Questions tagged [linear-algebra]
To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.
927
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Help with conmutator identity with angular momentum and vector [closed]
I need to prove this identity:
$ [ \textbf{J}^2, \textbf{J}\times \textbf{V}] = 2i\hbar( \textbf{J}^2 \textbf{V} - ( \textbf{J} \cdot \textbf{V}) \textbf{J}) $
Where $ \textbf{J}$ is an angular ...
- 11
3
votes
1
answer
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Similarity transformations in QFT
I am trying to understand the gaps in my knowledge that prevents me from completely understanding quantum field theory. Sometimes I ask pretty basic questions, but please excuse me if I make a blunder....
- 119
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1
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Density Matrices in Quantum Mechanics
I have a question about the physical meanings of various matrices expressed in Dirac bra-kets.
I take it that $\frac{1}{2}|A\rangle\langle A| + \frac{1}{2}|B\rangle\langle B|$ can be interpreted as a ...
- 884
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Choi-Jamialkoski Theorem in Phase damping channel [closed]
I am trying to replicate the solution I have to this problem provided by the instructor in the class where I am trying to use the Choi-Jamialkoski theorem to prove that Phase damping channel is ...
- 113
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2
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$F$ transforms like a tensor $\Rightarrow B$ transforms like a pseudo vector
Notation: In the following $*$ is the hodge operator from $\Lambda^1(\mathbb R^{1\times 3})\cong \mathbb R^{1\times 3}$ to $\Lambda^2(\mathbb R^{1\times 3})\cong A\subset\mathbb R^{3\times 3}$ (or its ...
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3
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Diagonalising the Hamilton operator, why does this magic work?
Let the Hamilton operator $H= \omega_1 a_1^\dagger a_1 + \omega_2 a_2^\dagger a_2 + \frac{J}{2} (a_1^\dagger a_2 + a_1 a_2^\dagger)$ be given, of course $a_j$ and $a_j^\dagger$ are the creation and ...
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Simultaneous Measurement of Anti-commutative Operators
In quantum mechanics, two variables $A$ $B$ can be observed simultaneously if they commute with each other, i.e. $[A, B]=0$. From what I learned from courses, this is established by two facts:
1: ...
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2
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0
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72
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What math do I need for physics? [closed]
I'm in 9th grade. I've gone through linear algebra, multivariable calculus, differential equations, and statistics. I'm attempting to get better at physics and maybe try out for the International ...
- 25
1
vote
1
answer
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Outer Product Other form [closed]
The outer product of a ket $|\psi\rangle$ with a bra $\langle\phi|$ according to the textbook Quantum Computing Explained by D. McMahon, behaves likes an operator. He illustrates this by applying an ...
- 31
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Parametrization of Unitary matrices
Is there any book to follow along with "The Unitary and Rotation Groups" by F.D. Murnaghan" for the first two chapters concerning the parametrization of general $n \times n$ unitary ...
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1
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Can someone redirect me to relevant mathematics? [closed]
These are two paragraphs from Chapter 3 of Principle of Quantum Mechanics by P.A.M. Dirac. I need to know what the relevant mathematics its referring specifically, I have some idea not proper enough ...
1
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1
answer
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Finding common eigenvectors for two commuting hermitian matrices [closed]
Let $A = \begin{bmatrix}
1 &0 &0 \\
0& 0& 0\\
0&0 &1
\end{bmatrix}$ and $B = \begin{bmatrix}
0 &0 &1 \\
0& 1& 0\\
1&0 &0
\end{bmatrix}$ ...
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0
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2
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Change of basis, matrix and operators
If $U$ is an unitary operator written as the bra ket of two complete basis vectors i.e
$U=\sum_{k}\left|b^{(k)}\right\rangle\left\langle a^{(k)}\right|$
Then
$U^\dagger=\sum_{k}\left|a^{(k)}\right\...
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1
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Kitaev Chain - Obtaining a real-orthogonal matrix that block-diagonlises the Kitaev Chain
I encounter a subtle problem regarding the Kitaev Chain. In Kitaev framework, he tried to express the Hamiltonian into real-orthogonal basis. Suppose the Majorana system is described by
$$
H = \frac{i}...
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1
answer
76
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Relation between diagonal and off-diagonal entries of Hermitian Operator
I am started doing a project in Quantum Chemistry and stumbled upon a problem which I can not seem to find the answer to.
As the title suggests, I am looking for a relation between the diagonal and ...
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1
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2
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79
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Understanding dot product in quantum mechanics [closed]
Let's say we have a two-state-system with state $\vert 1\rangle$ and state $\vert 2\rangle$. From my understanding one can assume the base vectors of this system to be $\vert1\rangle \mapsto (1,0)^\...
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1
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How to solve for the scattering solution of following Schrodinger equation?
Suppose you have non-relativistic fermions scattering off a delta function potential.
