock at Acme Mail Order. High school physics will give you the tension in the rope, a different set of high school equations will tell you how much extra tension is needed for a certain acceleration. A rough metric is add 10% for ...

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Two laws govern collisions: conservation of momentum and conservation of energy. Momentum is the product of mass and velocity, so we can write $$\sum m_i\cdot \vec{v_i} = const$$ Conservation or energy is a little bit trickier, since energy can be converted from one type to another. In an elastic collision, the kinetic energy is conserved, so $$\sum ... 1 To quantize a classical system, start from the Poisson bracket$$\{x_i, p_j\} = \delta_{ij}.$$This relation defines p_i as the momentum canonically conjugate to x_i and is equivalent to Hamilton's equations. Quantize by letting x_i, p_j be Hermitian operators on a Hilbert space, with commutator$$[\hat x_i, \hat p_j] = i\delta_{ij} $$(identity ... 1 Yes, you absolutely need to count the back pressure. Otherwise the force required would be essentially independent of flow rate or the properties of the fluid. For a given fluid, you need to assess the pressure needed inside the syringe to make the desired flow. The force on the plunger needs to overcome that. In many cases that will be the dominant ... 1 I hope this is useful as an answer to your question, because although this is not about your exact system it gives a helpful interpretation of what normal modes are. For the water molecule we can consider it as three masses linked by two identical springs. Similar to your system there are three normal modes, which can be represented as the following motions ... 1 If you let x_i be the position of the m_i, you can write a set of coupled equations$$\begin {pmatrix} m_1\ddot{x_1}\\ m_2\ddot{x_2}\\ m_3\ddot{x_3} \end {pmatrix}=A\begin {pmatrix} x_1\\ x_2\\ x_3 \end {pmatrix} where $A$ gives the forces from the springs. With more masses and springs you have more lines in the equation. If all the masses are the ...

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You might be best off creating a bunch of trial systems and evolving each one independently using the (much easier) Hamilton's equations, which are good old ordinary differential equations. After all this is how the Liouville equation was constructed in the first place by people like Gibbs---in the limit of an infinite number of trial systems, you get the ...

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