What is Operator: Definition and 1000 Discussions

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).
This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.

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  1. VVS2000

    I Understanding the Raising and Lowering Operator: A Scientific Analysis

  2. U

    I Orthogonality of Eigenvectors of Linear Operator and its Adjoint

    Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...
  3. V

    I Does associativity imply bijectivity in group operations?

    Quick question: do the group axioms imply that the group operator is bijective? More in general, does associativity imply bijectivity in general? I can think about a subgroup of S3 that only operates on 2 elements, but it is really isomorphic to S2. But is there some concept or term for a...
  4. karush

    MHB V.02.Binary operator by a*b=a+8b

    Define the binary operator * by: $a*b=a+8b$ Find each of the following (the only thing I knew to do here was plug in) [a.] $3*5\quad =3+8(5)=3+40=43$ [b.] $7*7\quad =7+8(7)=7+56=83$ [c.] $5*3\quad =5+8(3)=5+24=29$ [d.] $x*z\quad =x+8z$
  5. K

    Possible Results and Probabilities of a Measurement of Operator Q

    I first Normalise the wavefunction: $$ \Psi_N = A*\Psi, \textrm{ where } A = (\frac{1}{\sum {|a_n^{'}|^{2}}})^{1/2} $$ $$ \Psi_N = \frac{2}{7}\phi_1^Q+\frac{3}{7}\phi_2^Q+\frac{6}{7}\phi_3^Q $$ The Eigenstate Equation is: $$\hat{Q}\phi_n=q_n\phi_n$$ The eigenvalues are the set of possible...
  6. P

    I Unitary operator and eigenvalue

    Hello, I recently saw ##U|v\rangle= e^{ia}|v\rangle, \, a \in \mathbb{R}## and am wondering how to come up with this or how to show this. My first thought is based on the definition of unitary operators (##UU^\dagger = I##), I would show it something like this: ##(U|v\rangle)^\dagger =...
  7. Safinaz

    B Does the Laplace operator equal the Del operator squared?

    Hello , The Laplace operator equals ## \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} ## so does it equal as well nable or Del operator squared ## \bigtriangledown^2## ? where ## \bigtriangledown =\frac{\partial}{\partial...
  8. Hamiltonian

    I Using the Schrodinger eqn in finding the momentum operator

    I have read that the Schrodinger equation has no formal derivation we are simply applying the Hamiltonian operator on the wave function $$\hat H = i\hbar \frac{\partial}{\partial t} = \hat T + \hat V$$ here we substitute $$\hat T = \frac{\hat p^2}{2m}$$ where $$\hat p = -i \hbar...
  9. ubergewehr273

    Finding unitary operator associated with a given Hamiltonian

    Now from the relevant equations, $$U(t) = \exp(-i \omega \sigma_1 t)$$ which is easy to compute provided the Hamiltonian is diagonalized. Writing ##\sigma_1## in its eigenbasis, we get $$\sigma_1 = \begin{pmatrix} 1 & 0\\ 0 & -1\\ \end{pmatrix} $$ and hence the unitary ##U(t)## becomes...
  10. A

    I Problem involving a sequential Stern-Gerlach experiment

    An electron beam with the spin state ## |\psi\rangle = \frac{1}{\sqrt{3}}|+\rangle+\sqrt{\frac{2}{3}}|-\rangle##, where ##\{|+\rangle,|-\rangle\}## is the eigenstates of ##\hat S_z##, passes through a Stern-Gerlach device with the magnetic field oriented in the ##Z## axis. Afterwards, it goes...
  11. Haorong Wu

    I Derivatives for a density operator

    Hi. Suppose I have a state ##\left | \psi (0)\right >=\sum_m C_m \left | m\right >## evolving as $$\left | \psi (0+dz)\right>=\left | \psi (0)\right >+dz \sum_iD_i\left | i\right >=\sum_m C_m \left | m\right >+dz \sum_iD_i\left | i\right >=\sum_m( C_m+dz D_m)\left |m\right >.$$ Then the density...
  12. M

