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. Msilva

    Finding a matrix representation for operator A

    I need to find a matrix representation for operator A=x\frac{d}{dx} using Legendre polinomials as base. I would use a_{mn}=\int^{-1}_{-1}P_m(x)\,x\frac{d}{dx}\,P_n(x)\,dx, but I have the problem that Legendre polinomials aren't orthonormal \langle P_{i}|P_{l}\rangle=\delta_{il}\frac{2}{2i+1}. I...
  2. Icaro Lorran

    Can the operator Exp[-I*Pi*L_x/h] be faced as parity?

    Homework Statement The problem originally asks to evaluate ##exp(\frac{-i\pi L_x}{h})## in a ket |l,m>. So I am wondering if I can treat the operator as a parity operator or if I really have to expand that exponential, maybe in function of ##L_+## and ##L_-##. 2. The attempt at a solution If...
  3. S

    Interval of convergence of a linear operator

    Homework Statement A function of a hermitian operator H can be written as f(H)=Σ (H)n with n=0 to n=∞. When is (1-H)-1 defined? Homework Equations (1-x)-1 = Σ(-x)n= 1-x+x2-x3+... The Attempt at a Solution (1-H)-1 converges if each element of H converges in this series, that is (1-hi)-1...
  4. F

    Interaction picture - time evolution operator

    Hey all, I got some question referring to the interaction picture. For example: I have the Hamiltonian ##H=sum_k w_k b_k^\dagger b_k + V(t)=H1+V(t)## When I would now have a time evolution operator: ##T exp(-i * int(H+V))##. (where T is the time ordering operator) How can I transform it...
  5. S

    MHB Software for calculating eigenvalues and eigenfunctions of an integral operator

    Hi can someone direct me to a free software to calculate eigenvalues and normalized eigenfunctions of a linear integral operator. I am trying to solve a fredholm integral equation with degenerate kernel using it instead of linear equations thanks sarrah
  6. P

    The physical derivation of annihilation operator?

    From P. Meystre's book elements of quantum optics (Many labels of equations are wrong:H) Page 83, the annihilation operator and creation operator, which are helpful to discuss harmonic oscillator, are defined as ## a=\frac{1}{\sqrt{2\hbar\Omega}}(\Omega q+ip),\\...
  7. S

    MHB Norm of an integral operator

    I have a linear integral operator $K\psi=\int_{a}^{b} \,k(x,s) \psi(s) ds$ $L\psi=\int_{a}^{b} \,l(x,s) \psi(s) ds$ both are continuous I know how to obtain the eigenvalues of each alone. But how can I calculate the eigenvalues of the operator $I-{L}^{-1} K$ or at least the norm...
  8. K

    Eigenfunctions of the angular momentum operator

    Hi everyone, I tried to find the Eigenstate of the angular momentum operator myself, more specifically I tried to find a Function Y_{lm}(\theta,\phi) with L_zY_{lm}=mħY_{lm} and L^2Y_{lm}=l(l+1)ħ^2Y_{lm} where L_z=-iħ\frac{\partial}{\partial \phi} and...
  9. A

    Dispersion operator in quantum mechanics

    I am a beginner in quantum mechanics and I am confused about the operator ΔA defined to be ΔA Ξ A - <A>. Can someone please tell me how to interpret <A>? From what I can understand, <A> is the expectation value and is defined to be <Ψ|A|Ψ>. But that is just a scalar correct? How do subtract a...
  10. J

    Proving Isomorphism of Linear Operator with ||A|| < 1

    Hi, I have some trouble with the following problem: Let E be a Banach space. Let A ∈ L(E), the space of linear operators from E. Show that the linear operator φ: L(E) → L(E) with φ(T) = T + AT is an isomorphism if ||A|| < 1. So the idea here is to use the Neumann series but I can't really...
  11. P

    Derivative operator on both sides

    A basic question, not a homework problem. Say I have the expression: 5x = 10 Can I apply the derivative operator, d/dx, to both sides? d/dx(5x)=d/dx(10) would imply 5=0. I thought you can apply operators to both sides of an equation. Why can't you not do it in this case?
  12. ShayanJ

