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. Mentz114

    I Solving Operator Problem with Matrix Representation

    This matrix, I had hoped was a good candidate for (a representation of ) a unitary, self-adjoint operator \begin{align*} \hat{A}&= \frac{1}{D} \left[ \begin{array}{cc} a^2 & iy^2 \\\ -iy^2 & a^2\end{array} \right] \end{align*} ##a## and ##y## are real with ##D^2=a^2-y^2\ >\ 0## . ##\hat{A}##...
  2. E

    I Linear Algebra and Identity Operator Generalized to 3D

    I'm just getting into 3D quantum mechanics in my class, as in the hydrogen atom, particle in a box etc. But we have already been thoroughly acquainted with 1D systems, spin-1/2, dirac notation, etc. I am trying to understand some of the subtleties of moving to 3D. In particular, for any...
  3. K

    A Is There a Unique Hodge Star Operator for Any p-Vector in Differential Forms?

    I'm reading section 2.7 of Flanders' book about differential forms, but I have some doubts. Let ##\lambda## be a ##p##-vector in ##\bigwedge^p V## and let ##\sigma^1,\ldots,\sigma^n## be a basis of ##V##. There's a unique ##*\lambda## such that, for all ##\mu\in \bigwedge^{n-p}##,$$ \lambda...
  4. John Greger

    Showing That $\frac{d}{d_a} F_a(\hat{X}) \cdot \psi = F'(x) \psi$ at a=0

    Homework Statement Consider the operator ##F_a(\hat{X}) =e^{ia \hat{p} / \hbar} \cdot F(\hat{X}) e^{-ia \hat{p} / \hbar}## where a is real. Show that ##\frac{d}{d_a} F_a(\hat{X}) \cdot \psi = F'(x) \psi## evaluated at a=0. And what is the interpretation of the operator e^{i \hat{p_a} /...
  5. RicardoMP

    Bosonic operator eigenvalues in second quantization

    Homework Statement Following from \hat{b}^\dagger_j\hat{b}_j(\hat{b}_j \mid \Psi \rangle )=(|B_-^j|^2-1)\hat{b}_j \mid \Psi \rangle , I want to prove that if I keep applying ##\hat{b}_j##, ## n_j##times, I'll get: (|B_-^j|^2-n_j)\hat{b}_j\hat{b}_j\hat{b}_j ... \mid \Psi \rangle . Homework...
  6. Rabindranath

    Angular momentum operator for 2-D harmonic oscillator

    1. The problem statement I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian. The Attempt at a Solution I get...
  7. C

    I How Does the Rotation Operator Affect Spin in Quantum Mechanics?

    for compute: $$e^{\frac{iS_z\phi}{\hbar}}S_x e^{\frac{-iS_z\phi}{\hbar}}$$ so, if we use $$S_x=(\frac{\hbar}{2})[(|+><-|)+(|-><+|)]$$ $$e^{\frac{iS_z\phi}{\hbar}}(\frac{\hbar}{2})[(|+><-|)+(|-><+|)] e^{\frac{-iS_z\phi}{\hbar}}$$ so, why that is equal to...
  8. C

    Expanding the original commutator on the LHS

    Homework Statement Using [x,eiap]=-ħaeiap show that xneiap = eiap(x-ħa)n Homework Equations [x,eiap]=-ħaeiap From which it follows that, xeiap = eiap(x-ħa) The Attempt at a Solution [xn,eiap] = [xxn-1,eiap] = [x,eiap]xn-1 + x[xn-1,eiap]...
  9. thariya

    A Independence of Operator expectation values

    Hi! I want to know under what conditions the operator expectation values of a product of operators can be expressed as a product of their individual expectation values. Specifically, under what conditions does the following relation hold for quantum operators (For my specific purpose, these are...
  10. S

    I Historical basis for: measurement <-> linear operator?

