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

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29 views

Probabilities with a qubit

A two-state quantum system has orthonormal energy eigenstates ψ1 and ψ2, with energy eigenvalues E1 and E2 = E1 + ∆E (∆E > 0). These energy eigenstates form a complete set of wavefunctions for the ...
3
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2answers
77 views

Importance of Kronecker product in quantum computation

To get product state of two states $|\phi \rangle$ and $|\psi \rangle$, we use Kronecker product $|\phi \rangle \otimes |\psi \rangle$. Instead of Kronecker product $\otimes$, can we use Cartesian ...
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45 views

Principal axes: Are they unique?

Mathematically, principal axes are the eigenvectors of the Inertia Tensor. Physically, They are the axes that satisfy , Angular momentum || Angular Velocity || Principal Axis. Inertia tensor depends ...
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1answer
40 views

What makes a system linear?

Lately I have been interested in Image Processing, and I started by following this course: https://class.coursera.org/digital-001, which is quite awesome in my opinion. But in weeks 2, Linear ...
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15 views

Free groups and their appierance/emergence/applications in physics

While trying to master my knowledge on group theory in physics from more formal point of view, i noticed an entity called free group ( http://en.wikipedia.org/wiki/Free_group ). I'm aware that it is ...
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0answers
36 views

Addition in linear vector spaces [migrated]

In the definition of linear vector spaces, one of the axioms is that the addition must be commutative and associative. The addition of scalars and matrices are both commutative and associate. Can ...
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1answer
80 views

How do I derive the eigenvalues of the 1D Heisenberg model? (Bethe Ansatz)

I've been trying to work through Introduction to the Bethe Ansatz I (by Michael Karbach and Gerhard Muller) in spare time and I am having trouble deriving the eigenvalues given in equation (5) for ...
3
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2answers
107 views

The definition of transpose of Lorentz transformation (as a mixed tensor)

In the appendix of the textbook of Group Theory in Physics by Wu-Ki Tung, the transpose of a matrix is defined as the following, Eq.(I.3-1) $${{A^T}_i}^j~=~{A^j}_i.$$ This is extremely confusing for ...
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9 views

Dropping vertices of an overdetermined statics system graph

In statics, the problem of determining the tensions of K cables that connect a structure made of N points and keep all points in static equilibrium implies $ND$ systems of equations, where $D$ is the ...
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0answers
46 views

Learning how to use Levi-Civita symbol

I've recently started my second course in Quantum Theory and am now often required to prove more complex commutation relations. I'm aware that the Levi-Civita symbol often makes this sort of thing a ...
4
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2answers
839 views

These two operators commute…but their eigenvectors aren't all the same. Why?

The Hamiltonian $$H = \left[ \begin{array}{cccc} a & 0 & 0 & -b \\ 0 & 0 & -b & 0\\ 0 & -b & 0 & 0\\ -b & 0 & 0 & -a \end{array} \right] $$ commutes ...
1
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1answer
59 views

Forces exerted on legs of a table

This is actually for a computer science homework problem, but I haven't studied any physics in five years so I'm having a bit of trouble. I'm given a table of three legs with their coordinates and I ...
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0answers
81 views

Change of Basis For Pauli Matrix From Z Diagonal to X Diagonal Basis

I want to find a matrix such that it takes a spin z ket in the z basis, $$ \lvert S_z + \rangle_z $$ and operates on it, giving me a spin z ket in the x basis, $$ U \lvert S_z + \rangle_z = ...
2
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0answers
106 views

What is the physical meaning of complex eigenvalues?

I understand the mathematical origin of complex eigenvalues, and that complex eigenvalues come in pairs. But what is the meaning of the imaginary part? In particular I refer to an acoustic problem ...
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1answer
71 views

What does “projection of a vector” really mean?

Let $\vec{a}$ & $\vec{b}$ be two non-collinear, non-zero co-initial vectors having angle $\theta$ between them. The projection of $\vec{b}$ on $\vec{a}$ is given by the dot product of $\vec{b}$ ...
3
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1answer
87 views

Kraus operator rank

All quantum operations $\mathcal{E}$ on a system of Hilbert space dimension $\mathcal{d}$ can be generated by an operator-sum representation containing at most $\mathcal{d^2}$ elements. Extending ...
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1answer
31 views

How long does it take for $25~\text{mC}$ to pass a point if the current is $12.5~\text{mA}$? [closed]

How long does it take for $25~\text{mC}$ to pass a point if the current is $12.5~\text{mA}$? I = 12.5mA Q = 25mC t = ? The formula for this question was: I = Q/t, where I is amps, Q is Coulombs and ...
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0answers
45 views

When is the product of a hermitian unitary and another unitary hermitian?

