Questions tagged [linear-algebra]

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

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Raising and Lowering Operators of a Hamiltonian

Lets say that I have a Hermitian Hamiltonian $H$ with a non-Hermitian raising operator operator $A$ which satisfies \begin{equation} [H,A] = \Omega A, \quad \Omega \in \ \mathbb{R}_{>0}. \end{...
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Freedom in choosing elements/entries of an eigenvector

I want to understand why there is freedom in choosing entries of an eigenvectors on some instances. I will take up a particular Hamiltonian to explain this. $$H=H_0 \left[ {\begin{array}{ccc} 1 &...
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What does it mean to expand a function in its basis?

I was reviewing my quantum mechanics notes, and I was confused on what this expression meant: $$ |{\psi}\rangle = \sum_{i}|{\omega_i}\rangle\langle{\omega_i}|{\psi}\rangle $$ I understand that it's ...
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5answers
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Why does Quantum Mechanics use Linear Algebra? [closed]

I am currently doing Linear Algebra in hopes of one day tackling QM, and I need some motivation now to continue in this pursuit. The University I attend set this as a pre-requisite for QM. Now I have ...
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Doubt in a solved example from Quantum Mechanics: Concepts and Applications by Nouredine Zettili [closed]

Question 3.7 b) from Quantum Mechanics: Concepts and Applications by Nouredine Zettili, on page no. 188 (solved examples) - I understand all the solutions mentioned therein but can't figure out why ...
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Strange use of the mean value into the definition of operators

I am currently working on quantum mechanical wave packets and minimum uncertainty states, to be specific I am trying to prove that the minimum uncertainty state is represented by a gaussian. Anyway, I ...
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45 views

Unitary transformation of Dirac equation

Dirac equation is given by $$(i\gamma^\mu\partial_\mu-m)\psi=0.$$ The matrices $\gamma^\mu$ satisfy the relation $$\{\gamma^\mu,\gamma^\nu\}=\gamma^\mu\gamma^n+\gamma^\nu\gamma^\mu=2g^{\mu\nu},$$ ...
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5answers
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How to pick out eigenvectors after solving for eigenvalues?

I'm currently doing a bit of quantum mechanics, and I can't figure out how to pick out eigenvectors. Let me explain through an example. An operator $A= \begin{bmatrix} 1 &0 &0 \\ 0&0 &...
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1answer
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Why don't the specificities of quantum mechanics (like the necessity of complex number) appear in classical mechanics?

It's well known that classical mechanics is a crude approximation of reality, and that it can be derived from quantum mechanics. But if this is so, why is it not a linear theory, like quantum ...
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Simplifying a Bra-Ket Expression

Consider the following relations $$H_0|\psi_a\rangle = E_a|\psi_a\rangle$$ $$H_0|\psi_b\rangle = E_b|\psi_b\rangle$$ I am struggling then to understand why the following identity holds (its probably ...
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The Functional Determinants in Peskin and Schroeder (Eq.9.77)

I'm working on the Eq.9.77 in Peskin (page 304): To demonstrate this, we need only apply standard identities from linear algebra. First notice that, if a matrix $B$ has eigenvalues $b_i ,$ we can ...
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Solving Many DOF System with Symmetry (Eigenvector issue)

I'm working on dealing with the simple harmonic oscillations of a benzene atom. We're meant to solve it with symmetry. I can solve it by going the longer way via the Euler-Lagrange method and finding ...
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How to find simultaneous eigenstates for multiple non-commuting Hermitian operators?

Basic quantum mechanics told us that multiple commuting Hermitian operators have simultaneous eigenstates (as a complete basis for the Hilbert space). However, there are cases where the operator are ...
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1answer
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How the matrix representation of a Hamiltonian affects the eigenvalues?

Suppose we're given the following Hamiltonian: $$\hat{H}=\frac{\omega}{\hbar} \left(\hat{S}_+^2+\hat{S}_-^2\right)$$ Suppose also that we measure $\vec{S}^2$ and get $6\hbar^2$, i.e. reduced to the $s=...
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Quantum Computing: Preparation of the Bell state Notation [closed]

I was watching some lectures on qubits. They were talking about how to generate a Bell state. They described it as follows: Prepare state 00: $$\left |0 \right> \otimes \left |0 \right>$$ Apply ...
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1answer
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Understanding in perturbation of metric in tensor calculus

From the post demonstration of expand perturbations of metric, I understand the full demo below excepted for the last result, i.e : $$ \sqrt{-g}=\sqrt{-\operatorname{det} b}\left(1+\frac{1}{2} \...
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Defining an inner product over matrices and over vectors

In quantum mechanics, in Dirac notation an inner product is denoted as $\langle A|B\rangle$ and one fundamental postulate is given as follows: $\langle A|B\rangle = \langle B|A\rangle ^*$ If I were to ...
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Confusion in notation for completeness relation of simultaneous eigenkets

Given two compatible observables $A$ and $B$ with a common eigenbasis, the completeness relation is: $\newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|}$ $$ \sum_{i,j}\ket{a^i,b^j}\bra{...
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In quantum mechanics how the expression of average value of an observable is derived?