    I Parity Operator in 2D: Understanding Transformation & Spin

    Hello! What is the 2D (acting in spin space) representation of the parity operator. In principle we can make it a diagonal matrix with the right transformation and given that ##P^2=1## the matrix would be diag(1,1) or diag(1,-1). However spin shouldn't change under parity and using that it seems...
  13. D

    I Schrodinger Cat states in terms of Displacement Operator

    The coherent state can be written in terms of e^(αb†+α∗b)|0>. But how the even coherent state i.e. |α>+|-α> can be written in terms of displacement operator?
  14. D

    I Commutation of Beam splitter operator with Displacement operator

    I have a beam splitter operator (a†)b +(b†)a. Does it commute with exp(αâ†-α*â). Here a and ↠are creation and lowering operator
  15. Viona

    Operator acts on a ket and a bra using Dirac Notation

    Summary:: Operator acts on a ket and a bra using Dirac Notation Please see the attached equations and help, I Think I am confused about this
  16. stephen8686

    A Jaynes-Cummings Density Operator Evolution

    I am studying two level atoms interacting with fields in order to study Dicke Superradiance. From Loudon's book, the Optical Bloch Equations for a two level atom interacting with a field say (with rotating wave approx): $$\frac{d\rho_{22}}{dt}=- \frac{d\rho_{11}}{dt} = -\frac{1}{2}...
  17. Viona

    B The average value of S operator

    While reading in the book of Introduction to Quantum Mechanics by David Griffith in the section of Fine structure of Hydrogen: spin- orbit coupling, he said that the average value of S operator is considered to be the projection of S onto J. I could not understand why he assumed that. please...
  18. S

    I A strange definition for Hermitian operator

    In lecture notes at a university (I'd rather not say which university) the following definition for Hermitian is given: An operator is Hermitian if and only if it has real eigenvalues. I find it questionable because I thought that non-Hermitian operators can sometimes have real eigenvalues. We...
  19. S

    I Why is this 3D operator with degeneracies only giving me 2 eigenstates

    The matrix representation of a certain operator in a certain basis is $$\begin{bmatrix} 1 & 0 & 0 \\0 & 0 & -i \\ 0 & i & 0 \end{bmatrix} .$$ The eigenvalue problem leads to this equation $$0=det\begin{bmatrix} 1-\lambda & 0 & 0 \\0 & -\lambda & -i \\ 0 & i & -\lambda \end{bmatrix}...
  20. Q

    A How to systematically find the symmetry operator given a Hamiltonian?

    For instance,how to systematically derive the equns 2.2 & 2.5 given a Hamiltonian on the article below?; arxiv.org/pdf/0904.2771.pdf .
  21. M

    I Solving Proca Lagrangian w/ Extra Operator: Find Laws of Motion

    The euler lagrange equation I am using is: $$\frac {\partial^\beta \partial L}{\partial(\partial^\beta A^\alpha) }= \frac {\partial L} {\partial A^\alpha}$$ Now the proca lagrangian i am using is $$L= -\frac {1}{16\pi} F_{\alpha\beta} F^{\alpha\beta} + \frac {\mu^2} {8\pi} A_\alpha A^\alpha -...
  22. H

    B Is the Momentum Operator Hermitian? A Proof

    Momentum operator is ##p=-i\frac{d}{dx}## and its adjoint is ##p^\dagger=i\frac{d}{dx}##. So, ##p^\dagger=-p##. How is the momentum Hermitian?
  23. U

    Questions on field operator in QFT and interpretations

    For a real scalar field, I have the following expression for the field operator in momentum space. $$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{k}}e^{i\omega t}\right)$$ Why is it that I can discard the phase factors to produce the time...
  24. Marioweee

    Probability of measuring an eigenstate of the operator L ^ 2

    Calculate, with a relevant digit, the probability that the measure of the angular momentum $L ^2$ of a particle whose normalized wave function is \begin{equation} \Psi(r,\theta,\varphi)=sin^2(\theta)e^{-i\varphi}f(r) \end{equation} is strictly greater than ##12(\hbar)^2##...
  25. D

    A BRST operator Q in string theory and string field theory

    In string theory, physical states satisfy QBΨ = 0, where QB is the BRST operator. This equation of motion can be obtained from an action S = ∫ QBΨ*Ψ + Ψ*Ψ*Ψ There is a gauge invariance under δΨ = QBΛ. what is the framework in which the role of the BRST operator QB is understood in open string...
  26. F

    I What conditions are needed to raise a linear operator to some power?

