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. WannabeNewton

    Non-local differential operator

    Heyo. On page 4 of Srednicki's QFT text, the following equation is given (in an attempt to make the Schrodinger equation relativistic): ##i\hbar \partial_t \psi(x,t) = \sqrt{-\hbar^2c^2 \nabla^2 + m^2 c^4}\psi(x,t)## where ##\psi(x,t) = \left \langle x|\psi,t \right \rangle## is the position...
  2. G

    Green's function of the Klein-Gordon operator

    Again, from the Peskin and Schroeder's book, I can't quite see how this computation goes: See file attached The thing I don't get is how the term with (\partial^{2}+m^{2})\langle 0| [\phi(x),\phi(y)] | 0 \rangle vanishes, and also why they only get a \langle 0 | [\pi(x),\phi(y)] | 0 \rangle...
  3. caffeinemachine

    MHB Tricky Linear Algebra Question. To show that an operator is 'cyclic'.

    Hello MHB, I am stuck at this problem for quite a long time now. Problem. Let $F_p$ denote the field of $p$ elements, where $p$ is prime. Let $n$ be a positive integer. Let $V$ be the vector space $(F_p)^n$ over the field $F_p$. Let $GL_n(F_p)$ denote the set of all the invertible linear...
  4. M

    Learn Wigner Rotation, Tensor Operator & Two-Particle Helicity State

    Hi, Is there any good books which explain/calculate Wigner rotation, tensor operator, two-particle helicity state and related stuff in detail? Thanks.
  5. U

    Trace of Position and Momentum Operator

    Homework Statement Hi guys, I've not started a course on QM yet, but we are currently learning the maths used in QM. Show, by taking the trace of both sides show that finite dimensional matrix representations of the momentum operator p and the position operator x which satisfy [p, x] =...
  6. U

    Matrix Transformation of operator from basis B' to B

    Homework Statement Hi guys, actually this isn't a homework question, but rather part of the working in a textbook on Linear Algebra. Homework Equations The Attempt at a Solution I'm not sure why it's U*li instead of U*il. Shouldn't you flip the order when you do a matrix...
  7. G

    Total momentum operator for free scalar field

    Sorry for reopening a closed thread. But I have exactly the same doubt as this guy: https://www.physicsforums.com/showthread.php?t=346730 And the answer doesn't actually answer his question. I do get delta(p+p'), but they just help me in getting a_{p}a_{-p} and a_{p}^{\dagger}a_{-p}^{\dagger}...
  8. S

    How do these operations with Del operator work?

    How do these operations with Del operator work?? Homework Statement Let's say A and B are expressed by their cartesian components as: A = <P, Q, R> and B = <M, N, O> what would be the differente between (A.∇)B and B(∇.A) ? Homework Equations The Attempt at a Solution I tried...
  9. Telemachus

    Quantum Mechanics: Base on which an operator is given

    The thing is that often in the problems on quantum mechanics I've found that an operator is given, but not the base on which it is represented. I'll give an especific example in a moment. So then the problem asks me to find the eigenvalues and eigenvectors for a given operator and to express it...
  10. Q

    Time evolution operator versus propagate

    I am trying to understand the how the time evolution operator is used versus the Feynman propagate. My limited understanding is the following for which I am seeking clarity: 1. The time evolution operator is a unitary operator which enables us to calculate a probability amplitude of one...
  11. C

    Function notation for the derivative operator

    If we have the function f : x \mapsto f(x) = 3x^2, I am used to Lagrange's prime notation for the derivative: f' : x \mapsto f'(x) = 6x. I'm fond of this notation. But it has been mostly abandoned in my engineering courses in favor of Leibniz's notation, using differential operators such...
  12. michael879

    Solving the Mystery of Unitary Time Operator in QM

    This has been bugging me for a while, so I'm really hoping someone can give me a good answer. Please get as technical as necessary, I'm a 4th year HE physics grad student so I do know my stuff! Why is the time-reversal operator made anti-unitary in quantum mechanics? It is very straight...
  13. STEMucator

