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# 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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I am taking a fluid mechanics class and don't know very much physics. I was confused in class when the prof kept calling this derivative matrix of a fluid flow (A function from $\mathbb{R}^n\to\mathbb{... 1answer 102 views ### Ground state of electrons in diatomic molecule and probability to find electron near a shell at the ground state Question: The effective Hamiltonian of an outer-shell electron in a diatomic molecule is given by $$H = \left(\begin{array}{cc} E_1 & t \\ t & E_2 \\ \end{array} \right)$$ where$E_1$... 0answers 391 views ### Spectral decomposition of a time dependent operator Let$M$be an operator with spectral decomposition $$M(0) = \sum_{i = 1}^n \lambda_i \left|{m_i(0)}\right\rangle\left\langle{m_i(0)}\right|.$$ To find$M(t)$, I know that$M(t) = U^{\dagger} M(0) U$. ... 1answer 296 views ### Determinant of matrix exponential [closed] I am trying to find the determinant of the matrix $$\exp({i\sigma.\hat{n}\frac{\phi}{2}})$$ I expanded the above Matrix using the identity $$\exp({i\sigma.\hat{n}\theta})=\hat1\cos(\theta)+i\sigma.... 1answer 84 views ### Rotation by a given distance in given direction on a Bloch sphere I'm using the following matrix to rotate a states by a distance given by \theta in a direction given by \phi:$$ U = \left(\begin{array}{cc} \cos{\theta\over{2}} & -e^{- i\phi}\sin{\theta\... 1answer 259 views ### How to produce the basis states of a quantum computer? Suppose you have a one qubit system; the (traditional) basis states are$|0\rangle$and$|1\rangle$, and any state of the qubit can be described by a linear combination of these two. Now suppose you ... 2answers 867 views ### 2D harmonic oscillator: polar versus Cartesian eigenstates Consider the usual two-dimensional harmonic oscillator (2D HO) with the Hamiltonian $$H = -\frac{1}{2}\nabla_x^2 + \frac{1}{2}x^2 -\frac{1}{2}\nabla_y^2 + \frac{1}{2}y^2.$$ In Cartesian coordinates, ... 1answer 134 views ### Eigenvector eigenvalue [closed] If$A=\left(\begin{array}{cc} 1& 0 \\ 0 &-1\end{array}\right)$, what will be the eigenvalue and eigenvector? Let me know how to solve it. I am facing difficulty in finding the eigenvector. 4answers 1k views ### When writing this Lorentz transform as a matrix why do we take the transpose? I'm working on an assignment for an intro to relativity class and I've come up against this series of equations. The step I'm confused about is going from summation notation to matrix notation where ... 1answer 232 views ### What the primed quantities really are in this context? In this definition: The set of$N$quantities$V_j$is said to be the components of an$N$-dimensional vector$\mathbf{V}$if and only if their values relative to the rotated coordinate axes are ... 2answers 74 views ### In which context is this concept of vector being defined? In mathematics the concept of a vector can be made quite general and abstract with the idea of vector space. We define what is a vector space$V$and we say that a vector is an element of a vector ... 1answer 407 views ### Dimensions of a tensor product space Susskind, in his 'Theoretical Minimum' book about QM says that given two 2-dimensional spaces, the product state of two vectors, one from each space, is determined by four real parameters. But, he ... 2answers 7k views ### Why is the general solution of Schrodinger's equation a linear combination of the eigenfunctions? Here is a quote from Introduction to quantum mechanics by David J Griffiths: The general solution is a linear combination of separable solutions. As we're about to discover, the time-independent ... 0answers 179 views ### How to construct Hilbert spaces which are needed for quantum mechanics? I have some books on quantum mechanics. These explains that states of a system are unit vectors and observables are represented by self-adjoint linear operator on a Hilbert space, etc ... but don't ... 1answer 104 views ### How can I describe these “different kinds” of principal axes The following question regards the moment of inertia tensor of classical mechanics, given by $$\mathbf{I} = \begin{bmatrix} I_{xx}&I_{xy}&I_{xz}\\ I_{yx}&I_{... 0answers 161 views ### Determinant of a mixed rank-2 tensor Often dispersion relations in plasmas are found by setting the determinant of some quantity to equal zero. My question is, how does one do this when working covariantly with tensors (e.g. Broderick, A.... 1answer 541 views ### Why can operators be represented as matrices in quantum mechanics? I am studying introductory quantum mechanics in our undergraduate course. I saw that operators can be represented as matrices too. I can't figure out the proper reason. My attempt is: As operators ... 