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

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25
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4answers
5k views

Are matrices and second rank tensors the same thing?

Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified: Are matrices and second rank tensors ...
11
votes
1answer
613 views

What're the relations and differences between slave-fermion and slave-boson formalism?

As we know, in condensed matter theory, especially in dealing with strongly correlated systems, physicists have constructed various "peculiar" slave-fermion and slave-boson theories. For example, For ...
10
votes
3answers
1k views

What does the dual of a tensor mean (e.g. dual stress tensor in relativistic ED)?

I know what the dual of a vector means (as a map to its field), and I am also aware of of the definition a dual of a tensor as, $$F^{*ij} = \frac{1}{2} \epsilon^{ijkl} F_{kl}\tag{1}$$ I just don't ...
10
votes
1answer
253 views

Conceptual difficulty in understanding Continuous Vector Space

I have an extremely ridiculous doubt that has been bothering me, since I started learning quantum mechanics. If we consider the finite dimensional vector space for the spin$\frac{1}{2}$ particles, ...
8
votes
1answer
763 views

“An operator is hermitian”. Implications?

Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as: Every dynamical variable may be ...
7
votes
3answers
2k views

Difficulties with bra-ket notation

I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with ...
7
votes
1answer
326 views

How can (in Dirac's terminology) the product of two “real” linear operators be “not real”?

I'm puzzled about a statement from Dirac's book, The principles of quantum mechanics, (§8, p.28): As a simple examples of this result, it should be noted that, if $\xi$ and $\eta$ are real, in ...
7
votes
3answers
336 views

Use of 'complete' as in 'complete set of states' or 'complete basis'

Question. In the context of QM, I hear the phrases 'complete set of states' and 'complete basis' (among other similar expressions) thrown around rather a lot. What exactly is meant by 'complete'? ...
7
votes
3answers
737 views

What is a basis for the Hilbert space of a 1-D scattering state?

Suppose I have a massive particle in non-relativistic quantum mechanics. Its wavefunction can be written in the position basis as $$\vert \Psi \rangle = \Psi_x(x,t)$$ or in the momentum basis as ...
7
votes
2answers
270 views

Can we show that time is orthogonal to space?

It's easy to show that the time we measure is "in a different direction" from the space directions we measure. However, it's not immediately obvious to me that these directions are orthogonal. How do ...
6
votes
6answers
2k views

Linear Algebra for Quantum Physics

A week ago I asked people on this forum what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of ...
6
votes
2answers
112 views

Precise meaning of composition of ket and bra, e.g. $|\psi\rangle\langle\psi|$

I'm currently studying density matrices, and have been frequently coming across the construction $$|\psi\rangle\langle\psi| \,.$$ What is the formal meaning of this composition? I understand ...
5
votes
5answers
1k views

Show the Lorentz Transformation Matrices Have an Inverse

Assume the Lorentz transformations obey the relationship $$g_{uv}\Lambda^u_{p}\Lambda^v_\sigma = g_{p\sigma},$$ where $g_{uv}$ is the metric tensor of special relativity. How can one show, under that ...
5
votes
2answers
146 views

What does $\left|x,t\right>$ actually mean (Heisenberg picture)?

I am pretty much confused with this notation I believe. The Heisenberg states are denoted by $\left|x,t\right>$ and the Schrodinger states are given by $\left|x(t)\right>$. It seems like both of ...
5
votes
2answers
335 views

Is length/distance a vector?

I have heard that area is a vector quantity in 3 dimensions, e.g. this Phys.SE post, what about the length/distance? Since area is the product of two lengths, does this mean that length is also a ...
5
votes
3answers
227 views

Countable Matrix Representation

In my quantum mechanics class, my professor explained that the Hamiltonian along with position and momentum operators can be represented by matrices of countable dimension. This is especially usefull ...
5
votes
1answer
375 views

What do up-left orthogonality has in common with up-down and what is their relationship?