    What is the history of the concept that a measurement process is associated with a linear opeartor? Did it come from something in classical physics? Taking the expected value of a random variable is a linear operator - is that part of the story?
  11. Pouyan

    Eigenvector of raising operator

    Homework Statement show that the raising operator at has no right eigenvectors Homework Equations We know at|n> = √(n+1)|n+1> The Attempt at a Solution we define a vector |Ψ> = ∑cn|n> (for n=0 to ∞) at|Ψ>=at∑cn|n>=∑cn(√n+1)|n+1> But further I give up!:cry:
  12. Jd_duarte

    I Hermitian Operator Proof - Question

    Hi, I am questioning about this specific proof -https://quantummechanics.ucsd.edu/ph130a/130_notes/node134.html. Why to do this proof is needed to compute the complex conjugate of the expectation value of a physical variable? Why can't we just start with < H\psi \mid \psi > ?
  13. M

    Self-adjoint Operators and their Products: Solving for ##AB##

    Homework Statement Given two linear self-adjoint operators ##A,B##, is it true ##AB## is also self-adjoint. Homework Equations Self adjoint implies ##(A[f],g) = (f,A[g])## The Attempt at a Solution I'm not really sure. I'm stuck almost right away: ##(AB[f],g) = (A[B[f]],g) = (B[f],Ag) =...
  14. M

    Build Green's function of an integro-differential operator

    Homework Statement For a given ##a##, define $$B[u(x)] = u''(s) + \cos^2(a) u(s) - \frac{1}{2s_0}\int_{-s_0}^{s_0}(u''(s) + \cos^2(a) u(s) )\, ds,\\ s_0 = \frac{1}{\cos(a)}\arcsin(\cos a)$$ subject to boundary conditions $$u'(s_0) + \cot (a) \cos (a) u(s_0) = 0\\ -u'(-s_0) + \cot (a) \cos (a)...
  15. sams

    I A question about writing the notation of the nabla operator

    I have a simple question about the notation of the nabla operator in Vector Analysis. The nabla operator is a vector differential operator and it is written as: $$\nabla = \hat{x} \frac {∂} {∂x} + \hat{y} \frac {∂} {∂y} + \hat{z} \frac {∂} {∂z}$$ Is it okay if we accented nabla by a right...
  16. CharlieCW

    Eigenstates of helicity operator

    Homework Statement For massless particles, we can take as reference the vector ##p^{\mu}_R=(1,0,0,1)## and note that any vector ##p## can be written as ##p^{\mu}=L(p)^{\mu}_{\nu}p^{\nu}_R##, where ##L(p)## is the Lorentz transform of the form $$L(p)=exp(i\phi J^{(21)})exp(i\theta...
  17. S

    MHB Does the Norm of a Linear Integral Operator Equal Its Spectral Radius?

    Hello A simple question. I have a linear integral operator (self-adjoint) $$(Kx)(t)=\int_{a}^{b} \, k(t,s)\,x(s)\,ds$$ where $k$ is the kernel. Can I say that its norm (I believe in $L^2$) equals the spectral radius of $K?$ Thanks! Sarah
  18. CharlieCW

    Finding Eigenvalues of an Operator with Infinite Basis

    I just began graduate school and was struggling a bit with some basic notions, so if you could give me some suggestions or point me in the right direction, I would really appreciate it. 1. Homework Statement Given an infinite base of orthonormal states in the Hilbert space...
  19. A

    I Identity Operator for Multiple Particles

    Hi, For a particle in a box (so that the momentum spectrum is discrete), we can write the identity operator as a sum over all momentum eigenstates of a projection to that eigenstate: $$I=\displaystyle\sum\limits_{p} |p\rangle\langle p|.$$ I was wondering what the corresponding form of the...
  20. Muthumanimaran

    B Time average value of Spin operator

    From the book Introduction to Quantum Mechanics by Griffiths,. In the section 6.4.1 (weak field zeeman effect) Griffiths tells that the time average value of S operator is just the projection of S onto J while finding the expectation value of J+S $$S_{avg}=\frac{(S.J)J}{J^2}$$ How to prove this?
  21. Glenn Rowe

    A Evolution operator in QFT - why Schrodinger?

    I'm reading through a couple of books (Lahiri & Pal's "A First Book of Quantum Field Theory" and Greiner & Reinhardt's "Field Quantization" and have come to the derivation of the evolution operator which leads to the S-matrix. In both books, the derivation starts with the Schrodinger equation in...
  22. I

    I Where does the gradient operator come from?

    Can someone explain why the gradient of a function is just a vector made up of partial derivatives of the function?
  23. darida

    A First Variation of Jacobi Operator

    <Moderator's note: Moved from a homework forum.> Homework Statement From this paper. Let ##L## be the Jacobian operator of a two-sided compact surface embedded in a three-maniold ##(M,g)##, ##\Sigma \subset M##, and defined by $$L(t)=\Delta_{\Sigma(t)}+ \text{Ric}( ν_{t} , ν_{t}...
  24. T

    I Del operator in a Cylindrical vector fucntion

    Hi there I'm having a hard time trying to understand how come ∂r^/∂Φ = Φ^ ,∂Φ/∂Φ = -r^ -> these 2 are properties that lead to general formula. I've been thinking about it and I couldn't explain it. I understand every step of "how to get Divergence of a vector function in Cylindrical...
  25. P

    I Why do we need the position operator?

