Operator Definition and 1000 Threads

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. A

    When is the kernel of a linear operator closed?

    If you consider a bounded linear operator between two Hausdorff topological vector spaces, isn't the kernel *always* closed? I mean, if you assume singleton sets are closed, then the set \{0\} in the image is closed, so that means T^{-1}(\{0\}) is closed, right (since T is assumed continuous)? I...
  2. M

    Rotation Operator: Spin 1/2 vs Spin 1

    How does finding the rotation operator for a spin 1/2 particle differ from finding that of a spin 1 particle?
  3. P

    Number operator in the ground state

    Homework Statement Why does <0|\frac{1}{(2\pi)^3}∫ \hat{a}^{\dagger}(t,r) \hat{a}(t,r) d^{3} \textbf{k} |0> = \frac{1}{\pi^2}∫|β|^2 k^2 dk. Where \hat{a} and \hat{a}^{\dagger} and its conjugate are bogulobov transformations given by: \hat{a}(t,k) = \alpha(t)a(k) + β(t)b^{\dagger}(-k)...
  4. J

    Matrix Mechanics position and momentum operator

    Homework Statement Stationary states are related by 1/m*pij = Ei-Ej/(i*hbar) *xij where pij = ∫ψi(x)*p*ψj (x) and similarly for xij Homework Equations Classical correspondence for equation of motion is d<x>/t = <p>/m Schrodinger equation Ehrenfest Theorem Fourier Transform The...
  5. E

    Linear operator or nonlinear operator?

    Homework Statement Verify whether or not the operator L(u) = u_x + u_y + 1 is linear. Homework Equations An operator L is linear if for any functions u, v and any constants c, the property L(c_1 u + c_2 v) = c_1 L(u) + c_2 L(v) holds true. The Attempt at a Solution I feel...
  6. S

    Is the Expectation Value of the y-Component of Spin Represented by Sy?

    Hey, I'm having trouble interpreting a question, as the solutions say something different... Anyways the question part d) below: So we want to determine the expectation value of the y-component of the electron spin on the eigenstate of Sx, now I would of thought this was given by...
  7. S

    How Does the Raising Operator Work in Quantum Mechanics?

    Hey, I have a question on showing how the raising operator in QM raises a particular eigenstate by 1 unit, the question is showed below: I think I know how to do this but not sure if what I'm doing is sufficient: \hat{N}a^{\dagger}|n>=([\hat{N},a^{\dagger}]+a^{\dagger}\hat{N})|n>...
  8. G

    Del operator crossed with a scalar times a vector proof

    "Del" operator crossed with a scalar times a vector proof Homework Statement Prove the following identity (we use the summation convention notation) \bigtriangledown\times(\phi\vec{V})=(\phi \bigtriangledown)\times\vec{V}-\vec{V}\times(\bigtriangledown)\phi Homework Equations equation for...
  9. P

    Can the Stokes Parameters Make Photon Polarization Probability Zero?

    I have the following situation: About the polarization of the photon, I introduce the basis: Horizontal polarization $|\leftrightarrow>=\binom{1}{0}$ Vertical polarization $|\updownarrow>=\binom{0}{1}$ The density matrix in this problem is: $$\rho =\frac{1}{2}\begin{pmatrix} 1+\xi...
  10. B

    What defines an operator input/output for simple expressions

    Hello. I have some questions on operations. Suppose in the course of a derivation there is a mathematical statement of the form A+1=B+C then "+" is an operator acting on inputs "B" and "C". Question 1: Is the output of the operation "A" or the expression "B+C"? The reason I think the...
  11. I

    A little problem with charge operator

    I have a problem where it's said that the operator Q is likely to be: Q=\sum^3_{i=1}[\frac{1}{2}B_i + I_{3,i}] I have to apply this to the proton wave function which is the same as you can see in equation (3.20) here...
  12. R

    [QM] Total angular momentum rotation operator

    Homework Statement How to prove that for any representation of the spin, the state e^{-i{\pi}J_x/\hbar}|j,m\rangle is proportional to |j,-m\rangle The exponential term is the rotation operator where J_x is the x-component of the total angular momentum operator, and |j,m\rangle is an...
  13. S