I am familiar with the true (or general) notion of orthogonality, given in the Linear Algebra and Pythagoras theorem derived from the $\vec x \cdot \vec y = 0$. I have also recently got to know that ...
5
votes
2answers
279 views

Why are the solution coefficients for a harmonic oscillator proportional to minors of the determinant?

I'm studying the oscillations of systems with more than one degree of freedom from Landau & Lifshitz's Mechanics Third Edition (for those who have the book, my question corresponds roughly to ...
5
votes
1answer
229 views

Advanced atomic physics: From Liouville Equations to the Bloch equations

I'm trying to derive the Bloch equations from the Liouville equation. This should be possible according to this paper, where it discusses higher order Bloch equations (second order spherical tensors). ...
5
votes
1answer
84 views

When is an operator subspace the span of Kraus operators?

Let $A$ and $B$ be finite dimensional Hilbert spaces, and let $\mathcal{L}(A \to B)$ be the space of linear operators from $A$ to $B$. Say that a subspace $K \subseteq \mathcal{L}(A \to B)$ is a span ...
4
votes
5answers
409 views

Math of eigenvalue problem in quantum mechanics

I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
4
votes
2answers
138 views

Dimension of separable state

I'm from a pure mathematics background (BSc and MSc). I’m reading Quantum Computing: From Linear Algebra to Physical Realization by Nakahara. Please can you help me to understand how the dimension ...
4
votes
1answer
194 views

Trace of an operator matrix (Quantum computation and quantum information)

I'm reading the book Quantum computation and quantum information by Mike & Ike and I'm stuck at 2.60/2.61. There, the author says that, given the operator $A|ψ⟩⟨ψ|$, its trace is: $${\rm ...
4
votes
1answer
134 views

How are matrices used to represent quantities, and what is the meaning of a matrix?

So I'm reading this text on Quantum Mechanics, and it goes through a few chapters that I understand fairly well including probability. But then it says that all quantities, like position and energy ...
4
votes
1answer
96 views

Some question about symplectic transformation

I read Arnold's book Mathematical Methods of Classical Mechanics and come across with three problems in page 229. 1.Let $\lambda$ and $\bar{\lambda}$ be simple (multiplicity 1) eigenvalues of a ...
4
votes
2answers
132 views

Dimension of the space of solutions in an electric circuit

Consider an electric circuit with dc sources ( voltage and current) and resistors. Write down the equations. In the most general case, the solution of the system is not unique. The set of solutions ...
4
votes
1answer
155 views

Saturation of the Cauchy-Schwarz Inequality

Going to as little details as possible, here is a statement from Wald's text on QFT in curved spacetimes(I am not quoting the book) He considers two vector spaces ${\cal S}$ and ${\cal H}$. Note ...
3
votes
3answers
237 views

Why are the energy eigenstates realized in atomic transitions?

I have a question like "Why is it often assumed that particles are found in energy eigenstates?", it is a little different, though. When one solves the hydrogen atom, one can use a polynomial Ansatz ...
3
votes
1answer
344 views

Representation of operators in quantum mechanics

For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?
3
votes
1answer
206 views

The issue on existence of inverse operations of $a$ and $a^{\dagger}$

I have asked a question at math.stackexchange that have a physical meaning. My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
3
votes
2answers
135 views

Matrix elements of linear operators - orthonormal basis required?

In an early linear algebra class of mine, I learnt that a linear map $\mathcal{A}$ acting on a vector space could be represented by a matrix $A_{ij}$ according to the rule: $$\mathcal{A}({e_j}) = ...
3
votes
1answer
182 views

A tensor product of two spin-1 particles

I'm rather confused, and I was hoping if someone could help me figure out this (probably rather elementary) issue. I have two particles with spin 1, whose state I describe by $m_S$ and $m_I$ ...
3
votes
1answer
101 views

Non-Euclidean spaces in Quantum Mechanics

In quantum mechanics, I have been going through basics of the subject. In general the space of quantum states is Hilbert space (which is Euclidean - I presume). Being just curious, are there any ...
3
votes
1answer
156 views

A naive question about the Second Quantization?