It is an easy job to solve $H=-\partial_x^2+\epsilon \delta(x)$ by starting with an eigenfunction of the form $\...
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1
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1
answer
30
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What's the contraction for non-adjacent fields?
In section 8.2 of Coleman's QFT lectures, he introduces the definition of contraction of two fields,
where $T$ denotes time ordering and the colons normal ordering.
Then he proceeds to contraction in ...
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2
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Are the linear Lie groups matrices, tensors, or both?
In some ways, this is a question about notation.
In my experience, I have only seen the classical Lie groups — such as $\operatorname{GL}(n,\mathbb{R})$, $\operatorname{SL}(n,\mathbb{R})$, $\...
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3
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How to avoid paradoxes about time-ordering operation?
(Original title: is time-odering operator a linear operator?)
I'm confused with two formulas, one of which is
$$
\mathcal{T} \exp \left [-\frac{\mathrm{i}}{\hbar} \int_{t_0}^t \mathrm{d} t' \hat{H}_I(...
- 469
9
votes
1
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Are the instantaneous eigenstates of a time-dependent hamiltonian continuous?
I am trying to understand the adiabatic theorem. I can follow the proofs that are given in Wikipedia (https://en.wikipedia.org/wiki/Adiabatic_theorem) but there seems to be a hidden assumption.
For a ...
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1
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78
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Find projection operators degenerate energy eigensubspaces [closed]
A given system has Hamiltonian $H=\sum_{i=0}^{n}\sigma^{(i)}_{z}$, where $\sigma^{(i)}_{z}$ are the usual Pauli matrices. Now I want to find the corresponding $n+1$ projection operators corresponding ...
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1
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1
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172
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Checking if a state is an eigenstate of $L^2$ and $L_z$ without performing calculations
If I have a given state $\psi$ which is a linear combination of spherical harmonics, and I am asked if its an eigenstate of $L^2$ and $L_z$, is there a way to do it without using the eigenvalue ...
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0
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47
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Issue when trying to represent an operator as a matrix
We say that a ket $|V\rangle$ can be expresed in an orthonormal basis $(|e_1\rangle,|e_2\rangle,...|e_n\rangle)$ as :
$$|V\rangle = \sum_i^n v_i |e_i\rangle$$
where $v_i = \langle i|V\rangle $
for a ...
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1
vote
3
answers
311
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Confusion about how adjoint matrices operate on state vectors
My understanding is that for an inner product in state-space, since we want the value to always be a real number we say that $$\langle\psi|\phi\rangle= {\langle\phi|\psi\rangle}^* $$
where * denotes ...
- 370
1
vote
1
answer
93
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The abstract state of a particle
I recently started learning about quantum physics. In the book, Quantum physics by H.C. Verma, the author explains that there are many ways to represent the state of a particle. The wave function $\...
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1
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56
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How should I interpret eigenvectors in second quantization?
a) I would like to ask, if knowledge about eigenvectors in second quantization is important and what do they mean? Let's just say, I create Fock space [(NumberOfSites)x(Permutations) matrix], then I ...
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A linear algebra exercise from Griffiths "Introduction to quantum mechanics" [closed]
(Edited so that it obeys the rules of homework questions) I am stuck on this linear algebra problem from Griffiths's "Introduction to quantum mechanics". Can somebody give me some guidance? (...
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1
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A relation for adjoint representation of $U(N)$ acting on product of matrices and $SU(2)$ generators
Is the following relation true, and if so, what is the property that makes it so?
\begin{align}
\sum_{i=1}^3\mathrm{tr}\left([U^{-1}L_iU]\phi[U^{-1}L_iU]\phi\right) \stackrel{!}{=} \sum_{i=1}^3\...
0
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1
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85
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A problem with Griffiths "Introduction to Quantum Mechanics" linear algebra
I have a problem at understanding the way linear transformations are used in Griffiths Introduction to Quantum Mechanics. My knowledge about linear algebra is basic. He is making a reference to the ...
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0
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51
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Spherical harmonics How do they span eigenspaces?
When trying to find common eigenstates of $L^2$ and $L_z$, we find the eigenstate $Y_m^l (\theta, \phi)$
My question is, if $m_1$ and $\lambda_1 = l_1(l_1+1)$ both have multiplicity $3$, then there is ...
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0
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Question about the operators in quantum mechanics [duplicate]
I am confused about the operators in quantum mechanics and even the way they are used, their symbols, etc. Is there any book or anything that I can study so that I can fully understand them before ...
0
votes
1
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86
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Diagonalizing Operators Simultaneously [duplicate]
Suppose we have a Hamiltonian operator $\hat{H}$ and another operator $\hat{A}$ such that $[\hat{H},\hat{A}]=0$. Then, if the spectrum of $\hat{H}$ is non-degenerate, from my understanding the ...