    Normal Operator Proof: Proving ##T \in L(V)## is Normal

    Normal Operator Proof Homework Statement Prove an operator ##T \in L(V)## is normal ##⇔ ||T(v)|| = ||T^*(v)||##. Homework Equations (1) ##T \in L(V)## is normal if ##TT^*= T^*T##. (2) If T is a self-adjoint operator on V such that ##<T(v), v> = 0, \space \forall v \in V##, then...
  14. H

    Simplifying many body operator expressions

    Hello all, Is there any really good software, package or otherwise, that is helpful for simplifying long operator expressions, particularly with (anti-commuting) electron operators? For example, I'd like to be able to transform Hamiltonians, project onto subspaces with specific occupation...
  15. D

    Show that the complex conjugation operator is hermitian.

    Find its eigen values. Is this operator linear?
  16. L

    Multiplication between vector and vector operator

    How this is defined? ##\vec{r}\cdot \vec{\sigma}##? where ##\vec{r}=(x,y,z)## and ##\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)##. ##\sigma_i## are Pauli spin matrices.
  17. L

    Operator Theory Problem on Momentum Operator (QM)

    Homework Statement Given the operators \hat{x}=x\cdot and \hat{p}=-i\hbar \frac{d}{dx}, prove that: [\hat{x}, g(\hat{p})]=i\hbar \frac{dg}{d\hat p}Homework Equations None. The Attempt at a Solution I have very little idea on how to begin this problem, but I don't want a solution, I simply...
  18. omephy

    Unitary spacetime translation operator

    Srednicki eqn. (2.23) and (2.24) states: We can make this a little fancier by defining the unitary spacetime translation operator T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar) Then we have T(a)^{-1} \phi(x) T(a) = \phi(x-a) How do we get the second equation from the first equation?
  19. J

    Nabla operator and working with it

    While using the ∇ operator, most of the times we can treat it as a vector. I came across a few formulae(basically product rules).. ∇×(A×B)=(B.∇)A-(A.∇)B+A(∇.B)-B(∇.A) where A and B are vectors I wanted to know if there is any direct way of deriving it. By direct I mean assuming the basic...
  20. Ravi Mohan

    Condition for completeness of eigen vectors of an operator

    I am studying an article http://arxiv.org/abs/quant-ph/9907069 and having some problems understanding it. Is self adjointness of an operator a sufficient or necessary and sufficient requirement for its eigen vectors with the generalized eigenvectors (i don't know what are these) to form...
  21. D

    Relation between spectra of operator and spectrum of a fourier transfo

    Hello, Something I have some time wondering and still couldn't find the answer is to this question: if there is some relation between the Spectrum (functional analysis) and the Frequency spectrum in Fourier Analysis. Now that I think about it there seems to be a casuality the use of the...
  22. C

    Spherical tensor operators' commutation with lowering/raising operator

    I'm studying Shankar's book (2nd edition), and I came across his equation (15.3.11) about spherical tensor operators: [J_\pm, T_k^q]=\pm \hbar\sqrt{(k\mp q)(k\pm q+1)}T_k^{q\pm 1} I tried to derive this using his hint from Ex 15.3.2, but the result I got doesn't have the overall \pm sign on the...
  23. S

    What is the Relationship Between Modulo and Remainders in Math and Programming?

    I am confused about this. I have always thought that the modulo operator always has the result of a while number between 0 and the modulo divisor minus 1. I presume that the terms are called: a % b = c a : dividend b : divisor c : remainder a & b > 0 : a % b = b * [ ( a / b ) -...
  24. M

    Differential Operator Notation

    I am finding it unintuitive to follow a calculation in a certain notation I am not too familair with. To write down the equation $$ z(t) = - y(t) + \tau \frac{\partial y}{\partial t} $$ the following notation is employed $$ z(t) = -(1-\tau \frac{\partial }{\partial t}) y $$ So far so gud...
  25. G

    Explanation of exponential operator proof

    Can someone please explain the below proof in more detail? The part in particular which is confusing me is Thanks in advance!
  26. S

    Prove that v is an eigenvector of operator B

    Homework Statement Let B be the linear operator (1-x^{2}) \frac{d^2}{dx^2}-x\frac{d}{dx} Show that T_{4}(x) = 8x^{4} - 8x^{2} + 1 is an eigenvector of B, and find the corresponding eigenvalue. Attempt Righto, I find these rather difficult so a step by step solution would be nice but...
  27. L