In Dirac's Principles of QM following is stated: $$ \langle x | A + B | x \rangle = \langle x | A | x \rangle + \langle x | B |x \rangle $$ but $$ \langle x | AB | x \rangle \ne \langle x | A | x \...
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Index Placement Conventions in Group Theory

In his book on group theory, Wu-Ki Tung seems to utilize a few peculiar conventions regarding the placement of indices on vector and matrix symbols. For instance, if $|x\rangle=\sum_{i=1}^n|e_i\rangle ...
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About the behavior of the position and momentum operators

Following my book I came to know the following expressions for the position and momentum operators ($\hat{x},\hat{p}$): \begin{align}&\langle x|\hat{x}|\psi\rangle=x\psi(x) \ \ \ \ \ &(1)\\[1....
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Normal Modes and Normal Frequencies - Matrix Notation

Can somebody share with me sources where I can study about this topic ? I want to study the natural modes of a system with multiple degrees of freedom (with lagrangian formalism) but in terms of ...
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1answer
70 views

QM Basis Transformation Through Unitary Operator

I've a short question about basis transformations in QM. Suppose I have two bases $\{|{\phi_n}\rangle\}$ and $\{|{\phi_n'}\rangle\}$. For brevity, we can make them orthonormal. I know that any state ...
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In quantum mechanics why do we need the operators to be linear?

In quantum mechanics why do we need the operators to be linear or at best, antilinear?
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Max-relative entropy between a state and its marginals

Background The quantum relative entropy is defined for any quantum states $\rho, \sigma$ as $$D(\rho\|\sigma) = tr(\rho\log\rho) - tr(\rho\log\sigma)$$ For arbitrary choice of $\rho,\sigma$, the ...
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Why should bras be thought of as linear functionals?

Quoting from Ballentine's textbook on Quantum Mechanics: There are situations in which it is important to remember that the primary definition of the bra vector is as a linear functional on the space ...
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Ket notation in alternate forms

I have been told that I can describe a system by its wave number states: $$|k_1\rangle|k_2\rangle,$$ and that the following is true: $$|k_1\rangle|k_2\rangle=|k_1+k_2\rangle|k_1-k_2\rangle,$$ I am ...
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Why are bras and kets defined to be Hermitian adjoints of each other?

In the text The Physics of Quantum Mechanics by Binney and Skinner, the authors define $| \psi \rangle^\dagger \equiv \langle \psi |$ and $\langle \psi |^\dagger \equiv | \psi \rangle$. How can one ...
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1answer
41 views

Decoupling theory by diagonalising the Hamiltonian

I have a Hamiltonian of the form $H = 2k(\alpha \alpha^* -\beta \beta^*) -2\lambda (\alpha\beta^* + \beta \alpha^* )$ and I'd like to decouple the $\alpha$'s and $\beta$'s if possible. I know I need ...
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How to construct eigenvectors of angular momentum and directional parity (mirror)?

In the context of diatomic molecules I have seen it has come up that an eigenvalue $M_L\hbar$ of $L_z$ is doubly degenerate with respect to reflection about a plane containing the $z$ axis, whose ...
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For two commuting operators $A$ and $B$ and in absence of any degeneracy, is every eigenstate of $A$ is also an eigenstate of $B$ and vice-versa?

Two commuting operators $\hat{A}$ and $\hat{B}$ always share a complete set of common eigenfunctions. However, in the presence of degeneracy, every eigenstate of $\hat{A}$ need not an eigenstate of $\...
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Examples of antiunitary operator other than time reversal operator

It is well-known that time reversal operation is implemented as an anti-unitary operator. I wonder what are some other examples of anti-unitary operators that appear in the context of quantum ...
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Visualization of $n$-dimensional Hilbert spaces

I am learning quantum physics, and came across $n$-dimensional Hilbert spaces, is there any way one can visualize a $n$-dimensional space and the n components of the vectors existing in that space? P....
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Matrix representation of transformation defined on creation operators - Bogoliubov transform

