Operator Definition and 1000 Threads

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. V

    Spectrum of operator from L^2 to L^2

    Homework Statement Find spectrum and eigenvalues of operator from L^2(-1,1) to L^2(-1,1) T(f)(t) = ∫(t+s)^2f(s)ds The integral is taken over [-1,1] 2. The attempt at a solution I have already proven that this operator is self-adjoint and compact. However, I have now idea how to find...
  2. E

    Solve Inverse Problem: Find f(k) for sum_(k=1, to n) f(k) = F(n)

    I have general function sum_(k=1, to n) f(k) = F(n) ex.) sum_(k=1, to n) k = F(n) solution: F(n) = (1+n)n/2 But If I have inverse problem?: ex.) sum_(k=1, to n) f(k) = n^2 how to get f(k) ? Generaly, If I have F(n), how to get f(k) for sum_(k=1, to n) f(k) = F(n) Is...
  3. M

    Coordinate representation of the momentum operator

    The position operator in coordinate representation is: Xab=aδ(a-b) this is diagonal as expected The momentum operator turns out to look like Pab=-ih∂aδ(a-b) Now, this is not supposed to be diagonal because it does not commute with X. However it looks pretty diagonal to me. What am I...
  4. T

    How is the following operator linear?

    How is the following operator linear??! Homework Statement Is the following a linear vector function? F(r) = r - ix Homework Equations A function is linear if: F(r1 + r2) = F(r1) + F(r2) AND F(ar) = aF(r) The Attempt at a Solution F(r1 + r2) = (r1 + r2) - ix F(r1) +...
  5. N

    Inversion w.r.t. a sphere: Operator

    Hello everyone, I have enquired about inversion in a sphere here in past: https://www.physicsforums.com/showthread.php?t=440759 Although that time I could not come back to the discussion (apologies to Jason), later I went through some of the properties mentioned by him...
  6. P

    Proving Orthogonality of $\hat{A}|\Psi>$ with Anti-Unitary Operator $\hat{A}$

    Homework Statement \hat{A} is an anti-unitary operator, and it is known that \hat{A^2}= -\hat{I}, show that |\Psi> is orthogonal to \hat{A}|\Psi> Homework Equations I know that \hat{A} can be represented by a unitary operator, \hat{U}, and the complex conjugation operator, \hat{K}...
  7. H

    How can I find the momentum squared operator?

    Homework Statement This problem is about the momentum squared operator. First, I state how I saw the derivation for the momentum operator. Then I state how I attempt to (and fail to) derive the momentum squared operator using the same methods. Homework Equations <p> = ∫ ψ*(ħ/i ∂/∂x)ψ dx The...
  8. sunrah

    How Do You Differentiate e^(A*t) When A is a Constant Operator?

    Homework Statement calculate \frac{d}{dt}e^{\hat{A}t} where \hat{A} \neq \hat{A}(t) in other words operator A doesn't depend explicitly on t. Homework Equations The Attempt at a Solution \frac{d}{dt}e^{\hat{A}t} = (\frac{d}{dt}(\hat{A})t + \hat{A})e^{\hat{A}t} =...
  9. K

    Phase operator in quantum optics

    I recently got interested in some aspects of quantum optics and have a basic question. There is an uncertainty relation concerning the phase and the particle number of a mode. How is this phase observable defined? Where can I read about such basic concepts without dwelling deeply into higher...
  10. S

    Simple Linear Differential Operator Problem

    [b]1. (D+1)(D-x)(2e^x+cosx) [b]2. None [b]3. (D+1)(D-x)(2e^x+cos(x))=D(2e^x-sinx-2xe^x-xcosx)+1(2e^x-sinx-2xe^x-xcosx)=-4e^x+2e^x-xcosx-2cosx+xsinx-sinx The correct answer is 2e^x(xsinx-3sinx+2xcosx)
  11. S

    A compact, bounded, closed-range operator on a Hilbert space has finite rank

    Homework Statement Let H be an \infty-dimensional Hilbert space and T:H\to{H} be an operator. Show that if T is compact, bounded and has closed range, then T has finite rank. Do not use the open-mapping theorem. Let B(H) denote the space of all bounded operators mapping H\to{H}, K(H) denote...
  12. A

    Square of a hermitian operator in matrix form

    If we have a hermitian operator Q and we know it's matrix representation [Q], does that mean that [Q2] = [Q]2? For example, I'm pretty sure that's the case for p2 for a harmonic oscillator. We have p=ic(a+-a-) and so p2=c2(a+-a-)(-a++a-)*=c2(a+-a-)(a+-a-)=p p Which tells us that [p2]=[p]2...
  13. N