    Unbounded operator and expansion of commutator

    Consider two self-adjoint operators A and B with commutator [A,B]=C such that [A,C]=0. Now I consider an operator which is a function of A and is defined by the series ## F(A)=\sum_n a_n A^n ## and try to calculate its commutator with B: ## [F(A),B]=[\sum_n a_n A^n,B]= \\ \sum_n a_n...
  13. W

    Differential Linear Operator Problem not making sense

    Homework Statement I think there may be something wrong with a problem I'm doing for homework. The problem is: Solve the IVP with the differential operator method: [D^2 + 5D + 6D], y(0) = 2, y'(0) = \beta > 0 a) Determine the coordinates (t_m,y_m) of the maximum point of the solution as a...
  14. S

    How do you define the Eigenstates for the number operator?

    I was studying quantum states in quantum field theory and I came across the formula for defining eigenstates: |n> = [(a†)n / sqrt(n!)] * |0> However, my book did not actually define ground state |0> (meaning the book did not give some function or numbers or anything like that to define what...
  15. S

    Finding eigenstate for the annhilation operator

    Homework Statement Find the eigenvector of the annhilation operator a. Homework Equations a|n\rangle = \sqrt{n}|{n-1}\rangle The Attempt at a Solution Try to show this for an arbitrary wavefunction: |V\rangle = \sum_{n=1}^\infty c_{n}|n\rangle a|V\rangle = a\sum_{n=1}^\infty c_{n}|n\rangle...
  16. P

    Does the Sup/Inf Operator Apply in Vector/Matrix Conditions?

    Suppose y is a positive vector. Let p and x be two positive matrices with N rows, where ##p_j## and ##x_j## denotes the j:th row in these matrices, so that j = 1,…,N. Does the following hold: \inf_{k=1,...,N} [\sup_{l=1,...,N} [p_k(y-x_k)]] = \inf_{k=1,...,N} [p_k(y-x_k)] where ##p_k(y-x_k)##...
  17. ognik

    MHB Help with Matrix & Operator specifics

    Hi, those who have seen my recent posts will know that I am trying to put together a simple table that I am certain will help me get through a cloud of uncertainty that is proving a major obstacle to my on-going studies. I refer to...
  18. O

    MHB Quasi-nonexpansive operator

    Let (X,d) be a metric space. An operator $T:X\to X$ is said to be quasi nonexpansive if T has at least one fixed point in X and, for each fixed point p, we have $d\left(Tx,p\right)\le d\left(x,p\right)$ (1) And also we give a...
  19. A

    How to get position operator in momentum space?

    Hi, I wish to get position operator in momentum space using Fourier transformation, if I simply start from here, $$ <x>=\int_{-\infty}^{\infty} dx \Psi^* x \Psi $$ I could do the same with the momentum operator, because I had a derivative acting on |psi there, but in this case, How may I get...
  20. ognik

    MHB Matrix & operator summary

    Hi, I feel thoroughly muddled like I am drowning in a soup of terminology and notation, and I have assignment deadlines. So I have tried to compile a rough table that will give me a consistent base which I can use now - and add to going forward; trying also to stick with the usage in my...
  21. H

    Sturm-Liouville operator, operating on what?

    Consider a general Sturm-Liouiville problem ##[p(x) y'(x) ]' + [q(x) + \lambda \omega (x) ] y(x) = 0 ## ##\quad \Leftrightarrow \quad \hat{L} y(x) = \lambda \omega(x) y(x) \quad \text{with} \quad \hat{L} y(x) = -[p(x) y'(x)]' - q(x) y(x) ## where ##p(x), p'(x), q(x), \omega(x)## are...
  22. ognik

    MHB Please check ODE operator solutions

    1) Given $\mathcal{L}u=0$ and $ g\mathcal{L}u$ is self-adjoint, show that for the adjoint operator $ \bar {\mathcal{L}}, \bar{\mathcal{L}}(gu)=0$ Is it enough to say that if self-adjoint, then $ \mathcal{L}= \bar {\mathcal{L}} $. I assume g represents a function of x (so no inner products with...
  23. S

    Confusion about time-ordering operator

    Hi all, I have a severe confusion about the time-ordering operator. It is the best thing ever, I think, since it simplifies many proofs, due to the fact that operators commute (or anti-commute, but let's take bosonic operators for simplicity) under the time-ordering. However, sometimes I...
  24. Crush1986

    Momentum Operator Derivation Questions

    Hello, This is probably a very easy questions about the one-dimension momentum operator derivation. So you take the d<x>/dt to find the "velocity" of the expectation value. At one point in the derivation early on, you bring in the d/dt into the integral of the expectation value. The book I'm...
  25. R

    C/C++ Understanding the Ternary Operator in C++: How and When to Use It?