In this paper by Alba Cervera-Lierta the following transformation is performed: $$ a_k=u_kc_k+iv_kc^\dagger_{-k} \\ a^\dagger_k=u_kc^\dagger_k-iv_kc_{-k} $$ where $c_k$ and $a_k$ represent ...
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3answers
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Representing Quantum Gates in Tensor Product Space

I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example: Both qubits, $q_0$ and $q_1$ start in the ground ...
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General Relativity: change of coordinates in tangent space

For starters, in the context of the tangent space of a manifold in GR, we can derive that: $$g'_{\mu \nu}=\frac{\partial x^\rho}{\partial x'^\mu}\frac{\partial x^\sigma}{\partial x'^\nu}g_{\rho \sigma}...
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Can the Lorentz Transformation Matrix be Derived using the Change of Basis Formula?

So I watched a MinutePhysics video explaining the idea of Lorentz transforms geometrically and the way he described it sounded very similar to how 3Blue1Brown explained the idea of change of basis. I ...
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Critique on tensor notation

I am studying tensor algebra for an introductory course on General Relativity and I have stumbled upon an ambiguity in tensor notation that I truly dislike. But I am not sure if I am understanding the ...
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Physical significance of this operator in quantum mechanics

I have stumbled across this question and cannot seem to find an answer to it. Consider an operator $\textbf{A}$ with eigenkets $|{a_{i}\rangle}$ and distinct eigenvalues $a_{i}$ . One can check that ...
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The physical meaning of ${\rm Tr}()$ in quantum mechanics [duplicate]

I know what trace is in matrix algebra, but does it have any intuitive physical significance when acted on a quantum mechanical operator? What aspect of an operator $A$ does ${\rm Tr}(A)$ capture? Or ...
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Significance of Diagonalization in Degenerate perturbation Theory

I am studying Degenerate perturbation Theory from Quantum Mechanics by Zettili and i'm trying to understand the significance of diagonalizing the perturbed Hamiltonian. He uses the stark effect on the ...
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1answer
76 views

When does a Hermitian operator have real matrix elements?

I will use braket-notation, but my question is not specific to quantum mechanics. Instead, I would be interested in a general answer for operators in some Hilbert space. Let $H$ be a Hermitian ...
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Mutually un-biased bases [duplicate]

I was reading some lectures on quantum information theory and it came with the idea of " mutually unbiased bases" that I never heard about. But the definition is very simple: two basis $ B =...
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1answer
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Constructing the supertraceless portion of a connection over a supermanifold

Consider a tensor, $T$ of rank $(r,s)$ over a supermanifold, $M$ and take the supertrace over its indices $p$ and $q$ (DeWitt, p. 77, eq. 2.4.33): $$(-1)^{a_q(1+a_{p+1}+...+a_{q-1})}T^{a_1...a_{p-1}...
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Working out the properties of the Special Unitary Lie algebra $su(n)$ [closed]

Given the special unitary group, we can define $$ M = -iX = \begin{bmatrix} b_{11} & -ia_{12}+b_{12} & ... & -a_{1n} + b_{1n} \\ ia_{12}+b_{12} & b_{22} & ... & ... \\ ... &...
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Problem with matrices in Dirac notation

Let $|q\rangle$ be the eigenvectors of the position operator, let $|\psi\rangle$ be a state and let $\hat{p}$ be the momentum operator. In my book it's stated that i can interprete the quantity: $$\...
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1answer
39 views

Verifying the Gaussian Transformation of $exp\left\{\frac{1}{2}\sum_{i,j} S_i J_{ij} S_j\right\}$ from “Advanced Mean Field Methods”

The book Advanced Mean Field Methods mentions the following equation as a result of a "simple gaussian transformation". $$ exp\left\{\frac{1}{2}\cdot\textbf{s}^T \cdot \textbf{J} \cdot\textbf{s}\...
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Phase portrait with python using Jordan and eigenvalues

I want to make a phase portrait on python. I have 3 Ode's. I know the Jordan canonical form. And i know the eigenvalues. So i know when it is stable. Does anybody know how I can make a portrait using ...
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46 views

Measuring a state in a basis other than eigenbasis

Suppose I have a state expressed in its eigenbasis as follows. $\rho = \sum_i\lambda_i\vert i\rangle\langle i\vert$. It is now measured in some other basis $\{\vert x\rangle\}$ that is distinct from ...
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How do we construct the matrix representation for three Grassmann numbers? [duplicate]

I want to know how we find or construct the matrix representations for Grassmann numbers. For example, we can see from https://en.wikipedia.org/wiki/Grassmann_number: Grassmann numbers can always be ...

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