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1
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What specifically about the torque vector is perpendicular? Is the torque vector like this only so that it works smoothly with linear algebra?
What specifically about the torque vector is perpendicular?
Is the torque vector like this only so that it works smoothly with linear algebra?
The only explanation I get usually is "because it's ...
0
votes
2
answers
70
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Is there a way to carry out the eigenstates and eigenvalues of annihilation operator using only its matrix form?
Knowing the matrix elements of annihilation operator, can I solve the eigenvalue problem without using operator method?
I got stuck when I try to compute its eigenvalue, because the eigenvalues of a ...
1
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2
answers
108
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Why is the Schrödinger Equation valid for the component functions (wave function) of state vectors?
I'm new to quantum mechanics and confused about the way the Schrödinger equation is used (more general eigenvalue equations of observables).
Let's take the time-independent Schrödinger equation (...
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0
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106
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Eigenbasis of Hamiltonian and momentum operators
I was taught that, if two Hermitian operators commute, they share the same eigenbasis. Since the Hamiltonian and momentum operators commute, am I right in concluding that they share the same basis of ...
0
votes
1
answer
61
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Matrix formulation of the momentum operator
For a quantum state $\Psi=c_{1}\psi_{1}+c_{2}\psi_{2}$ with momentum eigenstates $\psi_{1}$ and $\psi_{2}$, the action of the momentum operator $\hat{p}$ is given by
$$\hat{p}\Psi=p_{1}c_{1}\psi_{1}+...
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0
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Weird question: elements of eigenvector as kinetic and potential energies
Assume we have $N$ particles each having some potential and kinetic energies. Denote the sum of kinetic energy as $\sum_i T_i = T$ and the sum of potential energy as $\sum_i V_i= V$. This is a closed ...
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Proving the uncertainty relation for the quantum covariance matrix
In quantum optics papers we often encounter this version of the Heisenberg uncertainly relation (for an $n$-mode quantum system):
$\sigma + \iota \Omega \ge 0$
Where $\sigma$ is the covariance matrix $...
- 336
0
votes
3
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Position of a point on a rigid body
I can not wrap my head around the following doubt:
How can we express (or prove the fact that) the position of a material point on a rigid body as the sum of a "traslational component" and ...
- 356
3
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0
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111
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Eigenkets of a two-state hamiltonian
I have a question related to this other question: Eigenenergies and eigenkets given the Hamiltonian. In it, OP is given the following hamiltonian:
$$
H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\...
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votes
2
answers
119
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Why are Eigenvectors of a 1D quantum ising hamiltonian real
I was modelling the 1D transverse quantum Ising model and made a Kronecker product loop to find the Hamiltonian of the system, for a given magnetic field configuration. Now, my question is that when I ...
2
votes
0
answers
63
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Measurement operator in a Bell experiment
I'm trying to figure out why a Bell experiment gives rise to the payoff (measurement) operator used in this paper on quantum game theory.
Two players are each in control of one half of an entangled ...
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1
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103
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Special relativity book which describes concepts using linear algebra notions
It seems so every idea of special relativity can be formulated quite nicely in Linear algebra notions such as the inner product matrix and change of basis matrices. However, I can't find a single book ...
-4
votes
1
answer
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Whats the meaning of the 1 Ket? [closed]
I am talking this one: $|1\rangle$. If I have 2 orthonormal states $|1\rangle$ and $|2\rangle$ in the 2D Hilbert space, does that imply the vector $\vec{\psi_n}=(1,2)$, if I would like to solve the ...
- 1
2
votes
1
answer
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Equation for the simultaneous eigenfunction of three operators in spherical coordinates
If I'm considering the three operators $H,L^2,L_z$ with the condition $[H,L^2]=[H,L_z]=[L^2,L_z]=0$, I can find a complete set of simultaneous eigenfunctions. If I study this problem in spherical ...
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0
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How the eigenvalue problem was solved?
In Gasiorowicz 3rd edition Chapter 3, I've tried to solve this problem
I checked the solution's manual,
When I tried to integrate it, the answer I got is
$$
\psi(x)=Ce^{x^2/2\lambda}
$$
Can you ...
0
votes
1
answer
67
views
Factorization of the wavefunction in a central Hamiltonian problem
I am trying to understand the topic of the title. If I consider a central Hamiltonian, so an Hamiltonian of the form $H=\frac{p^2}{2m}+V(r)$ what are the logical steps that lead me to the known result?...
- 921
0
votes
3
answers
76
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Quantum mechanics, are simultaneous eigenstates to be intended always as a tensorial product of two eigenstates?
The question is the one of the title, let $\hat{O}_1$ and $\hat{O}_2$ two commuting operators: $[\hat{O}_1,\hat{O}_2]=0$, there is an orthonormal basis formed by their simultaneous eigenstates.
These ...
- 921