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. Jason Bennett

    Lorentz algebra elements in an operator representation

    1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: Please see image [2] below. I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...
  2. M

    Momentum Operator for the real scalar field

    I think the solution to this problem is a straightforward calculation and I think I was able to make reasonable progress, but I'm not sure how to finish this... $$\begin{align*} \vec{P}&=-\int dx^3 \pi \nabla \phi\\ &= -\int\int\int dx^3\frac{dp^3}{(2\pi)^3 2e(p)} \frac{du^3}{(2\pi)^3}...
  3. Q

    A Bosonic Field operator in this exponential

    If i had a bosonic field ##\phi(x)## and I took the exponential in the following way to get the operator $$W=e^{\imath f \phi(x)}$$ where ##f## is a parameter what effect would this have when acting on the vacuum ##|0\rangle##? Is it analogous to the space translation operator? Will it transform...
  4. Luke Tan

    Solving Creation Operator Equation: Find Error in Calculation

    I have written the equation, with an unknown constant $$a^\dagger \lvert n\rangle = C_{n+1}\lvert n+1 \rangle$$ I then take the adjoint to get $$\langle n \rvert a = \langle n+1 \rvert C_{n+1}^\text{*}$$ I then multiply them to get $$\langle n \rvert aa^\dagger \lvert n \rangle = \langle n+1...
  5. Abhishek11235

    I Condition for delta operator and total time differential to commute

    While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...
  6. schwarzg

    A Simultanious eigenstate of Hubbard Hamiltonian and Spin operator in tw

    Please see this page and give me an advice. https://physics.stackexchange.com/questions/499269/simultanious-eigenstate-of-hubbard-hamiltonian-and-spin-operator-in-two-site-mod Known fact 1. If two operators ##A## and ##B## commute, ##[A,B]=0##, they have simultaneous eigenstates. That means...
  7. mesa

    What are the most important parts of the job for a reactor operator?

    Hey guys, what would you say are the most important aspects of the job of being a nuclear reactor operator?
  8. K

    I Annihilation/creation operator question

    I'm just curious what physical processes these operators represent. Since particles are created/destroyed in pairs, do they have to applied twice to describe an actual process?
  9. J

    I If T^2 = T, where T is a linear operator on V, T=I or T=0?

    I can't think of a counterexample.
  10. Charles Link

    Maxwell’s Equations in Magnetostatics and Solving with the Curl Operator

    Continue reading...
  11. adosar

    I Momentum operator in quantum mechanics

    The momentum operator for one spation dimension is -iħd/dx (which isn't a vector operator) but for 3 spatial dimensions is -iħ∇ which is a vector operator. So is it a vector or a scalar operator ?
  12. J

    I Matrix Representation of an Operator (from Sakurai)

    Look, I am sorry for not being able to post any LaTeX. But I am stuck at a place where I feel I should not be stuck. I can not figure out how to correctly do this. I can't seem to recreate the Pauli matrices with that form using the 3 2-dimensional bases representing x, y, and z spin up/down...
  13. N

    Explain this line "thread& operator = (const thread&) = delete;"

    here delete, thread, and const are obviously keywords, I saw this when the system libraries threw a breakpoint, not my code, I'm new at this and that line of code makes very little sense to me, afaik, (const thread&) should be a conversion operator, how can you set an operator to a statement...
  14. E

    Expectation value of operators and squeezing in the even cat state

    I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25. How do I derive the given equations?
  15. M

    I Finding the Matrix O for a 4x4 Operator Acting on a 4x1 Vector

    I have a 4x4 operator O. I apply it on a 4x1 vector A. Let's say A =[0.7; 0.4 ; 0.4; 0.3]. When O acts on A, I get B. Let's say B=[0.74 ; 0.56; 0.08 ; 0.36]. The problem is I don't know how to find O. Can you please help me. My basis are [1 ; 0 ; 0; 0], [0;1 ; 0 ;0] ... and so on. Thanks...
  16. P

