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

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How to take partial trace?

$L$ is a linear operator acting on hilbert space $V$ of dimension $n$, $L: V \to V$. The trace of a linear operator is defined as sum of diagonal entries of any matrix representation in same input ...
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votes
2answers
690 views

Finding the matrix representation of a superoperator

I am trying to express superoperator (e.g. the Liouvillian) as matrices and am having a hard time finding a way to do this. For instance, given the Pauli matrix $\sigma_y$, how do I find the matrix ...
3
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2answers
83 views

Lagrange Multipliers and Virtual Work: Are Joos & Freeman wrong?

I have come to suspect that the treatment of virtual work in configuration space using Lagrange multipliers given here "Theoretical Physics, by Georg Joos & Ira M. Freeman, pg 114" is not correct. ...
3
votes
1answer
64 views

Tensor indices and row and column labels of corresponding representation matrices

When reading undergraduate GR literature, I often see that the authors represent tensors ${\eta^\alpha}_{\beta}$, ${\eta^\beta}_{\alpha}$, $\eta_{\alpha \beta}$, $\eta^{\alpha \beta}$ as matrices. ...
3
votes
1answer
479 views

What is the physical interpretation of the dot/inner/scalar product of two vectors?

What is the physical interpretation of the dot/inner/scalar product of two vectors? See, if we multiply two scalars like 2*3 we say two times three is six. I also do understand multiplication of ...
3
votes
1answer
969 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
151 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
203 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
186 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
2k views

Jarlskog Invariant and its mathematical origin

CP violation is present in the weak interactions if There are no degeneracies in the up-quark/down-quark matrices The Jarlskog invariant $J=Im(V_{us} V_{cb} V_{ub}^* V_{cs}^*)$ is nonvanishing ...
3
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2answers
80 views

Interpretation of Eigenvalues and Eigenvectors of an hyperbolic conservation law $\partial_t W + A \partial_x W = 0$

I read in a article dealing with a hyperbolic partial differential equations this statement : For any system of hyperbolic partial differential equations (pde), expressed as (1) ...
3
votes
2answers
123 views

Proving the unitary relation of ensemble decompositions

In my class it was told that ensemble decompositions of a density operator $\rho$ are not unique, but that the ones that exist are related by a unitary operator. I'm trying to prove this, but I get ...
3
votes
1answer
124 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
74 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
601 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
76 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
266 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
326 views

Many Body Physics: Hamiltonian block structure and Symmetries

Consider a many body problem of a small cluster, e.g. the 'Hubbard-Cluster' (albeit the question may be of relevance for other Hamiltonians as well): $$\mathcal{H}=\sum_{<ij>\sigma} t_{ij} ...
3
votes
1answer
101 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
83 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
826 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
721 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
127 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
280 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 ...
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2answers
384 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?
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2answers
74 views

Understanding basics of tensors

I am trying to understand tensors to learn General relativity. In the book that I am reading they claim that if the basis of a vector space undergoes a linear transformation $T$ then the components of ...
3
votes
0answers
80 views

Geometry topics in physics [closed]

I'd like to learn modern physics at an advanced level, but since I've no access to university, I'm self-teaching, and appeal to the Internet for information about what to study and how. Currently, ...
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0answers
85 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 ?
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0answers
167 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 ...
2
votes
3answers
242 views

Dimension of vector resulting from tensorial product

I'm quoting what I found in a book about quantum computation: Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vectors, then ...
2
votes
3answers
185 views

Diagonalization Of $(\sigma_x+\sigma_y)$

Can this matrix $(\sigma_x\pm\sigma_y)$ be diagonalised? Clearly, if $\sigma_x$ is diagonalized by a similarity transformation $S_1\sigma_x{S_1}^{-1}$, then $\sigma_y$ can't be diagonalized by $S_1$, ...
2
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3answers
472 views

Vector space of $\mathbb{C}^4$ and its basis, the Pauli matrices

How do I write an arbitrary $2\times 2$ matrix as a linear combination of the three Pauli Matrices and the $2\times 2$ unit matrix? Any example for the same might help ?
2
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1answer
258 views

Infinite-dimensional Hilbert spaces in physical systems

Can anyone give an example of when infinite-dimensional Hilbert spaces are required to describe a physical system? The standard answer to this question is yes, and I'm sure some of you will be quick ...
2
votes
2answers
213 views

When is matrix representation of a Hermitian operator invertible?