1answer 93 views ### How to conclude a vector is space-like? I'm working with the Lorenztian inner product and would like to show that if a vector v is lightlike, so \langle v,v\rangle =0, and if \langle v,w\rangle =0, then either w is spacelike or w ... 1answer 997 views ### Pauli matrices in spherical coordinates In my work I currently have to work with the following partition function Z\propto e^{\vec{h}.\vec{\sigma}} where \vec{\sigma} are pauli matrices and h is some vector. So far I've been using ... 1answer 269 views ### Problem in deriving Schrodinger and Pauli equation from Dirac's Working out the non relativistic limit of the Dirac equation, we encounter this quantity: (\vec{\sigma} \cdot \vec{p}) and in my notes it says that$$ (\vec{\sigma} \cdot \vec{p})^2 = p^i p^j\sigma^... 1answer 78 views ### Calculating$U = e^{i\tau\cdot\alpha}$for$SU(2)$transformation [closed] How can I prove that if$U = e^{i\tau\cdot\alpha}$with$\alpha=\alpha(\sin\phi,\cos\phi,0)$, then $$U= \begin{equation} \left(\begin{array}{cc} \cos\alpha & e^{i\phi} \sin\alpha \\ - e^{-i\phi} ... 1answer 521 views ### Why transformations in quantum mechanics are linear? In quantum mechanics, when we want to introduce reference frame change, we do it such that \left|\psi'\right> = U\left|\psi\right>. Using the fact that \left<\psi|\psi\right>=1, we ... 1answer 74 views ### What is the implication of the existence of a non-singular matrix S: \gamma^\lambda S=S \gamma^m u? [closed]$$\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu}I_4$$Pauli's fundamental theorem states that given two sets of matrices \gamma^\mu and \gamma^\nu which obey the commutation rules (... 1answer 61 views ### Operator Linearity How can you prove rigorously that an operator is linear? For example the momentum operator p = -i\hbar\frac{\partial}{\partial x} or position operator x = x. 1answer 291 views ### Decomposition of a Pauli Gate [closed] I want to know how to perform a Pauli gate spectral decomposition. For example: \sigma_x = \left(\begin{matrix}0&1\\1&0\end{matrix}\right) How is it spectral decomposition performed? ... 1answer 41 views ### uncoupling pertubated 2-D harmonic oszillator I want to exactly solve a two dimensional harmonic oscillator with$$ \mathcal{H} = \dfrac{p_x^2}{2m} + \dfrac{p_y^2}{2m} + \dfrac{m\omega^2}{2}(x^2+y^2+2K xy)$$I understand that I have to find a ... 1answer 62 views ### Apparent analogies between statements from linear algebra and covariant tensor calculus When using covariant tensors in relativity or particle physics, there are some statements that seem like analogues of statements known from linear algebra. For example, if we have a symmetric real-... 2answers 373 views ### Lorentz tensor - always outer product of two four-vectors? A 2nd rank Lorentz tensor is defined as one that transforms as:$$T'=\Lambda T\Lambda^T$$It is clear that the quantity:$$\tilde A \tilde B^T$$where \tilde A and \tilde B are 4-vectors always ... 1answer 648 views ### Orthonormality and ket expansion for uncountable basis kets, such as the position eigenkets? This question is about a probable confusion of definitions that I may have somewhere. Also my math knowledge is not too big so I'll try not to get too abstract. Let's say I have a vector space made ... 1answer 104 views ### An assumption in the non-degenerate perturbation theory The perturbed Hamiltonian is$$ H = H_0 + g V , $$where g is the coupling parameter. The perturbed eigenvalue and eigenstate are of the form$$ E(g) = \sum_{r~=~0}^\infty g^r E_r ,\quad \left|... 1answer 204 views ### How can the energy eigenkets and position eigenkets of a quantum system both be complete? [duplicate] A quantum system is often given as a wavefunction of position. This is a vector in a continuous, infinite-dimensional vector space, with uncountable dimension. However we also know that the energy ... 1answer 617 views ### Active and passive transformation in Dirac notation - R. Shankar My questions are related with everything discussed in this post already. It's about the section on Active and Passive transformation in the 1st chapter (Linear Vector Spaces) of R. Shankar's ... 3answers 5k views ### How do one show that the Pauli Matrices together with the Unit matrix form a basis in the space of complex 2 x 2 matrices? In other words, show that a complex 2 x 2 Matrix can in a unique way be written as $$M = \lambda _ 0 I+\lambda _1 \sigma _ x + \lambda _2 \sigma _y + \lambda _ 3 \sigma_z$$ If$$M = \Big(\begin{... 