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Here some details \begin{equation*} \partial _{t}\mathbf{w}(x,t)+\mathbf{A}\cdot \partial _{x}\mathbf{w}(x,t)=0 \end{equation*} Let $\mathbf{A}$ be an $n\times n$ matrix. Then $\mathbf{w}$ must be $n$ -dimensional. Let us assume that $\mathbf{A}$ has real entries and $\mathbf{w }$ has real components. \begin{eqnarray*} \mathbf{w}(x,t) &=&\exp ...

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An eigenvector is simply a vector that is unaffected (to within a scalar value) by a transformation. Formally, an eigenvector is any vector $x$ such that for an operator $\Omega$, $\Omega x = \lambda x$ for some scalar constant $\lambda$. All operators of dimension $n$ have exactly $n$ eigenvectors/eigenvalues (though these are only all distinct if $\Omega$ ...

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The meaning of eigenvalue and eigenvector(or eigenstate if you want)depends on what operator and what observables you are considering. If the operator is now a hamiltonian, the eigenvalue you get will be the energy of the system, and the eigenvector tell you its "state" So for the SHO system,the eigenvalue of the hamiltonian is (n+1/2)hf=Energy and n ...

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I find Pulliam's notes for the Euler equations to be a pretty good introduction to this topic using the equations of fluid motion. The idea is that you start with a conservation law: $$\frac{\partial \vec{Q}}{\partial t} + \frac{\partial \vec{F}\left(\vec{Q}\right)}{\partial x} = 0$$ where $Q$ is your variable vector and $F$ is your flux function. You can ...

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The convention we pick here will interact with the convention we have for matrix multiplication in the following way: If we have matrices $A$ and $B$ and we use the usual convention that the matrix multiplication $AB$ multiplies the rows of $A$ with the columns of $B$ then we have either $$(AB)_{ij}=\sum_kA_{ik}B_{kj}\tag{#}$$ or ...

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It is definitely a Kronecker sum. Take the case where there are only two different states $+$ and $-$, then, for example, $$\hat H =E_+ \hat a^\dagger_+ \hat a_++E_- \hat a^\dagger_- \hat a_- .$$ What does $\hat a_+$ means ? Well, if we label the states with the number of excitations in the states $+$ and $-$ by $|n_+,n_-\rangle$, then we understand $\hat ... 0 Such a vector describes the quantum state of a spin-1 particle on a line, or any other particle with a position degree of freedom and 3 internal states. To start with, you can expand wavefunctions in a basis. E.g., if you have a wavefunction$\vert\psi\rangle$which depends on position, you can expand it in the position basis$\vert x \rangle_p\$, i.e., ...

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