Let's consider a single-particle(boson or fermion) with $n$ states $\phi_1,\cdots,\phi_n$(normalized orthogonal basis of the single-particle Hilbert space), and let $h$ be the single-particle ...
3
votes
2answers
170 views

Why is this not a realisable operation on a quantum system?

Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ ...
3
votes
1answer
61 views

Special relativity: how to prove that $g = L^t g L$?

We have $$X^\textrm{t}gX = 0 \iff X^\textrm{t}L^\textrm{t}gLX = 0,$$ where $X$ is a column vector of length four, $L$ is a non-singular $4 \times 4$ matrix, 't' denotes matrix transpose, and $$g = ...
3
votes
2answers
161 views

Eigenenergies and eigenkets given the Hamiltonian

For a two level system the Hamiltonian is: $$ H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\rangle\langle2|+|2\rangle\langle1|) $$ where $a$ is a number with the dimension of an energy. I need to ...
3
votes
1answer
60 views

Boson calculus and Maximum Weight State

I'm just going over a few past exams for tomorrow, and I've come across a question that I'm having quite a bit of difficulty with. Let $\left|0\right\rangle$ denote the Fock vacuum state so that ...
3
votes
1answer
169 views

Random Hankel matrix and eigenvalues distribution

I would like to know if there are any theoretical results on the distribution of the eigenvalues of Hankel matrices. I seek a result like the Marchenko–Pastur distribution for random matrices.
3
votes
1answer
46 views

How to define tensor contraction without referring to summation?

The textbook defines a tensor to be an element in $(T^*)^k×T^l→R$. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation ...
3
votes
1answer
53 views

Complex semi-definite programming

I'm doing some calculations and I want to simulate them in python or matlab (or whatever). However I use hermitian matrices and I don't really manage to find a library which enables me to calculate ...
3
votes
1answer
63 views

Mutually unbiased bases

This question can be formulated in two ways. Let there be two $d$-dimensional orthonormal bases $B_{1}$ and $B_{2}$. I refer to the elements of $B_{1}$ by $\lvert\nu_{i}\rangle$ and to the elements of ...
3
votes
1answer
243 views

Rotation about the $z$-axis on the Bloch sphere

I'm having some trouble with successfully working out a rotation about the $z$-axis on the Bloch sphere. Now, I know how this is performed, in principle. A rotation of the Bloch-sphere around an ...
3
votes
1answer
450 views

1D Ising Model (NN and NNN interactions) with 2 transfer matrices

I've tried an alternative solution for finding the partition function of this model. So is what I've done correct? If it isn't then please prove and explain why not. (I'm pretty sure I made a ...
3
votes
1answer
101 views

Are the symmetry operators well defined in the context of Projective Symmetry Group(PSG)?

Consider the Schwinger-fermion approach $\mathbf{S}_i=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$ to spin-$\frac{1}{2}$ system on 2D lattices. Just as Prof.Wen said in his seminal paper on PSG, the ...
3
votes
1answer
191 views

Algorithm for identifying planes in a Bravais Lattice

I have a lattice with Lattice Vectors $(\vec{t}_1,\vec{t}_2,\vec{t}_3)$ which are NOT orthogonal in general. How can I identify the atoms/unit cells that belong to a plane - that is normal to a given ...
3
votes
2answers
240 views

How to express continuous values as a matrix

Usually a quantity of a matrix is defined as the eigenvalues of the matrix. If so, how can anyone express continuous values, as in Schrodinger picture, into a matrix?
3
votes
0answers
42 views

How coordinate system shifting is related to similarity transformations?

I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?
3
votes
1answer
120 views

Determinant and adjunct of $k-\omega^2m$ in terms of natural frequencies

Given is a mechanical multiple degree of freedom system described by the following matrices and equation: mass matrix ${\bf{m}} = \left[\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 ...
3
votes
0answers
136 views

Fock Subspaces and Weight Vectors

This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons. I've got ...