    A Dipole moment operator and perterbations

    I am reading a PHD thesis online "A controlled quantum system of individual neutral atom" by Stefan Kuhr. In it on pg46, he has a Hamiltonian I am also reading a book by L. Allen "optical resonance and two level atoms" in it on page 34 he starts with a Hamiltonian where the perturbation is...
  17. QuasarBoy543298

    I Applying an observable operator on the current state

    hey :) assume I have an operator A with |ai> eigenstates and matching ai eigenvalues, and assume my system is in state |Ψ> = Σci|ai> I know that applying the measurement that corresponds to A will collapse the system into one of the |ai>'s with probability |<Ψ|ai>|2. with that being...
  18. lastItem

    I Calculating Momentum Operator Matrix Elements from <φ|dH/dkx|ψ>

    Is there a relationship between the momentum operator matrix elements and the following: <φ|dH/dkx|ψ> where kx is the Bloch wave number such that if I have the latter calculated for the x direction as a matrix, I can get the momentum operator matrix elements from it?
  19. M

    A Understanding the Spectrum of a Linear Operator

    Hi PF! What is meant by the spectrum of a linear operator ##A##? I read somewhere that if ##0## belongs in the spectrum, then ##A## is not invertible. Can anyone finesse this for me? I read the wikipedia page, but this was tough for me to understand. Perhaps illustrating with a simple example...
  20. M

    Proving No Eigenvalues Exist for an Operator on a Continuous Function Space

    Eigenvalues ##\lambda## for some operator ##A## satisfy ##A f(x) = \lambda f(x)##. Then $$ Af(x) = \lambda f(x) \implies\\ xf(x) = \lambda f(x)\implies\\ (\lambda-x)f(x) = 0.$$ How do I then show that no eigenvalues exist? Seems obvious one doesn't exists since ##\lambda-x \neq 0## for all...
  21. J

    A Which operator corresponds to the Green function in QFT?

    The Feynman propagator: $$D_{F}(x,y) = <0|T\{\phi_{0}(x) \phi_{0}(y)\}|0> $$ is the Green's function of the operator (except maybe for a constant): $$ (\Box + m^2)$$ In other words: $$ (\Box + m^2) D_{F}(x,y) = - i \hbar \delta^{4}(x-y)$$ My question is: Which is the operator that...
  22. A

    I Which operator for reflection in quantum mechanics?

    Hello, I know we have the parity operator for inversion in quantum mechanics and for rotations we have the exponentials of the angular momentum/spin operators. But what if I want to write the operator that represent a reflection for example just switching y to -y, the matrix in real space...
  23. Mutatis

    Show the formula which connects the adjoint representations

    That's my attempting: first I've wrote ##e## in terms of the power series, but then I don't how to get further than this $$ \sum_{n=0}^\infty (-1)^n \frac {Â^n} {n!} \hat B \sum_{n=0}^\infty \frac {Â^n} {n!} = \sum_{n=0}^\infty (-1)^n \frac {Â^2n} {\left( n! \right) ^2} $$. I've alread tried to...
  24. A

    I Is the Chirality Projection Operator Misused in This Scenario?

    Hello everybody! I have a doubt in using the chiral projection operators. In principle, it should be ##P_L \psi = \psi_L##. $$ P_L = \frac{1-\gamma^5}{2} = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} $$ If I consider ##\psi = \begin{pmatrix}...
  25. L

    I Definition of the potential energy operator

    In quantum mechanics, I can write the hamiltonian as ##\hat{H} = \hat{p}^{2}/2m + \hat{V}##. I am confusing with the definition of the operator ##\hat{V}##, who represents the potential energy. If the potential energy depend only on the position, is it correct write ##\hat{V} = V(\hat{x})##...
  26. chmodfree

    To prove that a coherent state is an eigenstate of the annihilation operator

    The definition of coherent state $$|\phi\rangle =exp(\sum_{i}\phi_i \hat{a}^\dagger_i)|0\rangle $$ How can I show that the state is eigenstate of annihilation operator a? i.e. $$\hat{a}_i|\phi\rangle=\phi_i|\phi\rangle$$
  27. M