    Hermitian Operator Expectation Values

    Hey, I have the following question on Hermitian operators Initially I thought this expectation value would have to be zero as the eigenvectors are mutually orthogonal due to Hermitian Operator and so provided the eigenvectors are distinct then the expectation would be zero... Though...
  14. V

    If V is a complex inner product and T is an operator on V such that <Tv,v> = 0

    The book I am going through says this : The below proposition is false for real inner product spaces. As an example, consider the operator T in R^2 that is a counter clockwise rotation of 90 degrees around the origin. Thus , T(x,y) = (-y,x). Obviously, Tv is orthogonal to v for every v in...
  15. A

    Eigenstates of the momentum operator

    For the free particle the solution to the SE are eigenstates of the momentum. You get something like: ψ = Aexp(ik(x-vt)) + Bexp(-ik(x+vt)) , where k is a constant And my book then says that first term represents a wave traveling to the right and the second a wave traveling to the left. But I...
  16. S

    Proving Diagonalizability of Adjoint Operator on Finite Inner Product Space

    I was looking for a hint on a problem in my professor's notes (class is over and I was just auditing). I want to show that if T:V→V is a linear operator on finite dimensional inner product space, then if T is diagonalizable (not necessarily orth-diagonalizable), so is the adjoint operator of...
  17. K

    Material/Fluid derivative operator questions

    http://upload.wikimedia.org/math/2/b/2/2b2fe1336915a03e04930c11b27f4585.png The above link shows the material derivative. Which is the derivative that follows a volume of fluid throughout its movement through a fluid. How is this derived from a chain rule? Is the v in that equation the...
  18. A

    What's Wrong with My Eigenvector Calculation?

    Homework Statement Homework Equations The Attempt at a Solution I don't know what's wrong with my work. I can't obtain the eigenvector provided in the model answer. My work Model Answer
  19. tomwilliam2

    Understanding the Divergence Operator for Time-Varying Vectors

    Homework Statement I'm trying to find the divergence of a vector field (a fluid flow vector), but the vector takes the form u = u(x,y,z,t) The Attempt at a Solution I only really know how to take the divergence of a time-independent vector, so I'm guessing I just take the partial...
  20. S

    Eigenvalues of a compact positive definite operator

    eigenvalues of a compact positive definite operator! Let A be a compact positive definite operator on Hilbert space H. Let ψ1,...ψn be an orthonormal set in H. How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where λ1≥λ2≥λ3≥... be the eigenvalues of A in decreasing order. Can...
  21. M

    Do the Creation Operator and Spin Projection Operator Commute?

    I have bumped into a term a^\dagger \hat{O}_S | \psi \rangle I would really like to operate on the slater determinant \psi directly, but I fear I cannot. Is there any easy manipulation I can perform?
  22. E

    A perturbation operator problem

    hi,my friends.I have a perturbation operator problem. v=ezE; why this formula is right?how to deduce it?a is bohr radius. thank you!
  23. N

    Dipole operator and correlations

    Hi I am reading a paper (http://arxiv.org/abs/0901.3105), where they after eq. (3) mention something I can't understand. First of all, (3) comes from the master equation of a collection of N atoms in a cavity. They say that (page 2, right after (3)): The last term describes the coupling of...
  24. DiracPool

    Hamiltonian Kinetic Energy Operator

    In the QM Hamiltonian, I keep seeing h-bar/2m instead of p/2m for the kinetic energy term. H-bar is not equivalent to momentum. What am I missing here?
  25. C

    Complex coefficents in density operator expansion?