    Properties of Differential Operator and Proper Formalism

    Let us take a function defined by y=cx To differentiate that, we use the operator d/dx \frac{d}{dx} y = \frac{d}{dx} cx^{-1} By the chain rule/implicit differentiation on the left and normal differentiation on the right we get, \frac{dy}{dx} = -1c x^{-2} What confuses me is the proper...
  14. D

    Linear operator exercise i can't understand

    Homework Statement Let x=(x1,x2,x3), Ax:=(x2-x3,x1,x1+x3), Bx:=(x2,2x3,x1) Find: (3B+2A2)x. Homework Equations The Attempt at a Solution Warning: I have no idea what I'm doing! (3B+2A2)x = 3Bx+2A2x 3Bx = (3x2,6x3,3x1) Now to find 2A2x. Considering that an index has a...
  15. L

    Operator on a set spaned the space

    Hi there, Let X be a Hilbert (Banach) space, and spanned by a set S, say. Let A be linear bounded operator on X into itself. Suppose that the operator is well known on S, that is Aa_i=b_i for all a_i\in S. First, is this operator unique on X? if yes, can we find Aa, for general element...
  16. K

    The time evolution operator (QM) Algebraic properties

    Homework Statement The hamiltonian for a given interaction is H=-\frac{\hbar \omega}{2} \hat{\sigma_y} where \sigma_y = \left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array} \right) the pauli Y matrix Homework EquationsThe Attempt at a Solution So from the time dependant schrodinger...
  17. S

    When is operator phi(x) an observable in QFT?

    In QFT of a real Klein-Gordon-Field, the field operator \phi(x) is an observable. Mathematically, this is the case because it is a sum (over all k) of a and a^\dagger and this yields a Hermitian operator. Physically, I can understand this because this equation would describe, for example, a...
  18. K

    Derivative of the inversion operator and group identity

    Homework Statement Let G be a Lie group, e be its identity, and \mathfrak g its Lie algebra. Let i be group inversion map. Show that d i_e = -\operatorname{id} . The Attempt at a Solution So this isn't terribly difficult if we have the exponentiation functor, since in that case e^{-\xi}...
  19. N

    Estimation of the Operator norm

    Homework Statement L : R^n → R is defined L(x1 , . . . , xn ) = sum (xj) from j=1 to n. The problem statement asks me to find an estimation for the operation norm of L, where on R the norm ll . llp, 1 ≤ p ≤ ∞, is used and on R the absolute value.The Attempt at a Solution from, ll Lv lly ≤...
  20. S

    Proving Adjoint of an Operator in Hilbert Space: Common Mistakes Explained

    Hi, I am wondering what mistake i might have made in the following. (v, Au) = [(v, Au)*]* = [(A†v, u)*]* = (u, A†v)* = ((A†)†u, v)* = (v, (A†)†u), where v and u are arbitrary vectors in a Hilbert space. That proves that A = (A†)†, doesn't it? My tutor says some of the steps are wrong...
  21. J

    Commutators of Angular momentum operator

    The letters next to p and L should be subscripts. [Lz, px] = [xpy − ypx, px] = [xpy, px] − [ypx, px] = py[x, px] −0 = i(hbar)py 1.In this calculation, why is [x, px] not 0 even they commute? 2.Why is py put on the left instead of the right in the second last step? i thought it should be...
  22. G

    How Does the Delta Emerge in the Time Evolution Operator's Exponential Form?

    If you have some Hamiltonian represented by a 2x2 matrix ## H = \left( \begin{array}{cc} 0 & \Delta \\ \Delta & 0 \end{array} \right) ## And you want to use the time evolution operator ## U = \exp ( - \frac{i}{\hbar} H t ) ## it says that ## U = \exp (- \frac{i \Delta}{\hbar} t) ## Why...
  23. S

    Inverse of the 1-D momentum operator

    Homework Statement I have to find the inverse of the 1-D momentum operator. Homework Equations The Attempt at a Solution Here's my solution: Pψ(x) = -iħ dψ/dx P-1[Pψ(x)] = P-1[-iħ dψ/dx] [P-1P]ψ(x) = -iħ [P-1 dψ/dx] Iψ(x) = -iħ [P-1 dψ/dx] ψ(x) = -iħ [P-1 dψ/dx] By...
  24. S

    Hermitian conjugate of an operator

    Hey guys, I'm doing a third year course called 'Foundations of Quantum Mechanics' and there's this thing in my notes I don't quite get. I was hoping to get your help on this, if you don't mind. It's about Hermitian conjugate operators. The sentences go (v, Au) = (A†v|u) <v|A|u> = <v|(A|u>)...
  25. C