    Each operator has a domain, so for a power of an operator to exist, the domain of the operator must remain invariant under the operation. Is that correct? mentor note: edited for future clarity
  27. Haorong Wu

    I Applying Heisenberg picture to density operator

    Suppose that a particle evolves from point A to point B. The state of the particle can be written as $$\rho=\sum \left | m\right >\rho_{mn}\left< n\right | .$$ Because the basis is evolving as the particle travels, I am considering applying the Heisenberg picture to the density operator. Let...
  28. T

    I Derivation of Eigenfunctions/Eigenvalues of the Momentum Operator

    Good afternoon all, In David Griffiths' "Intro to Quantum Mechanics", I'm looking through Example 3.2 on page 115 that shows how to get the eigenfunctions and eigenvalues of the momentum operator. I completely understand everything up until this part: ##\int_{-\infty}^{\infty} f_p'^*(x)...
  29. P

    I Is the current density operator derived from fundamental considerations?

    Hello, I found this article. In equation (1) the authors wrote that the current operator is given by : ## - \frac{\delta H}{\delta A} ##. I just would like to know if this relation is a just definition or if it can be derived from more fundamentals considerations ? Thanks !
  30. E

    A Operator Product Expansion as shown in Schwartz

    I have included here the screen shot of the page I am referring to.I am unsure of how this non-local Lagrangian of Eqtn(32.68) has been constructed. Have they just integrated the interaction Lagrangian densities over two different sets of points (x & y) ? If so, then why is there no P_L in...
  31. F

    Determine the norm of an operator Tf(t)

    I don't know how to start to find the bounded condition nor the norm. I thought about finding a maximal norm to show that it is bounded but I don't know how to continue.
  32. E

    B What is the Significance of the Laplace Operator in Vector Calculus?

    ##\frac {\partial \vec F} {\partial x} ## + ##\frac{\partial \vec F} {\partial y} ## = vector which gives me a direction of the greatest increase of the greatest increase of the function, where ##\vec F ## = gradient of the function. If I multiple the first by ##\hat i## and the second by ##\hat...
  33. regory

    B Understanding Circumflex Operators: Get Help Now

    Hi I have a problem for understanding the difference between an circumflex operator and non-circumflex operador. I'd appreciate your help
  34. Celso

    Operator with 3 degenerate orthonormal eigenstates

    With this information I concluded that the diagonal elements of ##\hat{A}## are equal to the eigenvalue ##a##, so ##\hat{A} = \begin{bmatrix} a & A_{12} & A_{13} \\ A_{21}& a & A_{23}\\A_{31} & A_{32} & a \end{bmatrix}## but I can't see how to go from this to the commuting relation, since I...
  35. MathematicalPhysicist

    A Understanding Local and Nonlocal Operators in Quantum Field Theory

    I am reading the claymath problem here: http://claymath.org/sites/default/files/yangmills.pdf on page 6, in the comments (section 5), they call a local operator to be an operator that satisifies: ##\mathcal{O}(\vec{x})=e^{-i\vec{P}\cdot \vec{x}}\mathcal{O}e^{i\vec{P}\cdot \vec{x}}## where...
  36. tanaygupta2000

    Matrix formulation of an operator

    I have successfully found the N by N matrix corresponding to the operator R. But the problem is, whenever I try to operate R on |bj> basis vectors, I am not getting |b(j+1)> as it should be. Instead, I am getting result as given in the question only by <bj|R = <b(j+1)| Matrix is not working...
  37. tanaygupta2000

    Expectation value of momentum operator

    I know that the eigenstates of momentum operator are given by exp(ikx) To construct a real-valued and normalized wavefunction out of these eigenstates, I have, psi(x) = [exp(ikx) + exp(-ikx)]/ sqrt(2) But my trouble is, how do I find the expectation value of momentum operator <p> using this...
  38. docnet

    Expectation value of kinetic energy operator

    The expectation value of the kinetic energy operator in the ground state ##\psi_0## is given by $$<\psi_0|\frac{\hat{p^2}}{2m}|\psi_0>$$ $$=<\psi_0|\frac{1}{2m}\Big(-i\sqrt{\frac{\hbar mw}{2}}(\hat{a}-\hat{a^{\dagger}})\Big)^2|\psi_0>$$ $$=\frac{-\hbar...
  39. A

    I What is the logarithm of the derivative operator?