    Show operator is compact/symmetric

    1) To show that ##K## is compact let ##\{ f_{n} \}_{n=1}^{\infty}## be a bounded sequence in ##L^{2}[0,1]## with ##\|f_{n}\| \le M##. For every ##\epsilon > 0##, there exists ##\delta > 0## such that ##|k(x,y)-k(x',y')| < \epsilon## whenever ##|x-x'|+|y-y'| < \delta##. Therefore, ##\{ Kf_{n}\}##...
  28. jdou86

    I Confusion about the Concept of Operators

    Dear all, I've been reading and got confused of the concept below have two questions question 1) For <ψ|HA|ψ> = <Hψ|A|ψ>, why does the Hamiltonian operator acting on the bra state and <ψ|AH|ψ> in this configuration it will act on the ket state? question 2) what does it mean for H|ψ> = |Hψ>...
  29. M

    Solving the Minimization Operator Problem

    Homework Statement Given a Hilbert space $$V = \left\{ f\in L_2[0,1] | \int_0^1 f(x)\, dx = 0\right\},B(f,g) = \langle f,g\rangle,l(f) = \int_0^1 x f(x) \, dx$$ find the minimum of $$B(u,u)+2l(u)$$. Homework Equations In my text I found a variational theorem stating this minimization problem...
  30. K

    Tensor Force Operator Between Nucleons: Spin & Position

    Homework Statement The tensor force operator between 2 nucleons is defined as ##S_{12}=3\sigma_1\cdot r\sigma_2\cdot r - \sigma_1\cdot \sigma_2##. Where r is the distance between the nucleons and ##\sigma_1##and ##\sigma_2## are the Pauli matrices acting on each of the 2 nucleons. Rewrite...
  31. C

    A D'Alembert operator in GR/DG

    Was not sure weather to post, this here or in differential geometry, but is related to a GR course, so... I am having some trouble reproducing a result, I think it is mainly down to being very new to tensor notation and operations. But, given the metric ##ds^2 = -dudv + \frac{(v-u)^2}{4}...
  32. Math Amateur

    I Tensors & the Alternation Operator .... Browder, Propn 12.25

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 12: Multilinear Algebra ... ... I need some help in order to fully understand the proof of Proposition 12.2 on pages 277 - 278 ... ...Proposition 12.2 and its proof read as...
  33. D

    I Vacuum projection operator and normal ordering

    I've been reading this book, in which the author expresses the vacuum projection operator ##\vert 0\rangle\langle 0\vert## in terms of the number operator ##\hat{N}=\hat{a}^{\dagger}\hat{a}##, where ##\hat{a}^{\dagger}## and ##\hat{a}## are the usual creation and annihilation operators...
  34. DuckAmuck

    B Question about how the nabla interacts with wave functions

    Is the following true? ψ*∇^2 ψ = ∇ψ*⋅∇ψ It seems like it should be since you can change the direction of operators.
  35. D

    I Parity operator and a free particle on a circle

    Hi. I have just looked at a question concerning a free particle on a circle with ψ(0) = ψ(L). The question asks to find a self-adjoint operator that commutes with H but not p. Because H commutes with p , i assumed there was no such operator. The answer given , was the parity operator. It acts...
  36. lalo_u

    Charge operator applied to matrix multiplets

    In the context of SM (##SU(3)_C\otimes SU(2)_L\otimes U(1)_Y##) the charge operator is ##Q_{SM} = T_3 + \frac{Y}{2}\mathbb{I}_2## and gives us the fermions charges. Here ##T_3=\frac{1}{2}\sigma_3## is the third ##SU(2)## generator. For example, assuming ##Y=-1## for the left lepton doublet...
  37. CptXray

    The operator of a distribution

    Homework Statement Let ##T## be a distribution in ##\mathcal{D}(\mathbb{R}^2)## such that: $$T(\phi) = \int_{0}^{1}dr \int_{0}^{\pi} \phi(r, \Phi)d\Phi$$ $$\phi \in \mathcal{D}(\mathbb{R}^2)$$ calculate ##r \frac{\partial{}}{\partial{r}} \frac{\partial{}}{\partial{\Phi}}T##. Homework...
  38. A