    PDE-Cayley Transform Of An Operator

    Homework Statement Prove that a unitary operator U acting on a Hilbert space H is the Cayley transform of some self-adjoint operator if and only if 1 is not an eigenvalue of U. Hope someone will be able to help Thanks ! Homework Equations The Attempt at a Solution At the...
  26. Rasalhague

    Topology generated by interior operator

    Given an interior operator on the power set of a set X, i.e. a map \phi such that, for all subsets A,B of X, (IO 1)\enspace \phi X = X; (IO 2)\enspace \phi A \subseteq A; (IO 3)\enspace \phi^2A = \phi A; (IO 4)\enspace \phi(A \cap B) = \phi A \cap \phi B, I'm trying to show that the set...
  27. I

    Conformal Field Theory: Evaluating the Vertex Operator on the Vacuum

    Homework Statement Evaluate \lim_{z \to 0}:e^{ik \cdot X(z)}:|0\rangle where X(z) is a free chiral scalar field in the complex plane. Homework Equations In Conformal Field Theory, the free chiral scalar field in the complex plane is given by: \begin{array}{rcl} X(z) &=& \frac{1}{2}q -...
  28. S

    What symbol is used for linear operator actions?

    Given a vector v\in H: R2->R3 which is a function from R2 to R3, and an operator G: H->H on the space H in which v lives, what is the most commonly used symbol to refer to this mapping, i.e., G(v)=G\otimes v or something else?
  29. Spinnor

    Box^2 A_μ = J_μ ; [Dirac operator] ψ = 0 or not?

    The four vector J_μ is the source of the Electromagnetic potential four vector A_μ. If I wanted a source term for the Dirac equation (not that I think there is such a thing) could it be (must it be) a spinor source term (what ever that means)? Related question, consider the two Feynman diagrams...
  30. P

    Simulating bitwise AND operator

    Hello, I have a problem where I must get value of a given binary bit in decimal number: f(decimal_number, bit_position) = bit_value For example, getting bit values in decimal 14 (binary 1110): f(14, 0) = 0 f(14, 1) = 1 f(14, 2) = 1 f(14, 3) = 1 In general it's easy: (14 &...
  31. S

    Question on the form of a vertex operator in a proof

    [Closed] Question on the form of a vertex operator in a proof Ok, never mind - I decided to find the solution in a different way.. This is a little too specialized anyway. (Is there a way to delete the thread?) Hi, I am reading paper [1] and I found that formula (33), \psi(xy)\psi^*(y)=\frac...
  32. Z

    Density operator for one part of a two-party state

    Homework Statement Let |\psi\rangle_{AB}=\sum_{i}\sum_{j}c_{ij}|\varphi_{i}\rangle_{A}\otimes|\psi_{j}\rangle_{B} be a normalized two party state, being \{|{\varphi_{i}}\rangle_{A}\} and \{|{\psi_{j}}\rangle_{B}\} basis of H_{A} and H_{B} respectively, with dimensions N and M. Find \rho_{A}...
  33. L

    The Parity Operator: Find the average value of the parity.

    Homework Statement A particle of mass m moves in the potential energy V(x)= \frac{1}{2} mω2x2 . The ground-state wave function is \psi0(x)=(\frac{a}{π})1/4e-ax2/2 and the first excited-state wave function is \psi1(x)=(\frac{4a^3}{π})1/4e-ax2/2 where a = mω/\hbar What is the average value of...
  34. N

    Continuous matrix = differential operator?

    Hello, Sorry if the question sounds silly, but can a continuous matrix be seen as a differential operator? First of all, let me state that I have no idea what a continuous matrix would formally mean, but I would suppose there is such an abstract notion, somewhere? Secondly, let me tell...
  35. L

    Some expressions with Del (nabla) operator in spherical coordinates

    Reading through my electrodynamics textbook, I frequently get confused with the use of the del (nabla) operator. There is a whole list of vector identities with the del operator, but in some specific cases I cannot figure out what how the operation is exactly defined. Most of the problems...
  36. S

    Showing a linear operator is compact

    Homework Statement Let [e_{j}:j\in N] be an orthonormal set in a Hilbert space H and \lambda_{k} \in R with \lambda_{k} \rightarrow 0 Then show that Ax=\sum_{j=1}^{\infty} \lambda_{j}(x,e_{j})e_{j} Defines a compact self adjoint operator H \rightarrow H The Attempt at a Solution...
  37. S