1answer 727 views ### Matrix representation of operators I have a question about the matrix representation of quantum operators, in one of the books I'm reading I found this: Let {\{\psi_{n}\}} be a complete orthonormal system and \bf A a operator. ... 2answers 435 views ### How to express \gamma^{\mu} \gamma^{\nu} as a linear combination of {1, \gamma^5, \gamma^{\mu}, \gamma^{u} \gamma^5, \sigma^{\mu \nu}}? ** EDIT: I think I have completely missed the mark on asking my question. Here is another try. I do not understand what a linear combination means in this situation. My naive desire is to have an ... 2answers 279 views ### Spectral theorem: matrices vs operators I'm a bit confused about some of the terminology being thrown around in my text. To start with: A diagonal representation for an opertor A on V is a representation A = \sum_i \lambda_i |i \... 2answers 99 views ### Operator functions: why is f(A) uniquely defined? In Nielsen and Chuang, they write: Let A = \sum_a a|a\rangle \langle a| be the spectral decomposition of A. Define f(A) = \sum_a f(a) |a \rangle \langle a|. Apparently this is uniquely defined. ... 1answer 311 views ### Problem with Bogoliubov transformations of an operator I have a set of Bogoliubov transformation as follows: \begin{equation} a(\beta) = a \cos \theta (\beta) - \tilde{a}^\dagger \sin \theta (\beta)\\ \tilde{a}(\beta) = \tilde{a} \cos \theta (\beta) + a^\... 1answer 205 views ### Expressing two particle state as a combination of single particle states Say I have a Fock space H with basis K = \{ | k \rangle \big| k \in \mathbb{N} \}. Then I consider the following single particle states:$$ | A \rangle = \sum_{k \in K} a_k | k \rangle, \tag{1}$$... 2answers 652 views ### Vectors ( Resolution of vectors ) How many components can a vector be resolved into? I think that it should be infinity because there can be infinite axes. Am I right? 1answer 40 views ### What is the relation between dimension of vector space and number of pairs of orthogonal vectors in that space In quantum mechanics when we say state of all states for a simple spin system is 2 dimensional. Q1. Does it mean that there are only two independent states (states which can be unambiguously ... 2answers 173 views ### Euler Rotations in Ordinary Space I'm reading LittleJohn's notes on Rotations in Ordinary Space on Quantum Mechanics. Link: http://bohr.physics.berkeley.edu/classes/221/1011/notes/classrot.pdf. I'm trying the last question given in ... 0answers 226 views ### Difference between tensor in mathematics and physical quantities In mathematics, we say that a an nth-rank tensor in m-dimensional space essentially takes n vectors of dimension n each and converts them into a scalar. Then what do we mean when we say that ... 0answers 1k views ### Raising and Lowering Indices using the Metric Tensor Given the next tensor: X^{\mu \nu}= \left(\begin{array}{cccc} 2 & 0 & 1 & -1 \\ -1 & 0 & 3 & 2 \\ -1 & 1 & 0 & 0 \\ -2 & 1 & 1 & -2 \\ \end{array}\... 1answer 5k views ### How to find the inverse metric? I am trying to find the inverse of the metric$$\mathrm ds^2 = \rho^2~\mathrm d\theta^2 -2a\sin^2\theta ~\mathrm dr~d\varphi + 2~\mathrm dr~\mathrm du + \rho^{-2}\left[\left(r^2+a^2\right)^2 -\Delta ... 0answers 64 views ### Physical significance of$s$-sparse and efficiently row computable Hermitian matrix In section II of Quantum algorithm for solving linear systems of equations by Harrow et al, the authors defined the matrix$A$as$s$-sparse and efficiently row computable.$s$-sparse means having at ... 0answers 30 views ### Propagating state information backwards through governing equations in a Kalman filter I'm working on a simple Kalman filter that estimates the position, velocity, and acceleration of a point mass using position measurements. If you know the position of something, you should also know ... 2answers 188 views ### Rotation between two Frames I have a simple three dimensional frame which I use to define a certain tensor$A$. I want to define a second frame which corresponds to a rotation of the first one define by the Euler angles ($\alpha,...
I am reading Martin Plenio's lecture notes on Quantum Mechanics from the Imperial College. In page 63 he wants to prove the relation $$[\hat{x},\hat{p}]=i\hbar$$ via \langle x | [\hat{x},\hat{p}]|\...
The classical rotational kinetic energy of a system is calculated as: $KE=\frac{1}{2}I\omega^2=\frac{1}{2}L\omega=\frac{L^2}{2I}$ where: $I=$ moment of inertia tensor, $\omega=$ angular velocity ...