    Hey, I recently had an exam where the quantum state were on the form |\psi\rangle = \frac{1}{\sqrt{2}} ( |+\rangle + i |-\rangle ) Here I formed the density operator for the pure state \rho(t) = |\psi\rangle \langle \psi| = \frac{1}{2} ( |+\rangle + i |-\rangle )( \langle +| - i \langle...
  26. J

    Second Quantization for Fermions: Creation Operator

    So, I'm studying Second Quantization for fermions and came across this equation. I was just wondering why there is a summation needed? And why do we do it with (i≠p).? Please can someone explain this to me? Reply and help is much appreciated.
  27. P

    Expectation operator - linearity

    Homework Statement Show that the expectation operator E() is a linear operator, or, implying: E(a\bar{x}+b\bar{y})=aE(\bar{x})+bE(\bar{y}) Homework Equations E(\bar{x})=\int_{-\infty}^{+\infty}xf_{\bar{x}}(x)dx With f_{\bar{x}} the probability density function of random variable x...
  28. A

    Identity Operator: Vector Expressions in Basis A

    I was wondering about this: The identity operator writes a vector in the basis that is used to express the identity operator: 1 = Ʃlei><eil But if you are to apply it to a vector in a given basis A should the lei> then be expressed in terms of their own basis or in terms of A?
  29. V

    Delta n: Exploring the nth Difference Operator

    We all know the greek letter delta is the mathematical symbol that represents "change in." I though about a new form of delta: Δn. Where n2 = the # of terms when you expand the delta operator. For example: the usual Δx = x2 - x1 But now: Δ2x = (X4-X3) - (X2-X1). We can see that for Δ2...
  30. F

    Eigenfunctions of spin operator

    What are the eigenfunctions of the spin operators? I know the spin operators are given by Pauli matricies (https://en.wikipedia.org/wiki/Spin_operator#Mathematical_formulation_of_spin), and I know what the eigenvalues are (and the eigenvectors), but I have no idea what the eigenfunctions of the...
  31. M

    Understanding Eigenvalue Measurement in Quantum Systems

    when we have a certain state ψ(t) and it is acted on by an operator A of eigenstates a, b, c and eigen vectors la>, lb>, lc> does it mean that after measuring A ( if the result was 'a'), the state lψ(t)> becomes in state la>?
  32. Z

    Spectrum of a linear operator on a Banach space

    Homework Statement I have a number of problems, to be completed in the next day or so (!) that I am pretty stuck with where to begin. They involve calculating the spectra of various different linear operators. Homework Equations The first was: Let X be the space of complex-valued...
  33. G

    Determining whether an operator is Hermitian

    Homework Statement Consider the set of functions {f(x)} of the real variable x de fined on the interval -\infty< x < \infty that go to zero faster than 1/x for x\rightarrow ±\infty , i.e., \lim_{n\rightarrow ±\infty} {xf(x)}=0 For unit weight function, determine which of the...
  34. W

    Expanding a translation operator

    I'm trying to understand the construction of the T(ε) operator and why it is equal to I-iεG/hbar. The textbook I'm using (Shankar) talks defines the translation operator with the phase factor: T(ε)\left|x\right\rangle=e^{i \epsilon g(x)/\hbar}\left|x+\epsilon\right\rangle and...
  35. L

    Understanding the Covariance of the Spin Projection Operator in Rest Frame?

    I cannot quite understand why expression \frac{1-\gamma_5 \slashed{s}}{2} is covariant? We defined it in the rest frame, and then said that because it is in the slashed expression, it's covariant, what does that mean? s is the direction of polarization, s \cdot s = -1
  36. M

    Solving Non-Homogeneous DEs: Finding the Annihilator for Particular Solutions

    Hello, I am having trouble when solving non-homogeneous DE's how to find the annihilator to find my particular solution. For example, if you have a DE that equals 24x^2cos(x), how do I find something that will annihilate this? It seems to me no matter how many derivatives you take, you...
  37. C

    Observables commute and time operator

    I just have two questions relating to what I have been studying recently. 1) I know that the total energy and momentum operators don't commute, while the kinetic energy and momentum operators do. Why is this the case? (explanation rather than mathematically). 2) One form of the HUP says that...
  38. I

    Using the rotation operator to solve for eigenstates upon a general basis

    Homework Statement I need to express the rotation operator as follows R(uj) = cos(u/2) + 2i(\hbar) S_y sin(u/2) given the fact that R(uj)= e^(iuS_y/(\hbar)) using |+-z> as a basis, expanding R in a taylor series express S_y^2 as a matrix Homework Equations I know...
  39. A