If I have a Hermitian operator $H:V \to V$ on a finite-dimensional vector space $V$, and I write down its matrix representation in some basis $B$ with matrix representation being $[H]_B$, then in what ...
2
votes
1answer
282 views

Extension of Lami's theorem

I was experimenting with the triple scalar product and forces in equilibrium when I came to this result: Consider 4 forces $ \pmb{F_i}$ for $i=1,2,3,4$. $\pmb{F_i}=F_i\hat{e_i}$ where $\hat{e_i}$ is ...
2
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2answers
802 views

Differences between wave function and set of orthonormal wave functions?

I'm reading a QM book. It first says for wave function: "The state of a physical system (or particle) is completely specified by an entity associated with it called a wave function, Ψ , that in ...
2
votes
1answer
58 views

Local fermionic symmetry and GS action

I have a trouble understanding an argument which I think has a simple answer but I am not getting it. The question is that if you don't impose local fermionic symmetry the GS action has only one term ...
2
votes
1answer
115 views

Characteristic polynomial of a Matrix

In fact, this problem is more likely to be a math problem. When I read a paper(http://arxiv.org/abs/0707.2875), the author includes the characteristic polynomial for a type of matrix $A_k$ with ...
2
votes
1answer
68 views

Diagonalisation: Schmidt vs eigenvalue - when to use which?

In physics we encounter diagonalisation of matrices or operators in a variety of areas. But there are different kinds, the main two being Schmidt decomposition and eigenvalue diagonalisation. The two ...
2
votes
1answer
160 views

Proving polarization identity for operators in complex vector spaces

In general for two operators to be equal, all their (matrix) elements must be equal $$A = B \rightarrow \langle \phi_1|A| \phi_2\rangle=\langle \phi_1|B| \phi_2\rangle$$ However, I am asked to show ...
2
votes
1answer
138 views

Proofs on operator algebra [closed]

I'd like to ask the community to please verify the first two proofs below and help me get through the last one since I seem to be stuck. Thank you in advance. Proof 1: Given two noncommutting ...
2
votes
1answer
68 views

Can All 4-D column matrices be given as tensor product of 2-D column matrices?

I am familiar with entanglement concept. But it feels bit weird to me that all possibilities of a system in a $4$-dimensional vector space cannot be given as tensor product of two $2$-dimensional ...
2
votes
2answers
150 views

Tensors as multilinear maps

Sean Carrol's in his book on GR introduces tensors as a multilinear map of a set of dual vectors and vectors onto R. I usually think of tensors as a multidimensional array of numbers with fixed ...
2
votes
1answer
473 views

why non orthogonal states are indistinguishable?

I want to know what does it mean by distinguishable quantum state from Mathematics perspective I mean mathematically. As a non physics background student could any one explain me why non orthogonal ...
2
votes
2answers
117 views

Expanding a ket in the position basis?

My textbook says that to find the ket $|ψ\rangle$ in the same position basis as the ket $|ø\rangle$ we do the following: $$|ψ\rangle=\int dø|ø\rangle \langle ø|ψ\rangle$$ Firstly can $|ø\rangle$ be ...
2
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1answer
99 views

Formular for interior product, example

In Nakahara's Geometry,Topology and Physics, the interior product is defined like this : $$i_X: \Omega^{r}(M) \rightarrow \Omega^{r-1}(M).$$ Where $ X \in X(M)$ and $\omega \in \Omega^{r}(M)$ ...
2
votes
3answers
330 views

Shankar's Active/Passive Change of Basis

I'm working my way through Shankar's Quantum Mechanics (7th printing, and I'm doing it alone, so I apologize if I have core concepts completely wrong). He has a section on Active and Passive ...
2
votes
1answer
101 views

Positive cone of operators: if two selfadjoints $a$, $b$ obey $a^2 + b^2 =1$, must they commute?

The question relates to the structure of the positive cone of operators, in C*-algebra. If $a$ and $b$ are selfadjoints such that $a^2 + b^2 = 1$ can one prove $a$ and $b$ commute? What one ...
2
votes
3answers
3k views

Spring-mass system with two springs and three masses [closed]

I'm trying to solve a system of springs and masses that is confusing me. First, the balls are all lined up linearly. Secondly, the ball in the middle has a smaller mass $m$ while the first and last ...
2
votes
1answer
33 views

Find force centroid from forces and moments [closed]

A force $\vec F = (F_x,F_y,F_z)$ is applied to a point that its position vector is $\vec r = (x,y,z)$ generating a moment $\vec M = (M_x,M_y,M_z)$. Find position vector of the point ($\vec r$) if ...