    What is ternary operator in c++ and its use?
  26. A

    Del operator and wave function

    I've been given the question "What is ∇exp(ip⋅r/ħ) ?" I recognise that this is the del operator acting on a wave function but using the dot product of momentum and position in the wave function is new to me. The dot product is always scalar so I was wondering if it would be correct in writing...
  27. lfqm

    Fixed amplitude of electric field operator in quantum optics

    Hi guys, I'm trying to understand why does the amplitude of the electric field operator in a cavity is fixed at \left ( \displaystyle\frac{\hbar\omega}{\epsilon_{0}V} \right )^\frac{1}{2} Every book I read says it is a normalization factor... but, normalizing an operator?, what is the meaning...
  28. drFredkin

    Are py and pz Compatible Operators? Exploring the Commutation Equation

    Homework Statement py and pz are components of the momentum . Do they compatible operators? Homework Equations compatible operators equation The Attempt at a Solution i think I have to use commutation
  29. duc

    Action of gradient exponential operator

    Homework Statement Find the action of the operator ## e^{\vec{a} . \vec{\nabla}} \big( f(\theta,\phi) . g(r) \big) ## where \nabla is the gradient operator given in spherical coordinates, f and g are respectively scalar functions of the angular part ## ( \theta, \phi) ## and the radial part ##...
  30. L

    Linear Algebra II - Change of Basis

    Homework Statement From Linear Algebra with applications 7th Edition by Keith Nicholson. Chapter 9.2 Example 2. Let T: R3 → R3 be defined by T(a,b,c) = (2a-b,b+c,c-3a). If B0 denotes the standard basis of R3 and B = {(1,1,0),(1,0,1),(0,1,0)}, find an invertible matrix P such that...
  31. FreeBiscuits

    Creation Operator is not a densely defined operator....

    Hi everyone, I am currently preparing myself for my Bachelor thesis in local quantum field theory. I was encouraged by my advisor to read the books of M. Reed and Simon because of my lag of functional analysis experience but I have quite often problems understand the “obvious” conclusions. For...
  32. Andre' Quanta

    The operator X for the position in QFT

    During a course of QFT my teacher said that in this theory is not possible to use the operator X for the position in order to construct with the momentum P and the spin S a set of irreducible operators that charachterize particles, and that we need a different point of wiev: the irreducible...
  33. naima

    Fields as operator valued distributions

    As fields ##\phi ## are ill defined at precise time and position i read that fields have to be smeared. So we have test functions f in bounded regions in space time. We have a Hilbert space and ##\phi (f) ## is an operator which acts on H. Maybe we can retrieved the usual wave function when it...
  34. H

    A question about commutaiton between operator

    Hamiltonian operator commutes with the linear momentum operator and for a particle in the box its wavefunction is Nsin(nπx/a) , N is the normalization constant But I found this wavefuntion is not a eigenfuntion for the momentum operator, why? Isn't the two operators commut with each other?
  35. Samuel Williams

    Linear Algebra - Transformation / operator

    Homework Statement Let T:V→V be a linear operator on a vector space V over C: (a) Give an example of an operator T:C^2→C^2 such that R(T)∩N(T)={0} but T is not a projection (b) Find a formula for a linear operator T:C^3→C^3 over C such that T is a projection with R(T)=span{(1,1,1)} and...
  36. T

    Error propagation when using modulus operator

    Sorry if the answer is very simple, but I just had a question regarding error propagation when using a modulo operator in intermediate steps. For example, I have ## \theta = arctan(\frac {A}{B}) ## and then I do ## \theta ## % ##2\pi## (modulo ##2\pi##). This gives me an answer between ##...
  37. JonnyMaddox

    Hamilton Operator for particle on a circle -- Matrix representation....

