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.

View More On Wikipedia.org
Recent contents Top users
  1. Danny Boy

    A Fundamental Theorem of Quantum Measurements

    The Fundamental Theorem of Quantum Measurements (see page 25 of these PDF notes) is given as follows: Every set of operators ##\{A_n \}_n## where ##n=1,...,N## that satisfies ##\sum_{n}A_{n}A^{\dagger}_{n} = I##, describes a possible measurement on a quantum system, where the measurement has...
  2. N

    Understanding the Del Operator in Vector Calculus

    F is a vector from origin to point (x,y,z) and û is a unit vector. how to prove? (û⋅∇)F=û only tried expanding but it's going nowhere
  3. SemM

    A How to find the domain of functions of an operator

    Hi, I have a strange nonlinear operator which yields non-Hermitian solutions when treated in a simple ODE, ##H\Psi##=0. It appears from a paper by Dr Du in a different posting, that an operator can be non-self-adjoint in one domain, but be self-adjoint in another domain defined by the interval...
  4. W

    How Does the Hamiltonian Affect the Time Evolution of a Qubit's Density Matrix?

    Homework Statement In a computational basis, a qubit has density matrix ## \rho = \left( \begin{array}{ccc} \frac{1}{2} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{2} \\ \end{array} \right)## At t=0. Find the time dependence of ##\rho## when the Hamiltonian is given by ##AI+BY##, ##A## and ##B##...
  5. W

    I Checking the effect an operator has on a state

    I've been reading these notes: http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere-rotations.pdf And on slide 11, to check what the rotation operator is doing, the state is pre- and post-multiplied by the operator so that the calculation performed is ##\rho' = R\rho R^{\dagger}## Why...
  6. SemM

    A The meaning of the commutator for two operators

    Hi, what is the true meaning and usefulness of the commutator in: \begin{equation} [T, T'] \ne 0 \end{equation} and how can it be used to solve a parent ODE? In a book on QM, the commutator of the two operators of the Schrödinger eqn, after factorization, is 1, and this commutation relation...
  7. M

    A Differential operator, inverse thereof

    Hi PF! I'm reviewing a text and the author writes where ##g## is an arbitrary function and ##B## is a differential operator. ##Bo## is a parameter. Then the author states the inverse of ##B## is where ##G## is the Green's function of ##B##. Can someone explain how we know this?
  8. S

    B Surjective/injective operators

    Hi, I found in Kreyszig that if for any ##x_1\ and\ x_2\ \in \mathscr{D}(T)## then an injective operator gives: ##x_1 \ne x_2 \rightarrow Tx_1 \ne Tx_2 ## and ##x_1 = x_2 \rightarrow Tx_1 = Tx_2 ##If one has an operator T, is there an inequality or equality one can deduce from this, in...
  9. hideelo

    A Extending a linear representation by an anti-linear operator

    When we consider representation of the O(1,3) we divide the representation up into the connected part of SO(1,3) which we then extend by adding in the discrete transformations P and T. However, while (as far as I know) we always demand that the SO(1,3) and P be represented as linear operators...
  10. S

    A Are bounded operators bounded indepedently on the function?

    Hi thanks to George, I found the following criteria for boundedness: \begin{equation} \frac{||Bf(x)||}{||f(x)||} < ||Bf(x)|| \end{equation} If one takes f(x) = x, and consider B = (h/id/dx - g), where g is some constant, then B is bounded in the interval 0-##\pi##. However, given that I...
  11. S

    A Is this operator bounded or unbounded?

    Hi, I have an operator which does not obey the following condition for boundedness: \begin{equation*} ||H\ x|| \leqslant c||x||\ \ \ \ \ \ \ \ c \in \mathscr{D} \end{equation*} where c is a real number in the Domain D of the operator H. However, this operator is also not really unbounded...
  12. S

    A Operator mapping in Hilbert space

    Hi, I have an operator given by the expression: L = (d/dx +ia) where a is some constant. Applying this on x, gives a result in the subspace C and R. Can I safely conclude that the operator L can be given as: \begin{equation} L: \mathcal{H} \rightarrow \mathcal{H} \end{equation} where H is...
  13. S

    A Can imaginary position operators explain real eigenvalues in quantum mechanics?