    Spacetime displacement operator in QFT

    I'm trying to fit together my understanding of quantum mechanics, quantum field theory, given my lacking maths education. In quantum mechanics we have a time displacement operator and a space displacement operator, which are respectively: \hat{T}(t) = e^{-i\hat{H}t} \hat{D}(\underline{x}) =...
  28. M

    Operator equations vs. field equations

    If my understanding is correct, the equations of QFT (Dirac, Klein-Gordon) govern the behavior of operator fields (assigning operator to each point in space). Does it mean there are no equations governing the behavior of fields (assigning a number / vector/ spinor to each point in space)? Is QFT...
  29. fluidistic

    Ladder operator for harmonic oscillator, I don't get a mathematical

    If the ladder operator ##a=\sqrt {\frac{m\omega}{2\hbar}}x+\frac{ip}{\sqrt{2m\hbar \omega}}## and ##a^\dagger=\sqrt {\frac{m\omega}{2\hbar}}x-\frac{ip}{\sqrt{2m\hbar \omega}}## then I get that the number operator N, defined as ##a^\dagger a## is worth ##\frac{m \omega...
  30. 7

    Replacing an operator of an angular momentum for a constant.

    While dealing with a circling particle in an spherical symetric potential our professor said that we can replace an operator of ##z## component of angular momentum ##\hat{L}_z## with the expectation value - he denoted it just ##L_z## - of the angular momentum if ##L_z## is constant. Why is that...
  31. S

    What is the Expected Value Operator and How is it Calculated?

    While reading the article:Law of the unconscious statistician, I came across a line and then a few lines after that, the expected value of a a function g(x) is said to be given by: ∫f(x)g(x)dx. However, if g(x) is not explicitly known, how does one calculate the integtral?
  32. L

    Statistical Operator: Explaining Temperature in Physics

    I have a question about statistical operator. In statistical physics you deal with temperature. So for example ##\hat{\rho}=\frac{1}{Z}e^{-\beta \hat{H}}## where ##\beta=\frac{1}{k_BT}##. In definition there is temperature. And also equivalent definition is ##\hat{\rho}=\sum_i w_i|\psi_i\rangle...
  33. Y

    Stuck on proof irreducible spherical tensor operator

    Hi everyone, I'm stuck on proving that a certain operator is an irreducible spherical tensor operator. These are tensor operators T^{k}_{q} with -k \leq q\leq k satisfying \mathscr{D} T^{k}_{q} \mathscr{D}^{\dagger} = \sum_{q'} \mathscr{D}^{k}_{q' q} T^{k}_{q'} where...
  34. O

    Grad, curl , div operator got any meaning?

    grad, curl , div operator got any meaning?? ∇x F (t) same as the first derivative of F with respect to t , or will get the gradient of F which is normal to the F ? ∇ dot F (t) , will get the scalar value of what?? lets say F is force , then can anyone please give me the meaning of those...
  35. V

    Representation of linear operator using series ?

    representation of linear operator using "series"? I was looking into the progression of quantum states with respect to time. From what I understood the progression of a state ## \left|\psi(t)\right> ## is given by: $$ \left|\psi(t)\right> = U(t)\left|\psi(0)\right> $$ I'm not sure if that's...
  36. Q

    Acceleration operator and the electron in a hydrogen atom

    I am wondering about acceleration in quantum mechanics. Let's consider spherically symmetric potential V(r). From the Heisenberg equation of motion, one finds the time derivative of the momentum operator \dot{\hat{p}}=\frac{i}{\hbar}\left[\hat{H},\hat{p}\right] = -\nabla V, from which we can...
  37. R

    Eigenvalues/Eigenstates of Spin Operator S in xz Plane

    Homework Statement Find the eigenvalues and eigenstates of the spin operator S of an electron in the direction of a unit vector n; assume that n lies in the xz plane. Homework Equations S|m>= h m|m> The Attempt at a Solution This question is from Zettili QM and they have...
  38. Fernando Revilla