I have a Hermitian unitary $\hat{H}$ and I want to know, if $\hat{U}$ is some other unitary, when is $\hat{H}\hat{U}$ a Hermitian unitary? Specifically, what are the conditions on $\hat{U}$? I know ...
1
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1answer
87 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 ...
3
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1answer
87 views

Deriving photon propagator

In Peskin & Schroeder's book on page 297 in deriving the photon propagator the authors say that $$\left(-k^2g_{\mu\nu}+(1-\frac{1}{\xi})k_\mu k_\nu\right)D^{\nu\rho}_F(k)=i\delta^\rho_\mu ...
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0answers
40 views

Examples of application of detour matrices in physics?

Are there any good examples of application of detour matrices in physics?
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2answers
120 views

Commuting operators and Direct product spaces

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? When can an eigenket $|\lambda$1$\lambda$2$>$ ...
4
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1answer
65 views

What information does the trace of a matrix give?

I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant ...
2
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1answer
47 views

Why eigenvector points to principal stress plane?

I can represent a tensor by a matrix. Suppose we are talking about a 2nd order tensor, and the matrix is therefore 3x3. If I find one eigenvector of that matrix; that vector represents normal vector ...
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0answers
18 views

How is it shown that the composition of two real operators is generally not real? [duplicate]

Dirac on page 28 of his QM book writes: Thus the conjugate complex of the product of two linear operators equals the product of the conjugate complexes of the factors in the reverse order. As ...
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23 views

Calculate static magnetic field in a volume of air with known sample points

So I know some calculus and I know some linear algebra, but do not really master electromagnetism (did a course ten years ago). There is a problem someone else has solved in a matlab script for me ...
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1answer
112 views

A question on partial trace and density matrix computation

Consider a Pure state of a two dimensional system $|\psi\rangle={1\over\sqrt{2}}(|e_1\rangle|e_1\rangle+|e_2\rangle|e_2\rangle)$ where $\{|e_i\rangle\}$ is an orthonormal basis. Could any one just ...
4
votes
2answers
169 views

Dimension of separable state

Please can you help me to understand how the dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$? This is the relevant passage: So far, we have assumed implicitly that the ...
1
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2answers
91 views

$\exp(i\alpha\hat {\bf n}\cdot{\bf \sigma} )=\cos\alpha I+i(\hat {\bf n}\cdot{\bf \sigma})\sin\alpha$

Could anyone tell me $\hat {\bf n}\cdot{\bf \sigma}$ is defined in such way? In the book they have not defined what is $n_z,n_x,n_y$. It is from Quantum Computing: From Linear Algebra to Physical ...
3
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1answer
68 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 ...
2
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0answers
59 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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5answers
402 views

Physical applications of matrices and determinants

Other than notation devices, I don't see any direct application of matrices/determinants in physics. For example, they are just a different way to write a partial derivative and determinants find if ...
3
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3answers
252 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
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1answer
61 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 ...
1
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0answers
106 views

Link between Quantum and Classical Mechanics [duplicate]

In classical mechanics we have momentum as generator of translation by following definition: $$f(x+\delta x)=f(x)+[f(x),p]\delta x+....$$ I was wondering whether using this relation and commutation ...
2
votes
2answers
160 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 ...
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2answers
48 views

Straight line on 8-axis [closed]

How would a straight line appear in a 8-axis co-ordinate system. Will it continue to look like a straight line or will it appear like zig-zag line. We can see that straight line appears like a dot in ...
0
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1answer
90 views

Showing Dirac equation's Lorentz invariance and use of unitary matrix $U$

Dirac equation is $i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0 $ To show its Lorentz invariance, we convert spacetime into $x'$ and $t'$ from $x$ and $t$ and then $( iU^\dagger \gamma^\mu ...
2
votes
2answers
96 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
votes
1answer
50 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 $\omega \in \Omega^{r}(M)$ ...
3
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0answers
51 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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2answers
87 views

Idempotent Operators

If $P$ is an idempotent operator, $P^2 = P$ and we have a vector $|\psi\rangle$, $P\neq1$, and the relation $$PL |\psi\rangle = L|\psi\rangle,$$ what conclusions can we draw about $L$, which is a ...
2
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1answer
132 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 ...
1
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2answers
40 views

Terminologies for moment of inertia

Perhaps someone can suggest the right terms for the following mathematical objects related to moment of inertia? A inertia tensor $I$. $$I \equiv \begin{bmatrix} I_{1,1} & I_{1,2} & I_{1,3} ...
2
votes
1answer
51 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 ...
5
votes
1answer
241 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). ...
2
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3answers
102 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 ...
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3answers
122 views

Eigenvectors of Pauli $\sigma_x$ matrix

I solve for eigenvectors of Pauli X matrix. I get {1,-1} and {1, 1}. In exercises I solve, there is an answer $\frac{1}{\sqrt{2}} $ * {1,1} and $\frac{1}{\sqrt{2}} $ * {1,-1}. Where does that ...
4
votes
1answer
162 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 ...
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1answer
80 views

Manipulation with braket notation

I am still getting used to the braket notation. Is this manipulation correct? $$ \frac{-\hbar^2}{2m}\int_{-\infty}^{\infty}\frac{\partial^2\phi_n^*}{\partial z^2} \phi\,dz = ...