    Hello, some operators seem to "add up" and give real eigenvalues only if they are applied on the imaginary position, ix, rather than the normal position operator, x, in the integral : \begin{equation} \langle Bx, x\rangle \end{equation} when replaced by:\begin{equation} \langle Bix...
  14. tarkin

    Does this operator commute with the Hamiltonian operator?

    Homework Statement Show that the mean value of a time-independent operator over an energy eigenstate is constant in time. Homework Equations Ehrenfest theorem The Attempt at a Solution I get most of it, I'm just wondering how to say/show that this operator will commute with the Hamiltonian...
  15. PsychonautQQ

    A What Is the Role of the Boundary Operator in p-Chains?

    Can someone help me to understand what the boundary operator on a p-chain is doing exactly? Or boundary operators in general? I really need to develop a better intuition on the matter.
  16. A

    A Transformation of position operator under rotations

    In the momentum representation, the position operator acts on the wavefunction as 1) ##X_i = i\frac{\partial}{\partial p_i}## Now we want under rotations $U(R)$ the position operator to transform as ##U(R)^{-1}\mathbf{X}U(R) = R\mathbf{X}## How does one show that the position operator as...
  17. M

    MHB Adjoint operator and orthogonal projection

    Hello, I want to show $ T^{*}(Pr_{C}(y)) = Pr_{T^{*}(C)}(T^{*}y)$ where $T \in B(H)$ and $TT^{*}=I$ , $H$ is Hilbert space and $C$ is a closed convex non empty set. but i don't know how to start, or what tricks needed to solve this type of problems. also i want know how to construct $T$ to...
  18. M

    I Measurement Values for z-component of Angular Momentum

    Given a wave function $$\Psi(r,\theta,\phi)=f(r)\sin^2(\theta)(2\cos^2(\phi)-1-2i*\sin(\phi)\cos(\phi))$$ we are trying to find what a measurement of angular momentum of a particle in such wave function would yield. Attempts were made using the integral formula for the Expectation Value over a...
  19. T

    I Anihilation operator expression: is there a typo here?

    For those experienced with this stuff, Weinberg argues (Weinberg, QFT, Volume 1) that an expression for the anihilation operator acting on a state vector when all particles are either all bosons or all fermions is $$a(q) \Phi_{q_1 q_2 ... q_N} = \sum_{r=1}^{N} ( \pm 1)^{r+1} \delta (q - q_r)...
  20. binbagsss

    String Theory, Number Operator , Mass of States

    Homework Statement I have the following definition of the space-time coordinates Homework Equations Working in a certain gauge we can also do: From which we can find: Where ##N_{lc} ## sums over the transverse oscillation modes only. The Attempt at a Solution [/B] MY QUESTION: I...
  21. A

    Eigenvalue of Exchange Operator in Hartree-Fock: 2e-

    Homework Statement Homework Equations The Attempt at a Solution [/B] ##\hat{S}_1.\hat{S}_2 = (S(S+1) - S_1(S_1 + 1) - S_2(S_2 + 1))/2## therefore singlet: ##\psi_s = \frac{\phi_a (1)\phi_b(2)(\alpha(1)\beta(2) - \alpha(2)\beta(1))}{\sqrt(2)}## So for singlet, ##\mathcal{V} = -\frac{K...
  22. A

    A Qualitative Explanation of Density Operator

    Hey all! I am prepping myself for a quantum course next semester at the graduate level. I am currently reading through the Cohen-Tannoudji Quantum Mechanics textbook. I have reached a section on the density operator and am confused about the general concept of the operator. My confusion stems...
  23. W

    I Hermitian Operators: Referencing Griffiths

    I have a few issues with understanding a section of Griffiths QM regarding Hermitian Operators and would greatly appreciate some help. It was first stated that, ##\langle Q \rangle = \int \Psi ^* \hat{Q} \Psi dx = \langle \Psi | \hat{Q} \Psi \rangle## and because expectation values are real...
  24. N

    Lowering Operator Simple Harmonic Oscillator n=3

    Homework Statement Show that application of the lowering Operator A- to the n=3 harmonic oscillator wavefunction leads to the result predicted by Equation (5.6.22). Homework Equations Equation (5.6.22): A-Ψn = -iΨn-1√n The Attempt at a Solution I began by saying what the answer should end...
  25. binbagsss

