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. M

    What Determines the Direction of a Dipole Operator in Quantum Optics?

    Hi, In quantum optics, when we talk about atom field interaction with a classical field and quantized atom, we say that the Hamiltonian has an interaction part of the form \hat{d}.\vec{E} where d is the dipole operator. For a two level atom, the dipole operator has only off diagonal elements...
  2. C

    Unitary operator + Lorents transformations (question from Peskin)

    Hi. I am trying to understand a statement from Peskin and Schroeder at page 59 they write; "The one particle states |\vec p ,s \rangle \equiv \sqrt{2E_{\vec p}}a_{\vec p}^{s \dagger} |0\rangle are defined so that their inner product \langle \vec p, r| \vec q,s\rangle = 2 \vec E_\vec{p}...
  3. N

    Show That g(y)=proj_x y is a Linear Operator.

    Homework Statement Let x be a fixed nonzero vector in R^3. Show that the mapping g:R^3→R^3 given by g(y)=projxy is a linear operator. Homework Equations projxy = \left(\frac{x\cdot y}{\|x\|}\right)x My book defines linear operator as: Let V be a vector space. A linear operator on V is...
  4. B

    Help understanding position operator eigenfunction derivation

    I'm having trouble understanding the derivation of the the position operator eigenfunction in Griffiths' book : How is it "nothing but the Dirac delta function"?? (which is not even a function). Couldn't g_{y}(x) simply be a function like (for any constant y) g_{y}(x) = 1 | x=y...
  5. W

    Quantum Theory: Operator Exponentiation

    Homework Statement Let \left|x\right\rangle and \left|p\right\rangle denote position and momentum eigenstates, respectively. Show that U^n\left|x\right\rangle is an eigenstate for x and compute the eigenvalue, for U = e^{ip}. Show that V^n\left|p\right\rangle is an eigenstate for p and...
  6. B

    Expression with two vectors and del operator

    Homework Statement (A.∇)B What does this mean and how do I go about trying to expand this (using cartesian components)?
  7. T

    Linear algebra problem (standard matrix for a linear operator)

    Homework Statement Determine the standard matrix for the linear operator defined by the formula below: T(x, y, z) = (x-y, y+2z, 2x+y+z) Homework Equations The Attempt at a Solution No idea
  8. K

    From unitary operator to hamiltonian

    Hi there, If the evolution operator is given as follows U(t) = \exp[-i (f(p, t) + g(x))/\hbar] where p is momentum, t is time. Can I conclude that the Hamiltonian is H(t) = f(p, t) + g(x) if no, why?
  9. C

    Operator algebra with commutators

    Homework Statement Consider the operator A and its Hermitian adjoint A*. If [A,A*] = 1, evaluate: [A*A,A] Homework Equations standard rules of linear algebra, operator algebra and quantum mechanics The Attempt at a Solution [A,A*] = AA* - A*A = 1 A*A = (1+AA*) [A*A,A] =...
  10. K

    Understanding Unitary Operator Evolution in Quantum Mechanics

    Hi there, I am reading a book in which the unitary evolution operator is U = \exp(-i H/\hbar) where H is the given Hamiltonian. But in another book, I found that the evolution operator is general given as U = \exp(-i \int H(t) dt / \hbar) which one is correct and why there are two...
  11. H

    Shape Operator for Schwarzschild spacetime in 2-dim

    Hello: I would like to understand how to compute the shape operator (and eigenvalues etc) for a complex example like the Schwarzschild spacetime. It's easy for a submanifold in Euclidean space, but I don't know how to do it for the more advanced examples like the schwarzschild spacetime in...
  12. T

    Derivation of Laplace Operator in Spherical and Cylindrical Coordinates

    Hey Guys, Does anyone know where I can find a derivation of the laplace operator in spherical and cylidrical coordinates?
  13. PerpStudent

    Rationale of the position operator?

    Why is the position operator of a particle on the x-axis defined by x multiplied by the wave function? Is there an intuitive basis for this or is it merely something that simply works in QM?
  14. M

    Discretization of the divergence operator

    I work with a grid-based code, this means that all of my quantities are defined on a mesh. I need to compute, for every point of the mesh the divergence of the velocity field. All I have is, for every cell of my mesh, the values of the 3-d velocity in his 26 neighbors. I call neighbors the...
  15. S

    Determinant of Transpose Operator

    I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's definition of the determinant expressed in terms of an alternating bilinear form but am having...
  16. M

    Eigenvalue of position operator and delta function.