    MHB Self-adjoint operator (Bens question at Yahoo Answers)

    Self-adjoint operator (Ben's question at Yahoo! Answers) Here is the question: Here is a link to the question: Self-adjoint and properties? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  39. C

    Total momentum operator in peskin/schroeder

    I'm trying to work out the total momentum operator on page 22 of Peskin/Schroeder for myself, and I'm a little confused about the last few steps. Assuming I went through the first few steps correctly, I've arrive at this expression: $${\bf P}=\frac12\int\frac{d^3p}{(2\pi)^3}{\bf...
  40. 7

    How do we know that wave f. is the eigenfunction of an operator H?

    I am kind of new to this eigenvalue, eigenfunction and operator things, but i have come across this quote many times: First i need some explanation on how do we know this? All i know about operator ##\hat{H}## so far is this equation where ##\langle W \rangle## is an energy expected value...
  41. W

    C/C++ [C++] How Operator Overloading Works

    Hi all; I'm trying to learn about classes and objects, here is a program that demonstrate operator overloading, i cannot understand how it works, when i tried this: //classes #include <iostream> using namespace std; class CVector { public: int x,y; CVector(){}...
  42. S

    C# Programming Operator precedence

    Homework Statement What is the value of x = 5 + (9 * 5) * (3 ^ 3/2 - 20).Homework Equations Operator precedence.The Attempt at a Solution x = 5 + (45) * (3 ^ 3/2 - 20). Is the "^" here the exclusive or operator? In that case wouldn't x have 2 possible values?
  43. M

    Equivalence of differential operator terms in action

    Hi guys, I'm sure I'm being very stupid here but I'm reading through notes which contain various actions for fields, most of which are very similar, however there is some discrepancies with the way differential operators are shown acting on the fields and I can't for the life of me work out...
  44. A

    Spectrum of a linear operator on a Banach space

    I'm trying to understand the spectrum and resolvent of a linear operator on a Banach space in as much generality as I possibly can. It seems that the furthest the concept can be "pulled back" is to a linear operator T: D(T) \to X, where X is a Banach space and D(T)\subseteq X. But here are a...
  45. DrClaude

    Action of a linear operator on vectors

    Not really a homework problem, just doing some self-studying. Homework Statement Let ##| a \rangle## by any vector in an ##N##-dimensional vector space ##\mathcal{V}##, and ##\mathbf{A}## a linear operator on ##\mathcal{V}##. The vectors $$ | a \rangle, \mathbf{A} | a \rangle...
  46. S

    Proof that HK is hermitian operator only if HK=KH

    Let H and K be hermitian operators on vector space U. Show that operator HK is hermitian if and only if HK=KH. I tried some things but I don't know if it is ok. Can somebody please check? I got a hint on this forum that statements type "if only if" require proof in both directions, so here...
  47. ShayanJ

    Understanding the Use of Controlled Not Gates in Quantum Computing

    In an article I'm reading, the author defines an operator as below: \hat{U}_{CNOT}(\theta)=\exp{(-i \theta \hat{U}_{CNOT})}=\hat{1} \cos{\theta}-i \hat{U}_{CNOT} \sin{\theta} Where \hat{U}_{CNOT} is the controlled not gate(http://en.wikipedia.org/wiki/Controlled_NOT_gate). Then the...
  48. F

    Measuring the physical quantity corresponding to an operator.

    Homework Statement Homework Equations N/A The Attempt at a Solution I am having trouble getting my head around these questions, the first part a) wasn't too tricky, I used the fact that eigenfunctions of a Hermitian operator \hat{O} are orthogonal and got my normalisation...
  49. Fernando Revilla

    MHB Amy's question at Yahoo Answers (Self adjoint operator)

    Here is the question: Here is a link to the question: Show a linear transformation is self adjoint? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  50. J.Hong

    Operator algebra of chiral quasi-primary fields

    Studying conformal field theory, I tried to derive general expression for the commutation relations of the modes of two chiral quasi-primary fields. At first, I expressed the modes \phi_{(i)m} and \phi_{(j)n} as contour integrals over each fields, and took commutation relation. I used...
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