    I found this article which claims to have found the logarithm of derivative and even gives a formula. But I tried to verify the result by exponentiating it and failed. Additionally, folks on Stackexchange pointed out that the limit (6) in the article is found incorrectly (it does not exist)...
  40. S

    Spherical components of a rotated operator

    The operator is the ##T_{xyz}## component of the rank 3 tensor ##T=\vec{r}\otimes\vec{r}\otimes\vec{r}## whose Cartesian components are ##T_{ijk}=r_ir_jr_k##. This tensor ##T## also has spherical components ##T_{q}^{(k)}## where ##k=0,1,2,3##, which in principle can be related to their Cartesian...
  41. docnet

    Prove the rotational invariance of the Laplace operator

    Hello, please lend me your wisdom. ##\Delta u=\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u## ##Rx=\left<r_{11}x_1+...r_{1n}x_n+...+r_{n1}x_1+...+r_{nn}x_n\right>## ##(\Delta u)(Rx)=(\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u)\left<r_{11}x_1+...r_{1n}x_n...
  42. L

    A Hamiltonian commutes with a parity operator -- What does that mean?

    If Hamiltonian commutes with a parity operator ##Px=-x## are then all eigenstates even or odd? Is it true always or only in one-dimensional case?
  43. R

    MHB Compute Image Gradient Intensity - No Noise

    Hello guys, if I have an image with 11x11 pixels and in the center of the image is a square of 5x5 pixels, with the gray level of the background 0 and the gray level of the square is 50. How can I compute the result of the magnitude of edges(intensity of the contour) or better said the gradient...
  44. H

    A Position is no more an operator in QFT

    In quantum mechanics there is no operator for time (problem with unbounded energy). position is no more an operator in field theory. was there still a problem in QM?
  45. F

    Deriving Casimir operator from the Lie Algebra of the Lorentz Group

    Hello everyone, I am new here, so please let me know if I am doing something wrong regarding the formatting or the way I am asking for help. I did not really know how to start off, so first I tried to just write out all the ##\mu \nu \rho \sigma## combinations for which ##\epsilon \neq 0## and...
  46. LCSphysicist

    What is the role of the projector operator in vector manipulation?

    I don't know what is the answer, so i am not sure when to stop the computation or not. The far i reached was ## <k|j> \sum_{j} c_{j}|k>##. That is, the action of the projector operator is, obviously, project the state in |k>. Now, the coefficients was changed. So now what i have to do?
  47. jtbell

    C/C++ Puzzle with C++ constructor and overloaded operator

    I mentioned in another thread that I've been playing with a C++ class for 3D vectors, to review overloaded operators, constructors, etc. All those functions display output so I can follow which ones get called, and when. The class generates a serial number for each vector so I can tell them...
  48. Y

    Question about operator overloading

    In operator overload I often see the parameter is always a reference parameter. I want to find out why. I read the Copy Constructor in the book. I want to verify whether I understand this correctly as it's really important. I use "xx" just to represent some operator like =, >> etc. class ABC...
  49. Y

    Exploring Operator Overloading w/ Jamie King's Code

    I want to start a thread on Operator overloading for related question. I watched this youtube video and I like the way Jamie King write the operator overloading function: This is the program I copied from him: #include <iostream> using namespace std; struct Vector {int x; int y;}; //Below is...
  50. Supantho Raxit

    A What is a good basis for coupled modes in a resonator?

    Suppose, there is an electro-optical modulator that can couple the neighboring modes in an optical ring resonator. The Hamiltonian for the system looks something like this^^ (see the attached image). Here we sum over all modes m and 𝜙0 is a parameter. What will be a good set of basis for the...
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