    Green's method- linear differential operator

    Homework Statement 2. Homework Equations 3. The Attempt at a Solution [/B]- So with the (from what i interpret of the notes this is needed) the same boundary conditions when time is fixed, we can relate the 'fundamental problem'- the initial condition ##t=0## given by a delta...
  26. T

    Solving Hermitical Operator Homework Questions

    Homework Statement Homework Equations The Attempt at a Solution Should I do this or I can just simplify it like this ? And also what would the integral of f(r) equal to at -inf<r<0?
  27. Z

    Rotation Operator: Interaction between Two-Level Atom in {|g>, |e>} Basis

    Hi, I'm working on the interaction between a two level atom (|g>, |e>) In my exercise we have to use the rotation operator : R(t)=exp[i(σz+1)ωt/2] with σz the pauli matrix which is in the {|g>,|e>} basis : (1 0) (0 -1) If i want to represent my rotation operator in the {|g>,|e>} basis. Then...
  28. M

    A Can the convolution operator be diagonalized using the Fourier transform?

    Hi there, I am also familiar with Hilbert spaces and Functional Analysis and I find your question very interesting. I agree that the Fourier transform is a powerful tool for analyzing LTI systems and diagonalizing the convolution operator. As for your question about whether the same can be...
  29. Z

    I Why the velocity operator commutes with position (Dirac equation)

    ##\hat{v}_i=c\hat{\alpha}_i## commute with ##\hat{x}_i##, ##E^2={p_1}^2c^2+{p_2}^2c^2+{p_3}^2c^2+m^2c^4## But in classical picture,the poisson braket...
  30. arpon

    I Experiment: Spin Rotation Operator

    How do we experimentally apply the operator ## \exp{\left(-i\phi\frac{ S_z}{\hbar}\right)}## on a quantum mechanical system? (Here ##S_z## is the spin angular momentum operator along the z-axis) For example, on a beam of electrons?
  31. S

    I Is the Laplacian Operator Different in Radial Coordinates?

    Hi, I have that the Laplacian operator for three dimensions of two orders, \nabla ^2 is: 1/r* d^2/dr^2 (r) + 1/r^2( 1/sin phi d/d phi sin phi d/d phi + 1/sin^2 phi * d^2/d theta^2) Can this operator be used for a radial system, where r and phi are still valid, but theta absent, by setting...
  32. arpon

    I Energy operator and the Hamiltonian operator: Are they same?

  33. X

    I Pauli Spin Operator Eigenvalues For Two Electron System

    I'm studying for a qualifying exam and I see something very strange in the answer key to one of the problems from a past qualifying exam. It appears the sigma^2 for a two electron system has eigenvalues according to the picture below of 4s(s+1) while from my understand of Sakurai it should have...
  34. davidge

    I Spherical Harmonics from operator analysis

    I found an interesting thing when trying to derive the spherical harmonics of QM by doing what I describe below. I would like to know whether this can be considered a valid derivation or it was just a coincidence getting the correct result at the end. Starting making a Fundamental Assumption...
  35. saadhusayn

    A Confusion regarding the $\partial_{\mu}$ operator

    I'm trying to derive the Klein Gordon equation from the Lagrangian: $$ \mathcal{L} = \frac{1}{2}(\partial_{\mu} \phi)^2 - \frac{1}{2}m^2 \phi^2$$ $$\partial_{\mu}\Bigg(\frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)}\Bigg) = \partial_{t}\Bigg(\frac{\partial \mathcal{L}}{\partial...
  36. J

    A Discrete measurement operator definition

    Consider the Gaussian position measurement operators $$\hat{A}_y = \int_{-\infty}^{\infty}ae^{\frac{-(x-y)^2}{2c^2}}|x \rangle \langle x|dx$$ where ##|x \rangle## are position eigenstates. I can show that this satisfies the required property of measurement operators...
  37. P

    A If [A,B]=0, are they both functions of some other operator?

    In other words, if we are told that A and B commute, then does that mean that there exists some other operator X such that A and B can both be written as power series of X? My instinct is yes but I haven't been able to prove it.
  38. F

    I Does a field operator always commute with itself?

