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

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53 views

Energy levels in close-proximity of each other in time-independent degenerate perturbation theory

I've applied second order time-independent degenerate perturbation theory corrections to the energy with the method presented in Modern Quantum Mechanics by J.J. Sakurai. I shortly summaries this ...
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0answers
40 views

Why is the matrix representation in the same basis not same for a density operator?

I have a $\rho : V \to V$ density operator of a $n$ dimensional space $V$ and $\{i\}=\{i_1,i_2..i_n\}$ is an orthonormal basis of this space. The density operator is defined as $$\rho=\sum ...
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2answers
88 views

Relation between Vector space $V$ and its dual $V^{*}$ [closed]

I asked the same question in Math.SE, but I was suggested to ask it here as well. I am studying relativity, and as you know the theory extensively uses the notion of covariant and contravariant ...
4
votes
1answer
155 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 ...
11
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1answer
824 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 ...
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2answers
83 views

What's the degree of freedom of this kind of matrix?

We first have a unitary matrix $$\{a_{ij}\}\quad(n\times n)$$ I know how to calculate its degree of freedom, which is $n^2$ if we consider a real variable as one degree of freedom. Now we have a ...
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1answer
35 views

The Simon's Algorithm, confusing equation

I'm approaching the Simon's Algorithm and have troubles with understanding a logic in an introduction. Above the eq. 6.5.4 they introduce that set S which has 2 elements. As far as I understand, ...
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2answers
151 views

Is there a simple proof that Kirchhoff's circuit laws always provide an exactly complete set of equations?

Suppose I have a complicated electric circuit which is composed exclusively of resistors and voltage and current sources, wired up together in a complicated way. The standard way to solve the circuit ...
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1answer
37 views

Dealing with tensor products in an exponent

I am looking at the following problem and I am struggling to follow the steps involved. Consider the non-interacting Hamiltonian $$H_{AB}=H_A\otimes I_B+I_A\otimes H_B$$ So I'm trying to prove that ...
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2answers
83 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 ...
2
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1answer
59 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 ...
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2answers
96 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 ...
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192 views

Linear response theory for Gross Pitaevskii equation

I am trying to linearize the following GP eq: \begin{equation} i\partial_{t}\psi(r,t)=\left[-\frac{\nabla^{2}}{2m}+g\left|\psi(r,t)\right|^{2}+V_{d}(r)\right]\psi(r,t) \end{equation} The ansatz for ...
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3answers
102 views

Can a component of vector be greater than the vector itself?

...we have at our disposal an infinite variety of ways of resolving a given force into components. . . . The fact that any component may happen to be larger than the vector itself doesn't ...
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1answer
59 views

Finding all decompositions of mixed states

Some quantities, such as the entanglement of formation, are defined using a quantity that is minimized over all possible decompositions of a mixed state. A closed form can be found for this in some ...
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1answer
137 views

Second Rank Tensors [duplicate]

I'm a little confused, for the twentieth time, on what tensors are. I thought they were a generalization of matrices-but then they have special transformation rules. I'm looking for a concise ...
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2answers
97 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 ...
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21 views

distance random matrix

In some physics problems it is sometimes useful to define a distance matrix for a system of particles with positions denoted by $x_1$, ..., $x_N$. Then the matrix would be given by ...
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2answers
77 views

The abstract space of metrics in GR

I know this is a general (har har) question, but has any work been done understanding the mathematical space the allowed metrics in GR form? (I guess it'd be called a tensor space???)
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47 views

What is the relationship between the formal definition of a tensor and the frequently discussed notion of a “higher order matrix”?

I've been doing some self study on the principles of tensors & manifolds in preparation for a first course in general relativity. I tend to learn better when presented with the full mathematical ...
6
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2answers
282 views

Infinite dimensional vector spaces vs. the dual space

I just happened across this over on Math Overflow. It references the following theorem from linear algebra: A vector space has the same dimension as its dual if and only if it is finite ...
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1answer
45 views

Decompose a massive four vector in to two massless ones

I'm trying to decompose a massive four-vector, $p_1$ with $p_1^2=m^2\neq0$, in to two massless ones, $k_i$ with $k_i^2=0$. But I'm having trouble find basis vectors $k_i$ such that I can always ...
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0answers
23 views

Free groups and their appearance, emergence and applications in physics

While trying to develop my knowledge of group theory in physics from a more formal point of view, I noticed an entity called a free group. I'm aware that it is extremely important in pure mathematical ...
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3answers
3k views

Perturbation theory with degeneracy even after 1st order

Most textbooks on basic quantum mechanics tell you that when your initial Hamiltonian $H_0$ has degenerate states, then before you can do (time independent) perturbation theory with a perturbation ...
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1answer
44 views

Hamiltonian acting on sum operator

I am following a derivation in a book. It is implementing a state $|{\psi}\rangle$ into the eigenvalue equation $\hat{H}|{\psi}\rangle=E|{\psi}\rangle$. The $|{\psi}\rangle$ term contains a ...
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2answers
179 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
37 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 ...
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2answers
164 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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203 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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69 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
42 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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1answer
113 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 ...
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1answer
102 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 ...
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2answers
220 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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15 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
66 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 ...
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2answers
896 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 ...
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4answers
3k 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 ...
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201 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 = ...
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1answer
107 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 $\vec{b}$ ...
4
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2answers
179 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 ...
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1answer
33 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
47 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 ...
2
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2answers
289 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 ...
4
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1answer
156 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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46 views
4
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1answer
141 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 ...
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2answers
157 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$2$>$ ...
29
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4answers
8k 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 ...
3
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1answer
76 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 ...