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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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4
votes
3answers
589 views

What does tensor mean in a physics context?

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{...
0
votes
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 $E_1$ ...
0
votes
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 $M(t) = U^{\dagger} M(0) U$. ...
-2
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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....
1
vote
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\...
0
votes
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 ...
0
votes
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, ...
-5
votes
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.
6
votes
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
8
votes
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 ...
0
votes
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 ...
1
vote
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_{...
0
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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....
4
votes
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 ...
1
vote
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 $w$ ...
0
votes
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 ...
1
vote
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^...
0
votes
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} ...
9
votes
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 ...
0
votes
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 (...
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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 $x = x$.
0
votes
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? ...
-1
votes
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 ...
1
vote
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-...
0
votes
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 ...
0
votes
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 ...
0
votes
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|...
1
vote
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 ...
2
votes
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 ...
2
votes
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{...
1
vote
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. ...
3
votes
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 ...
4
votes
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 \...
3
votes
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. ...
3
votes
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^\...
3
votes
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: $$ | A \rangle = \sum_{k \in K} a_k | k \rangle, \tag{1}$$...
-1
votes
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?
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
226 views

Difference between tensor in mathematics and physical quantities

In mathematics, we say that a an $n$th-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 ...
1
vote
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}\...
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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,...
2
votes
2answers
361 views

Another derivation of canonical position-momentum commutator relation

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}]|\...
0
votes
1answer
794 views

calculating rotational kinetic energy of a diatomic molecule

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 ...