    In quantum field theory (QFT), the requirement that physics is always causal is implemented by the microcausality condition on commutators of observables ##\mathcal{O}(x)## and ##\mathcal{O}'(y)##, $$\left[\mathcal{O}(x),\mathcal{O}'(y)\right]=0$$ for spacelike separations. Intuitively, I've...
  39. Wrichik Basu

    B QFT for Beginners: Operators & Their Physical Significance

    I'm a beginner in QFT, starting out with D. J. Griffiths' book in this topic. I have a question on the operators used in QM. What are operators? What is the physical significance of operators? I can understand that ##\frac {d}{dt}## to be an operator, but how can there be a total energy...
  40. binbagsss

    Modular forms, Hecke Operator, translation property

    Homework Statement I am trying to follow the attached solution to show that ##T_{p}f(\tau+1)=T_pf(\tau)## Where ##T_p f(\tau) p^{k-1} f(p\tau) + \frac{1}{p} \sum\limits^{p-1}_{j=0}f(\frac{\tau+j}{p})## Where ##M_k(\Gamma) ## denotes the space of modular forms of weight ##k## (So we know that...
  41. DeathbyGreen

    I Creation operator and Wavefunction relationship

    Hello, I've noticed that some professors will jump between second quantized creation/annihilation operators and wavefunctions rather easily. For instance \Psi_k \propto c_k + ac_k^{\dagger} with "a" some constant (complex possibly). I'm fairly familiar with the second quantized notation, and...
  42. binbagsss

    Quantum mechanics Hermitian operator

    Homework Statement I have the criteria: ## <p'| L_{n} |p>=0 ##,for all ##n \in Z ## ##L## some operator and ## |p> ##, ## |p'> ##some different physical states I want to show that given ## L^{+}=L_{-n} ## this criteria reduces to only needing to show that: ##L_n |p>=0 ## for ##n>0 ##...
  43. L

    What Is the Dimension of Eigenvectors for Operators in Quantum Mechanics?

    Homework Statement ##H=\frac{J}{4}\sum_{i=1}^2 \sigma_i^x \sigma_{i+1}^x## Homework Equations ##\sigma^x ## is Pauli matrix and ##J## is number.[/B]The Attempt at a Solution For ##i=1## to ##3## what is dimension of eigen vector? I think it is ##8##. Because it is like that I have tri sites...
  44. Adgorn

    Proof regarding the image and kernel of a normal operator

    Homework Statement Show that if T is normal, then T and T* have the same kernel and the same image. Homework Equations N/A The Attempt at a Solution At first I tried proving that Ker T ⊆ Ker T* and Ker T* ⊆ Ker, thus proving Ker T = Ker T* and doing the same thing with I am T, but could not...
  45. Adgorn

    Proving the square root of a positive operator is unique

    Homework Statement The problem relates to a proof of a previous statement, so I shall present it first: "Suppose P is a self-adjoint operator on an inner product space V and ##\langle P(u),u \rangle## ≥ 0 for every u ∈ V, prove P=T2 for some self-adjoint operator T. Because P is self-adjoint...
  46. Danny Boy

    A Forming a unitary operator from measurement operators

    If we consider a measurement of a two level quantum system made by using a probe system followed then by a von Neumann measurement on the probe, how could we determine the unitary operator that must be applied to this system (and probe) to accomplish the given measurement operators.
  47. C

    I Does Bra-Ket Notation Clarify the Gradient Operator in Quantum Mechanics?

    I'm trying to understand gradient as an operator in Bra-Ket notation, does the following make sense? <ψ|∇R |ψ> = 1/R where ∇R is the gradient operator. I mean do the ψ simply fall off in this case? Equally would it make any sense to use R as the wave function? <R|∇R |R> = 1/R
  48. redtree

    I Green's function and the evolution operator

    The Green's function is defined as follows, where ##\hat{L}_{\textbf{r}}## is a differential operator: \begin{equation} \begin{split} \hat{L}_{\textbf{r}} \hat{G}(\textbf{r},\textbf{r}_0)&=\delta(\textbf{r}-\textbf{r}_0) \end{split} \end{equation}However, I have seen the following...
  49. T

    I Question regarding the charge conjugation operator

    So I am aware the charge conjugation operator changes the sign of all internal quantum numbers. But I was wondering how it acts on a state such as ## \left|\pi^{+} \pi^{-} \right>## when the individual ##\pi's## are not eigenstates of C. I believe the combination of the ##\pi's## has eigenvalue...
  50. B

    I Angular momentum operator commutation relation

    I am reading a proof of why \left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z Given a wavefunction \psi, \hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...
Back
Top