    As I understand it, |Ψ|2 gives us the probability density of the wavefunction, Ψ. And when we integrate it, we get the probability of finding the particle at whichever location we desire, as set by the limits of the integration. But when we use the position operator, we have integrand Ψ*xΨ dx...
  26. mertcan

    Computational High order nonlinear operator split method

    Hi, I really need to have sources related to high order operator split method for nonlinear pdf ode equations. Could you provide me with sources about that books files videos links...??
  27. renec112

    Expectation value of raising and lower operator

    I am practicing old exams. I tried my best but looking at an old and a bit unreliable answer list, and i am not getting the same result. Homework Statement At time ##t=0## the nomralized harmonic oscialtor wavefunction is given by: ## \Psi(x,0) = \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i...
  28. T

    Characteristic equation in terms of the Laplace operator

    Homework Statement The stability of a spinning body may be explored by using equation (3.40), with no torque components present. It will be assumed here that the spin is about the z -axis and has a rate ωZ = S. Homework Equations $$I_{xx}\dot{ω} - (I_{yy}-I_{zz})Sω_y = 0$$ $$I_{yy}\dot{y} -...
  29. fluidistic

    I How to pick an operator onto which the wavefunction collapses

    Imagine a system of 1 particle in a superposition of eigenstates of some operator(s). If one were to make a measurement of a property of that particle, how is the operator (or observable) "picked" so that the wavefunction collapses into an eigenstate of said operator? In other words, how do one...
  30. SchroedingersLion

    I Liouville operator time dependency

    Hi guys, I have a two questions concerning basics of classical mechanics. It is to my current computational physics lecture. As I don't know how to write formulas here, I had to include the equations and the situation as a picture. I would appreciate the answers. Regards edit: quality...
  31. mertcan

    A Higher Order Operator Splitting Method

    Hi, in the link https://www.researchgate.net/profile/Andrew_Sornborger/publication/220662120_Higher-order_operator_splitting_methods_for_deterministic_parabolic_equations/links/568ffaab08aec14fa557b85e/Higher-order-operator-splitting-methods-for-deterministic-parabolic-equations.pdf and equation...
  32. Anatoly

    A New Discovery: τ - Maths on Facebook

    Where do you post mathematical discoveries? Recently I've founded and posted on facebook: https://i.imgur.com/EGbrnuN.png https://i.imgur.com/bNenU8Z.png https://i.imgur.com/aZ5Esss.png https://i.imgur.com/eCiqyvG.png τ is similar to Σ and Π but it appears to be in itself.
  33. kvothe18

    I Mean value of the nuclear tensor operator

    Does anyone know how can you prove that the mean value of the tensor operator S12 in all directions r is zero? S12 : http://prntscr.com/j3gn40 where s1, s2 are the spin operators of two nucleons.
  34. M

    Finding the position operator in momentum space

    Homework Statement Given ##\hat{x} =i \hbar \partial_p##, find the position operator in the position space. Calculate ##\int_{-\infty}^{\infty} \phi^*(p) \hat{x} \phi(p) dp ## by expanding the momentum wave functions through Fourier transforms. Use ##\delta(z) = \int_{\infty}^{\infty}\exp(izy)...
  35. S

    Expectation Value and Probabilities of Spin Operator Sy

    Homework Statement (a) If a particle is in the spin state ## χ = 1/5 \begin{pmatrix} i \\ 3 \\ \end{pmatrix} ## , calculate the expectation value <Sy>(b) If you measured the observable Sy on the particle in spin state given in (a), what values might you get and what is the probability of...
  36. S

    Normalised eigenspinors and eigenvalues of the spin operator

    Homework Statement Find the normalised eigenspinors and eigenvalues of the spin operator Sy for a spin 1⁄2 particle If X+ and X- represent the normalised eigenspinors of the operator Sy, show that X+ and X- are orthogonal. Homework Equations det | Sy - λI | = 0 Sy = ## ħ/2 \begin{bmatrix} 0...
  37. hilbert2

    A Is the Inverse Momentum Operator an Essential Tool in Quantum Mechanics?