    Showing that the range of a linear operator is not necessarily closed

    Homework Statement Let T: \ell^{2} \rightarrow \ell be defined by T(x)=x_{1},\frac{1}{2}x_{2},\frac{1}{3}x_{3},\frac{1}{4}x_{4},...} Show that the range of T is not closed The Attempt at a Solution I figure that I need to find some sequence of x_{n} \rightarrow x such that...
  38. H

    Solving the Angular Momentum Operator for j=1

    Homework Statement Consider the angular momentum operator \vec{J_{y}} in the subspace for which j=1. Write down the matrix for this operator in the usual basis (where J^{2} and J_{z} are diagonal). Diagonalize the matrix and find the eigenvalues and orthonormal eigenvectors. Homework...
  39. H

    Find the inner product of the Pauli matrices and the momentum operator?

    Homework Statement Show that the inner product of the Pauli matrices, σ, and the momentum operator, \vec{p}, is given by: σ \cdot \vec{p} = \frac{1}{r^{2}} (σ \cdot \vec{r} )(\frac{\hbar}{i} r \frac{\partial}{\partial r} + iσ \cdot \vec{L}), where \vec{L} is the angular momentum operator and...
  40. D

    Proving Boundedness of Operator T in L^p(-2,2)

    Homework Statement The operator T maps from L^p(-2,2)\rightarrow L^p(-2,2) is defined (Tf)(x) = f(x) x Show that the operator maps from L^p(-2,2) into the same. Homework Equations p is a natural from 1 to infinity. Holders inequality Substitution integrals The Attempt at a Solution I look at...
  41. fluidistic

    Why Use Spherical Coordinates for Hydrogen Atom's Hamiltonian in 3D?

    If I consider the problem of for example the hydrogen atom. I.e. a central force problem with an effective potential V(r) that depends only of r, the distance between the positively charged nucleous and the negatively charged electron. In the Schrödinger's equation, one considers the...
  42. Shackleford

    Unitary Operator: Proof & Counterexample

    Here's the definition. Let T be a linear operator on a finite-dimensional inner product space V. If \|\vec{T(x)}\| = \|\vec{x}\| \\ for all x in V, we call T a unitary operator. The question is asking about for all x in some orthornormal basis for V. Isn't that the same as for all x in V?
  43. V

    How Does the Divergence Operator Apply in the Transport Theorem?

    I'm reading about the transport theorem in my vector calculus book. They state the following at the beginning of the section: ======================================================== Let F be a vector field on R^3. Let c(x, t) denote a flow line on F starting at location x and continuing out...
  44. P

    MHB Is the Sturm-Liouville Operator Symmetric in Inner Product Spaces?

    What does it mean for Sturm-Liouville operator to be symmetric w.r.t an inner product? I was reading in a book that it is symmetric but that was about a certain integral being zero and inner products had not even been mentioned.
  45. L

    Can an operator without complete eigenstates be measured?

    Say you have some operator A with an incomplete set of eigenstates, but the state of the system is such that it happens to be expressible as a sum (possibly infinite, or integral, whatever) of the eigenstates of A, and let's say the eigenvalues are real and whatever is necessary...we may assume...
  46. T

    What Happens to the Momentum Operator as Planck's Constant Approaches Zero?

    I have a doubt making a little of thinking of basic notions of QM and I think that the answer should be very simple but I can't make up my mind. So, here I go: I usually hear that when we want to go from to Quantum Mechanics to Classical Mechanics, one have to make h go to 0 and then the magic...
  47. J

    Eigenvalue Spectrum of this Operator

    Hello I have this Hamiltonian: \mathcal{H} = \alpha S_{+} + \alpha^{*}S_{-} + \beta S_{z} with \alpha, \beta \in \mathbb{C} . The Operators S_{\pm} are ladder-operators on the spin space that has the dimension 2s+1 and S_{z} is the z-operator on spin space. Do you know how to get (if...
  48. M

    Calculating Statistical Operator $\hat{\rho}$

    \hat{\rho} = \begin{bmatrix} \frac{1}{3} & 0 & 0 \\[0.3em] 0 & \frac{1}{3} & 0 \\[0.3em] 0 & 0 & \frac{1}{3} \end{bmatrix} If I have this statistical operator I get i\hbar\frac{d\hat{\rho}}{dt}=0 So this is integral of motion and...
  49. M

    Number Operator in Matrix Form

    Hi- I have a basic QM problem I am trying to solve. We are just starting on the formalities of Dirac notation and Hermitian operators and were given a proof to do over Spring Break. I am stuck on how to set up the operators and wave equation in matrix and vector form to complete the proof as...
  50. S

    Hermitian Operator: Definition & Overview

    what is it?
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