    Liouville operator in Robertson Walker metric

    Homework Statement I'd like to calculate the form of Liouville operator in a Robertson Walker metric. Homework Equations The general form is $$ \mathbb{L} = \dfrac{\text{d} x^\mu}{\text{d} \lambda} \dfrac{\partial}{\partial x^\mu} - \Gamma^{\mu}_{\nu \rho} p^{\nu} p^{\rho}...
  39. LarryS

    I Stationary states vs. the unitary time evolution operator

    In QM, states evolve in time by action of the Time Evolution Unitary Operator, U(t,t0). Without the action of this operator, states do not move forward in time. Yet even stationary states, like an eigenstate of energy, still contain a time variable – they oscillate in time at a fixed...
  40. E

    I Inner product of a vector with an operator

    So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ | \hat A ψ \rangle = \langle ψ | \hat A ^* \hat A | ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counter-intuitive...
  41. jishnu

    Doubt regarding a basic Python operator

    Hi everyone, I am beginner in python programming. So many doubts are being generted in the learing process Can anyone please explain me how the bitwise NOT (~) operator actually works on values. I have attached a screen short of my textbook (unofficial) with this post and I am confused how...
  42. A

    A Representing harmonic oscillator potential operator in. Cartesian basis

    My question is given an orthonormal basis having the basis elements Ψ's ,matrix representation of an operator A will be [ΨiIAIΨj] where i denotes the corresponding row and j the corresponding coloumn. Similarly if given two dimensional harmonic oscillator potential operator .5kx2+.5ky2 where x...
  43. M

    Quantum Teleportation Homework: Deriving EPR Pair & Measuring Spin 1/2 Particles

    Homework Statement This isn't exactly a problem but rather a problem in understanding the derivation of the phenomenon, or more precisely, one step in the derivation. In the following we will consider the EPR pair of two spin ##1/2## particles, where the state can be written as $$ \vert...
  44. E

    Divergence operator for multi-dimensional neutron diffusion

    Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...
  45. S

    Finding the adjoint operator

    1. The problem statement, all variables and given/k Find the adjoint operator L* to the first order differential operator L = curl defined to domain omega. The full problem is attached. Homework EquationsThe Attempt at a Solution I've checked online. I am getting two different answers. Is the...
  46. D

    Number operator on wavefunction

    Homework Statement Consider the state $$\psi_\alpha = Ne^{\alpha \hat a^\dagger}\phi_0, $$ where ##\alpha## can be complex, and ##N = e^{-\frac{1}{2}|\alpha|^2}## normalizes ##\psi_\alpha##. Find ##\hat N \psi_\alpha##. Homework Equations $$\hat N = \hat a^\dagger \hat a$$ $$\hat a\phi_n =...
  47. S

    Prove Sturm-Liouville differential operator is self adjoint.

    Homework Statement Prove Sturm-Liouville differential operator is self adjoint when subjected to Dirichlet, Neumann, or mixed boundary conditions. Homework Equations l = -(d/dx)[p(x)(d/dx)] + q(x) The Attempt at a Solution I have no idea. If someone can give me a place to start that would...
  48. John Greger

    I Obtain simultaneous eigenfunctions?

    Let's consider two observables, H (hamiltonian) and P (momentum). These operators are compatible since [H,P] = 0. Let's look at the easy to prove rule: 1: "If the observables F and G are compatible, that is, if there exists a simultaneous set of eigenfunctions of the operators F and G, then...
  49. S

    Prove that the exchange operator is Hermitian

    Homework Statement [/B] Let P be the exchange operator: Pψ(1,2) = ψ(2,1) How can I prove that the exchange operator is hermitian? I want to prove that <φ|Pψ> = <Pφ|ψ>Homework Equations [/B] <φ|Pψ> = <Pφ|ψ> must be true if the operator is hermitian. The Attempt at a Solution [/B] <φ(1,2) |...
  50. 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}##...
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