    Explaining Pure States and Stationary States in Quantum Mechanics

    I know that the average momentum <p> is defined as m\frac{d}{dt}<x>. But why is this also equal to : \intψ*\frac{h}{(2\pi)i}\frac{\partial ψ}{\partial x}dx ? the integral goes from negative inf to inf, * indicates conjugate,ψ the wavefunction. Also, why is it in general that for any average...
  40. J

    MHB Is this operator diagonalizable？

    Let M be the space of all 2 × 2 complex matrices, satisfying 〖(X)bar〗^t = -X (skew-hermitian). Consider M as a vector space over R. Define a bilinear form B on M by B(X,Y) = -tr(XY) (1) Show that B takes real values, is symmetric and positive definite. (2) For any A ∈ M , define the...
  41. R

    Discover the Spin of an Electron Using Angular Momentum Operator and Eigenvalues

    I should Use the fact that in general the eigenvalues of the square of the angular momentum operator is J(J + 1)h and show the spin of the electron. I have J= L+S and J2 = L2+ S2 Homework Statement But how could i find the spin of the electron
  42. R

    Why are the only possible values of an operator its eigenvalues?

    Like the title says, why are the only possible values of an operator its eigenvalues? reading shankar right now and I'm having difficulty understanding why this has to be the case, given some operator/variable Ω
  43. V

    Linear Algebra : Proving that Every map is an identity operator

    Suppose T belongs to L(V,V) where L(A,W) denotes the set of linear mappings from Vector spaces A to W, is such that every subspace of V with dimension dim V - 1 is invariant under T. Prove that T is a scalar multiple of the identity operator. My attempt : Let U be one of the sub spaces of V...
  44. V

    Change of the Del operator in two particle interactions

    Change of the "Del" operator in two particle interactions Ok,so John Taylor's Classical Mechanics has this small subtopic "energy interactions between 2 particles".And,in that,hes defined a "del1" operator as the vector differential operator with respect to particle 2 at the origin.Hence,the...
  45. C

    Commutation Relationships and Operator Functions

    There are 2 operators such that [A,B] = 0. Does [F(A),B]=0 ? Specifically, let's say we had the Hamiltonian of a 3-D oscillator H and L^2. We know that L^2 = Lx^2+Ly^2+Lz^2, and it is known that [H,Lz] = 0. Can we say that since H and Lz commute, H and Lz^2 also commute, by symmetry H and...
  46. N

    Pseudospin Operator: Exploring Multi-Atom Systems

    Hi Often in the context of multi-atom systems, such as in cavity QED, it is customary to introduce a so-called "collective pseudospin operator". An example of this is for the inversion for some atom j, \sigma_{j, z}, which becomes \sum_{j} \sigma_{z, j} = \sigma_z To me this seems very...
  47. T

    Is Becoming a Reactor Operator the Right Path for Me?

    I want to be on the cutting edge of nuclear engineering, but I am afraid that I might not have the genius necessary to do it. I'm in my first semester of taking NucE classes, and my Fluid Mechanics class is tearing me up! Not to mention my Fundamentals of Nuclear Science/Engineering class is...
  48. L

    Normalized Eigenvectors of a Hermitian operator

    Hi all Homework Statement Given is a Hermitian Operator H H= \begin{pmatrix} a & b \\ b & -a \end{pmatrix} where as a=rcos \phi , b=rsin \phi I shall find the Eigen values as well as the Eigenvectors. Furthermore I shall show that the normalized quantum states are: \mid +...
  49. P

    Operator with strictly positive eigenvalues

    Homework Statement Consider a Hilbert space with a (not necessarily orthogonal) basis \{f_i\} Show that G=\sum_i |f_i\rangle\langle f_i| has strictly positive eigenvalues. Homework Equations The Attempt at a Solution I know that G=\sum_i |f_i\rangle\langle f_i| is hermitian...
  50. K

    Unitarity of Time-evolution Operator

    I am reading a quantum mechanics book. I did not clearly understand one particular idea. When the book talks about the time-evolution operator U(t,t_0), it says that one very important property is the unitary requirement for U(t,t_0) that follows from probability conservation. My question is...
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