    I'd like to show that if there exists some operator \overset {\wedge}{x} which satisfies \overset {-}{x} = <\psi|\overset {\wedge}{x}|\psi> , \overset {\wedge}{x}|x> = x|x> be correct. \overset {-}{x} = \int <\psi|x> (\int<x|\overset {\wedge}{x}|x'><x'|\psi> dx')dx = \int <\psi|x>...
  17. P

    Help with the conventions used for curl operator

    http://web.mit.edu/6.013_book/www/chapter2/2.4.html I was going through the curl derivation on the above link. How does equation 3 turn out? Δy is the incremental length. But how do you decide whether it is +Δy/2 or -Δy/2. And why is the line integral taken Δz when the change is in the y...
  18. D

    Time evolution operator in terms of Hamiltonian

    Homework Statement "Show that if the Hamiltonian depends on time and [H(t_1),H(t_2)]=0, the time development operator is given by U(t)=\mathrm{exp}\left[-\frac{i}{\hbar}\int_0^t H(t')dt'\right]." Homework Equations i\hbar\frac{d}{dt}U=HU U(dt)=I-\frac{i}{\hbar}H(t)dt The Attempt at a...
  19. F

    Matrix of. Linear operator question

    I am trying to figure out what the matrix of this linear operator would be: T:M →AMB where A, M, B are all 2X2 matrices with respect to the standard bases of a 2x2 matrix viz. e11, e12 e21 and e22. Any ideas? Il know it should be 2X8 matrix. I am trying to teach myself Abstract algebra using...
  20. H

    What is the Correct Value of the Square of a Momentum Operator?

    know I'm missing something obvious. for a momentum operator p = -iħ d/dx if I square the -iħ part I get (+1)ħ2 but I believe the correct value (as in the kinetic energy of the Hamiltonian) is -ħ/2m d2/dx2. how is the value of the term -ħ/2m where the square of -i = +1? Thanks!
  21. X

    Angular momentum operator identity J²= J-J+ + J_3 + h*J_3 intermediate step

    Homework Statement I do not understand equal signs 2 and 3 the following Angular momentum operator identity: Homework Equations \hat{J}^2 = \hat{J}_1^2+\hat{J}_2^2 +\hat{J}_3^2 = \left(\hat{J}_1 +i\hat{J}_2 \right)\left(\hat{J}_1 -i\hat{J}_2 \right) +\hat{J}_3^2 + i...
  22. K

    Some inconsistency on operator expectation value

    Consider two Hermitian operator A, B; Define [A,B]=iC, then operator C is also Hermitian. we calculate the expectation value with respect to |a>, one eigenstate of A with the eigenvalue a. From the left side, we have: <a|[A,B]|a>=<a|(AB-BA)|a>=(a-a)<a|B|a>=0, while on the right side...
  23. M

    Displacement Operator definition

    Hi, Could someone explain how the following two definitions of the displacement operator are equal? The first is the standard one 1) e^{\alpha a^{\dagger}-\alpha*a} But what about this one? This is from a Fock state decomposition of a coherent state. 2) e^{\frac{-|\alpha^{2}|}{2}}e^{\alpha...
  24. M

    Singularity in Laplacian operator

    I am trying to understand the following basic problem, \partial_{xx} f^\alpha (x) = \alpha (\alpha-1) \frac{1}{f^{2-\alpha}} \partial_x f + \alpha \frac{1}{f^{1-\alpha}} \partial_{xx} f So it is not hard to see that if f tends to zero the laplacian becomes undefined (im not sure if i...
  25. USeptim

    Wave function collapse by orbital angular momentum operator Lz

    I have some doubts about the implications of the orbital angular operators and its eigenvectors (maybe the reason is that I have a weak knowledge on QM). If we choose the measurement of the z axis and therefore the Lz operator, the are the following spherical harmonics for l=1...
  26. G

    Is There a Method for Finding A When Given B and ∇XA?

    Let's say I have this relation B=∇XA I know B, now I want A. What ever do I do? Is there some tried and true method out there?
  27. P

    Angular momentum operator acting on |j,m>

    Homework Statement Prove that e^{-i \pi J_x} \mid j,m \rangle =e^{-i \pi j} \mid j,-m \rangle Homework Equations J_x \mid j,m \rangle =\frac{\hbar}{2} [\sqrt{(j-m)(j+m+1)} \mid j,m+1 \rangle + \sqrt{(j+m)(j-m+1)} \mid j,m-1 \rangle] The Attempt at a Solution Expanding e^{-i \pi J_x}...
  28. M

    Solution to 2nd order ODE using the D operator method with 2 trig terms on RHS

    Hey, I have the DE y'' -2y' + 3y = xsin(x) + 2cosh(2x) Using the D operator as D = \frac{dy}{dx} this becomes (D2 -2D +3)y = xsin(x) + 2cosh(2x) so yp = \frac{1}{p(D^2)} operating on xsin(x) + 2cosh(2x) (i think) So i know if this was say \frac{1}{p(D^2)} operating on...
  29. D

    Interpretation of the determinant of an operator in complex vector space

    There's a geometric interpretation of the determinant of an operator in a real vector space that I've always found intuitive. Suppose we have a n-dimensional real-valued vector space. We can plot n vectors in an n-dimensional Cartesian coordinate system, and in general we'll have an...
  30. D

    Matrix rep of operator acting on bras

    If A is a linear operator, and we have some ordered basis (but not necessarily orthonormal), then the element Aij of its matrix representation is just the ith component of A acting on the jth basis vector. We can also represent the action of A on a ket as the matrix product of A's matrix with...
  31. L