    Hey JO. The Hamiltonian is: H= \frac{p_{x}^{2}+p_{y}^{2}}{2m} In quantum Mechanics: \hat{H}=-\frac{\hbar^{2}}{2m}(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial x^{2}}) In polar coordinates: \hat{H}=-\frac{\hbar^{2}}{2m}( \frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}...
  38. D

    Total angular momentum operator for a superposition

    Hi all, Quick quantum question. I understand the total angular momentum operation \hat{L}^2 \psi _{nlm} = \hbar\ell(\ell + 1) \psi _{nlm} which means the total angular momentum is L = \sqrt{\hbar\ell(\ell + 1)} But how about applying this to an arbitrary superposition of eigenstates such as...
  39. jk22

    Measurement operator for Chsh experiment

    The experiment is described at p26 of http://arxiv.org/abs/quant-ph/0402001 In this experiment we see that we sum measurement results and not measure the sum. Is then the quantum measurement operator not : $$S=A\otimes B\otimes\mathbb{1_{64}}-\mathbb{1_{4}}\otimes A\otimes...
  40. fricke

    Do momentum and kinetic energy operators always commute for a free particle?

    For particle in the box wave function, it is the eigenfunction of kinetic energy operator but not the eigenfunction of momentum operator. So, do these two operators commute? (or it has nothing to do with commutator stuff?) How about for free particle? For free particle, the wave function is...
  41. dhalilsim

    D'Alembert operator is commute covariant derivative?

    For example: [itex] D_α D_β D^β F_ab= D_β D^β D_α F_ab is true or not? Are there any books sources?
  42. C

    Tensor Products of Operator

    In a multi-particle system, the total state is defined by the tensor product of the individual states. Why is it the case that operators, say position of 2 particles, is of the form X⊗I + I⊗Y and not X⊗Y where I are the identities for the respective spaces and X and Y are the position operators...
  43. P

    Momentum Operator In Curved Spacetime

    Hello, I'm sorry if this question sounds silly, but in QM the Momentum Operator is ##{\hat{p}}=-i{\hbar}{\nabla}## . In Relativistic QM in Flat Space, this operator can be written ##{\hat{P}_{\mu}}=-i{\hbar}{\partial}_{\mu}## . Would it be correct, then, to say that in curved spacetime the...
  44. Fantini

    MHB Calculating the effect of an operator on an arbitrary state

    Hello. I need help with the following: Suppose a basis set of states $\varphi_i$. Calculate the effect of the operator $\widehat{R} \equiv \Pi_i \left( \widehat{Q} -q_i \right)$ on an arbitrary state $\Psi$, assuming that the equation $\widehat{Q} \varphi_i = q_i \varphi_i$ is satisfied. I'm...
  45. S

    MHB Operator Form for Volterra Linear Integral Equations

    Hi I know Fredholm linear integral equations i.e. $y(x)=f(x)+\lambda\int_{a}^{b} \,k(x,s)y(s)ds$ can be written compactly in the operator form as $y=f+\lambda Ky$ this facilitates many proofs. Usually there is a condition on $\lambda$ for the convergence of say successive approximations method...
  46. P

    Matrix form of Density Operator

    Hi All, I have spent hours trying to understand the matrix form of Density Operator. But, I fail. Please see page 2 of the attached file. (from the book "Quantum Mechanics - The Theoretical Minimum" page 199). Most appreciated if someone could enlighten me this. Many thanks in advance. Peter Yu
  47. I

    Eigenvalue of lowering operator

    How to prove that eigenvalue of lowering operator is zero?
  48. B

    QM: Expectation value of raising and lowering operator

    Homework Statement Using J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle Derive that : \langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)] Homework Equations J_- = J_x - iJ_y J_+ = J_x + iJ_y The...
  49. I

    Quantum Mechanics - Lowering Operator

    Homework Statement let A be a lowering operator. Homework Equations Show that A is a derivative respects to raising operator, A†, A=d/dA† The Attempt at a Solution I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can...
  50. D

    Displacement Operator: Exponential of Parameter & Vector?

    Hi . I've just encountered something called the displacement operator which is the exponential of a parameter multiplied by a vector but I thought the argument of an exponential had to be a scalar. Is this not true ?
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