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4

This might not be what Nakahara has in mind, but one can make sense of this using the idea of projective Hilbert spaces. Let $\mathcal{P}(\mathcal{H})$ denote the projective space associated to the "normal" space $\mathcal{H}$. The subset of separable states is not a subvectorspace in the proper sense, as Holographer notes. Yet it can be understood as a ...


3

Note that the space of separable states is not a vector space, and in particular not a subspace of the total Hilbert space: the sum of two separable states is unlikely to be separable. So dimension here means something more general than vector space dimension. Having said that, I would disagree with the author on his dimension! I would say that the space of ...


3

As stated, $\mathbf{n}$ is a unit vector and $n_x$, $n_y$ and $n_z$ are its cartesian components. $\mathbf{n}$ is just a vector pointing in an arbitrarily direction with magnitude 1. Taking $\mathbf{n} \cdot \mathbf{\sigma}$, we have \begin{equation} \mathbf{n} \cdot \mathbf{\sigma} = n_x\sigma_x + n_y \sigma_y + n_z \sigma_z \\ = n_x \left(\begin{array}{cc} ...


3

What you really want to know are the definitions of the $\sigma_i$ --- these are the Pauli matrices: $$ \sigma_x = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \qquad \sigma_y = \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} \qquad \sigma_z = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$$ Hopefully you can see now how the equation ...


0

Other than notation devices ... Matrices are far more than mere notational devices, but even if they were, don't deride notational devices! More compact notation simplifies writing, simplifies reading, simplifies thinking, and because of that, it enables new ways of thinking. Think of the progress from marks on a stick to representing numbers by ...


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Surprised nobody mentioned optics, so I will. Matrices are used extensively in geometric optics and polarization. Examples include the ABCD matrix method for representing the effects of optical elements (e.g lenses and mirrors) in ray optics, and the Mueller matrices/Stokes vectors to represent the effects of polarizers and plates on the polarization of ...


1

More use of matrices: The moment of inertia tensor needed to describe the rotational motion of rigid bodies The Pauli matrices for spinn-1/2 (but that example is perhaps included in the Lie group example already mentioned).


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Applications of matrices: Matrix (aka quantum) Mechanics, obviously Mechanics of deformable solids (where matrices describe stresses) Statics (most in engineering contexts), where matrices describe stresses. Symmetries (where matrices describe rotations/scaling/translations etc..) Coordinate transformations, where matrices describe the transformation a ...


3

Lie groups are fundamental for talking about anything related to symmetries in physics on a level of some rigor, and every finite-dimensional Lie group is a matrix group. Consequently, the trace as a basic matrix operation shows up anywhere where invariance on the adjoint action of the group is needed, and the matrices are everywhere. The Slater determinant ...


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You may use PICOS for Python: "PICOS is a user friendly interface to several conic and integer programming solvers, very much like YALMIP under MATLAB." Since the version 1.0.1, it is possible to do complex semidefinite programming with Picos: http://picos.zib.de/v101dev/complex.html


1

Suppose your initial state is $\lvert 2\rangle$ and that the states $\lvert 0 \rangle$ and $\lvert 1 \rangle$ have lower energies than $\lvert 2 \rangle$. Assuming that there is no so called selection rule that prevents $\lvert 2 \rangle$ from emitting a photon and end up in $\lvert 0 \rangle$ or $\lvert 1 \rangle$, then the final state will be $$ \lvert 2 ...


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The expectation value of energy is something else than the energy in a particular experiment. With your choice of the initial states, the photons emitted (negative difference) or absorbed (positive difference) will have energies either $$ E_1-E_0 \text{ or } E_2-E_0 \text{ or } E_1-E_5 \text{ or } E_2-E_5 $$ If each of the four transitions were equally ...


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As you stated already, a measurement of the energy of the hydrogen atom must return an energy eigenvalue. Measuring before and after a transition gives us two energies $E_n$ and $E_m$. This is always true, regardless of the fact that the expectation value of the energy before measuring might not be a difference of $E_p$ and $E_q$ for some $p,q$: The actual ...



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