    Dirac's Quantum Mechanics - the definition of the time evolution operator

    Dirac's "Quantum Mechanics" - the definition of the time evolution operator I'm reading Dirac's "Principles Of Quantum Mechanics" to learn more about the formal side of the subject. I have a question about the way he defines the time evolution operator in the book. Either there's a mistake or...
  32. P

    Understanding Hermitian Operators

    Hi. In a question I needed to figure out whether -\frac{i\hbar}{m} \hat{p} is hermitian or not. Since the constant doesn't matter this is similar to whether i \hat{p} is hermitian or not. I thought that since \hat{p} is hermitian, then i times it would not be, since it would not...
  33. F

    Linearly independent sets within repeated powers of a linear operator

    Homework Statement Suppose that T:W -> W is a linear transformation such that Tm+1 = 0 but Tm ≠ 0. Suppose that {w1, ... , wp} is basis for Tm(W) and Tm(uk) = wk, for 1 ≤ k ≤ p. Prove that {Ti(uk) : 0 ≤ i ≤ m, 1 ≤ j ≤ p} is a linearly independent set.Homework Equations The Attempt at a Solution...
  34. M

    Using the D operator technique to solve a trig based DE

    hi everyone, im currently trying to teach myself the D operator technique, as opposed to the 'guess method' which i don't really like. I stumbled upon this on yahoo answers: (D^2 + 1)y = 4 cos x - sin x Find the complementary function by solving the auxiliary equation: m² + 1 = 0...
  35. K

    Understanding Exponential of Operator: Expansion and Eigenvalues Explained

    Hi there, I learn from the text that the exponential of an operator could be expanded with a series such that e^{\hat{A}} = \sum_{n=0}^{\infty} \frac{\hat{A}^n}{n!} So if the eigenvalue of the operator \hat{A} is given as a_i e^{\hat{A}}|\psi\rangle will be a matrix with diagonal elements...
  36. M

    General question about crossover operator for a genetic algorithm

    Hey guys, this is my first time working on a genetic algorithm. It seems to me that the algorithm is primarily defined by how you choose to define your crossover operator and fitness function. Let's say the crossover takes two parents and produces one child. Is it necessary/good/bad/etc that the...
  37. R

    Vector Identity (del operator)

    i am completely lost as to how to go from p^ \frac{1}{m}∇p to \frac{m}{m+1} ∇p^\frac{m+1}{m}
  38. X

    Trace and its square of mixed state density operator using integral

    Homework Statement I want to show that tr\left(\hat{\rho}_{mixed}\right)=1 tr\left(\hat{\rho}_{mixed}^{2}\right)<1 when \hat{\rho}_{mixed}=\frac{1}{2\pi}\int_{0}^{2\pi}d \alpha \hat{\rho}(\psi) Homework Equations tr\left(\psi\right)= \sum_{n}\langle n|\psi|n\rangle...
  39. E

    What is the Meaning of Expectation and Deviation of an Operator?

    What does the expectation and deviation of an operator mean?? The way I understood it was every observable has a operator to it and the expectation of the observable uses the operator to calculate the deviation ... for ex :: <p>=integral( (si)* momentum operator (si) ) dx ... so what does...
  40. O

    What is the purpose of this matrix operator?

    Hi, A fuller view can be found here: (i) What is this operator? (ii) What does this operator mean if Sp is a 5x5 matrix and (Z'Z)-1 is a 3x3 matrix?Thanks a lot.
  41. N

    The divergence operator in a rotated reference frame

    One can easily prove that \nabla \cdot f is invariant under a rotation of the reference frame, however I would like to prove that the divergence operator itself is invariant (same principle, different approach). In other words I want to prove that \mathbf \nabla = \mathbf e_x...
  42. E

    Eigen functions of momentum operator

    Hello, I am going through the book Introduction to QM by D.Griffiths. In the third chapter the book says the eigen functions of the momentum operator do not belong to the Hilbert space ... But the only condition that a vector belongs to the Hilbert vector space is that the integral...
  43. M

    Position operator for a system of coupled harmonic oscillators

    Hi I would like to know if I have a system of coupled harmonic oscillators whether the standard position operator for an uncoupled oscillator is valid, i.e. x_i = \frac{1}{\sqrt{2}} (a_i+a_i^\dagger)? where i labels the ions. To give some context I am looking at a problem involving a...
  44. C

    What is the symbol of an integral operator?

    Could someone please give a definition what the symbol of an integral operator is? Does it have asymptotic expansions as symbols of differential operators? I know about symbols of differential operators but, shame on me, heard nothing about the symbol of an integral operator. Any help in...
  45. U

    Complex Powers of an Elliptic Operator

    There is a paper written by R.T. Seeley in the Proceedings of Symposia Pure Mathematics that I've seen cited by several papers I've been reading, but I can't find it anywhere. The citation given is R. SEELEY, Complex Powers of an Elliptic Operator, “Singular Integrals (Proc. Symp. Pure Math...
  46. 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...
  47. 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...
  48. 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...
  49. 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) +...
  50. 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...
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