# Questions tagged [linear-algebra]

To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.

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### Harmonic Oscillator Eigenket Notation

I'm reading the $3^{\mathrm{rd}}$ edition of Sakurai and Napolitano's Modern Quantum Mechanics, and I have a brief question about the notation used to describe the eigenstates of the harmonic ...
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### Is amount of substance fundamentally a scalar quantity? (in the mathematical sense of scalar)

Reading the SI (and ISO) standard for units and quantities, I'm currently puzzled by something very subtle. If I can see and understand why we talk about scalars, vectors, and tensors in the context ...
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### Distinguishing different senses of 'vector' [closed]

Two mutually orthogonal unit vectors acting at a point $p$, produce a resultant, whereas the two orthogonal unit basis vectors at the origin do not, why?
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### Coherent state basis

I'm learning about coherent states in a more in depth lesson the the quantum harmonic oscillator. Coherent states are eigenstates of the lowering operator. In my head this is just saying: since any ...
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1 vote
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### Quaternions as rotation generators

The following exercise appears in Geometric Algebra for Physicists by Chris Doran and Anthony Lasenby in section 1.8. The unit quaternions $i, j, k$ are generators of rotations about their ...
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### The Heisenberg Picture [closed]

In the Heisenberg picture, observables, rather than states, undergo unitary evolution. Can an observable turn into an $un$observable as a result of unitary time evolution? That is, If $U$ is a unitary ...
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### Why are the coefficients of vectors (that are not eigenstates) the probability amplitudes?

In the adiabatic approximation, we assume that: are the solutions of the eigenvalue problem: We know however that the ψn(t) 's here may not be actual solutions to the time dependent Schrodinger ...
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### Covariant vectors and change of basis matrix

I'm studying the basics of special relativity and I've been having some difficulties with the concept of contravariant and covariant vectors. Now, at least in the basic way in which we introduce them ...
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### Hamiltonian as a matrix and its elements [closed]

Let us consider an electron in an infinitely deep one-dimensional potential well of thickness L with zero potential energy at the bottom of the well. The normalised eigenfunction solutions to this can ...
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### Distributing operators inside of the bra and kets confusion

I'm reading Griffiths and he has this section where he states that $|\hat{Q}f\rangle$ is mathematical nonsense and that really we should write $\hat{Q}|f\rangle$, where the latter makes more sense to ...
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### Quantum Particles on a circle and Circulant matrices [closed]

I have $n$ particles on a circle with the Hamiltonian \begin{equation} H = \sum_{n=1}^N \frac{p_n^2}{2m} + \frac{1}{2}m\omega^2 \sum_{n=1}^N (x_{n+1}-x_n)^2 \end{equation} I need to find the energy ...
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### Binary number as quantum observable variable

I am currently reading this article about quantum computations on set of N two-state ions. There author associates ground state with number zero $|0>$ and excited state of ion with number $|1>$ ...
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### Why is the trace of the outer product of two states equal to the inner product of the two states?

Why is it that, given two quantum states $|\psi_1\rangle$, $|\psi_2\rangle$, $$\mathrm{Tr}(|\psi_1\rangle\langle\psi_2|) = \langle\psi_2|\psi_1\rangle \quad$$ I went through the equation with the ...
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$\newcommand{\hp}{\hphantom{#1}}$ We have the entangled state of two pairs of qubits: $$|\psi \rangle =\frac{1}{2}|0011\rangle-\frac{1}{2}|0110\rangle-\frac{1}{2}|1001\rangle+\frac{1}{2}|1100\... • 159 0 votes 1 answer 86 views ### What do special unitary groups SU(n) represent geometrically? It's frequently said that special orthogonal group SO(n) represent rotations in n-dimensional space. What do SU(n) groups represent? • 657 2 votes 4 answers 138 views ### Quantum mechanics: "Representation" vs. "basis" I am confused about the difference between the terms "representation" and "basis" of a state or operator. For example, Let us have eigen-kets of Hamiltonian H denoted by |\phi_n\... • 719 4 votes 1 answer 404 views ### Notion of Co- and Contravariance in Dirac-Notation \newcommand{\bra}{\left<#1\right|}\newcommand{\ket}{\left|#1\right>}\newcommand{\bk}{\left<#1\middle|#2\right>}\newcommand{\bke}{\left<#1\middle|#2\middle|#3\right>} A (... • 113 0 votes 1 answer 34 views ### Use Index Notation properly when indices are already used in identifying which bases is the matrix metric calculated with respect to I am wondering how to apply the usual linear algebra to the rather unfamiliar case of 'matrices' with indices in special relativity or even general relativity. In particular, consider$$f=\sqrt{-\det\...
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From the first postulate of quantum mechanics we known that the vector $|\psi\rangle$ is the mathematical entity that says, intuitively, "in a time $t$, the (state of a) system is a vector". ...