    In QM, the inverse distance operator ##\hat{r}^{-1}## appears often because of the association to Coulomb potential. The operator of inverse momentum, ##\frac{1}{\hat{p}}## is a lot more rare. In the book "Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers...
  38. Peter Morgan

    A Field quantization and photon number operator

    [Moderator's note: This thread is spun off from a previous thread since it was getting into material too technical for the original thread. The quote at the top of this post is from the previous thread.] Field quantization doesn't require a photon picture. A measurement device that creates a...
  39. renec112

    First order pertubation of L_y operator

    Hi, I am trying to solve an exam question i failed. It's abput pertubation of hydrogen. I am given the following information: The matrix representation of L_y is given by: L_y = \frac{i \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1...
  40. K

    C/C++ Copy Assignment Operator for CarCounter Class

    Write a copy assignment operator for CarCounter that assigns objToCopy.carCount to the new objects's carCount, then returns *this. Sample output for the given program: Cars counted: 12 #include <iostream> using namespace std; class CarCounter { public: CarCounter(); CarCounter&...
  41. Pattarasak

    I How to create the operator of unknown beamsplitter

    Hello everyone, I'm undergraduate and my project is an experiment in the field of quantum optics. For now, I have an Unknown beamsplitter in my lab and I want to calculate the operator of this beamsplitter in matrix form (this BS is not perfected equipment because the reflected beam is not...
  42. S

    I Understanding the Parity Operator in Dirac Field Theory

    Hello! I am a bit confused about matrices dimensions in the second quantization of the Dirac field. The book I am using is "An Introduction to Quantum Field Theory" by Peskin and Schroder and I will focus in this question mainly on the Parity operator which is section 3.6. The field operator...
  43. Gene Naden

    A Angular momentum operator derived from Lorentz invariance

    I am working through Lessons in Particle Physics by Luis Anchordoqui and Francis Halzen; the link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf. I am on page 11, equation 1.3.20. The authors have defined an operator ##L_{\mu\nu} = i( x_\mu \partial \nu - x_\nu \partial \mu)##...
  44. BigDig123

    I Is the total Spin operator a vector

    Hello, I am learning about Excited states of Helium in my undergrad course. I was wondering if the total spin operator Ŝ is a vector quantity or not. Thanks for your help.
  45. C

    A Angular Moment Operator Vector Identity Question

    In my EM class, this vector identity for the angular momentum operator (without the ##i##) was stated without proof. Is there anywhere I can look to to actually find a good example/proof on how this works? This is in spherical coordinates, and I can't seem to find this vector identity anywhere...
  46. TeethWhitener

    I Srednicki QFT Chapter 4 time-evolved operator

    In chapter 4 of Srednicki's QFT book (introducing the spin-statistics theorem for spin-0 particles), he introduces nonhermitian field operators (just taking one as an example): $$\varphi^+(\mathbf{x},0) = \int \tilde{dk}\text{ }e^{i \mathbf{k}\cdot\mathbf{x}}a(\mathbf{k})$$ and time-evolves them...
  47. SemM

    I Proving that an operator is unbounded

    Hi, after reading on boundedness and unboundedness, I like to prove that an operator, S, is unbounded. However I am not sure this is good enough? \begin{equation} ||S\psi|| \leqslant c ||\psi|| \end{equation} \begin{equation} ||S\psi|| = \bigg( \int_a^b (S \psi)^2 dx \bigg)^{1/2} \end{equation}...
  48. N

    I Representing a Hamiltonian in an operator form

    Given a Hamiltonian in the position representation how do I represent it in operator form? for example I was asked to calculate the expectancy of the Darwin correction to the Hydrogen Hamiltonian given some eigenstate (I think it was |2,1> or something bu that doesn't matter right now), now I...
  49. L

    A Operator Identity: Quantum Mechanics Explanation w/ References

    In Quantum mechanics, when we have momentum operator ##\vec{p}##, and angular momentum operator ##\vec{L}##, then (\vec{p} \times \vec{L})\cdot \vec{p}=\vec{p}\cdot (\vec{L} \times \vec{p}) Why this relation is correct, and not (\vec{p} \times \vec{L})\cdot \vec{p}=\vec{p}\cdot (\vec{p} \times...
  50. SemM

    A Hilbert-adjoint operator vs self-adjoint operator

    Hi, while reading a comment by Dr Du, I looked up the definition of Hilbert adjoint operator, and it appears as the same as Hermitian operator: https://en.wikipedia.org/wiki/Hermitian_adjoint This is ok, as it implies that ##T^{*}T=TT^{*}##, however, it appears that self-adjointness is...
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