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

    Average of any operator with Hamiltonian

    Homework Statement Prove that for any stationary state the average of the commutator of any operator with the Hamiltonian is zero: \langle\left[\hat{A},\hat{H}\right]\rangle = 0. Substitute for \hat{A} the (virial) operator:\hat{A} = \frac{1}{2}\sum\limits_i\left(\hat{p}_ix_i...
  2. S

    Implications of an arbitrary phase for momentum operator

    In quantum mechanics, the phase of the wavefunction for a physical system is unobservable. Therefore, both ψ = ψ(x) and ψ' = ψ(x)eiθ are valid wavefunctions. For ψ = ψ(x), we have the following: \widehat{x}ψ = xψ \widehat{p}ψ = λψ For ψ' = ψ(x)eiθ, we have the following...
  3. mishima

    [Processing] += operator, function equivalent?

    Hi, I was curious how I could turn any expression that looks like: x += (100- x) * 0.01; into a function that could be graphed.
  4. DavideGenoa

    Compact operator in reflexive space compact

    Hi, friends! I find an interesting unproven statement in my functional analysis book saying the image of the closed unit sphere through a compact linear operator, defined on a linear variety of a Banach space ##E##, is compact if ##E## is reflexive. Do anybody know a proof of the statement...
  5. DavideGenoa

    Existence of surjective linear operator

    Dear friends, I read that, if ##A## is a bounded linear operator transforming -I think that such a terminology implies that ##A## is surjective because if ##B=A## and ##A## weren't surjective, that would be a counterexample to the theorem; please correct me if I'm wrong- a Banach space ##E##...
  6. A

    Can Any Linear Operator Be Expressed Using Hermitian Components?

    Homework Statement Show that any linear operator \hat{L} can be written as \hat{L} = \hat{A} + i\hat{B}, where \hat{A} and \hat{B} are Hermitian operators. Homework Equations The properties of hermitian operators. The Attempt at a Solution I am not sure where to start with this...
  7. I

    C/C++ Assigning numOnes with Modulo Operator

    A cashier distributes change using the maximum number of five dollar bills, followed by one dollar bills. For example, 19 yields 3 fives and 4 ones. Write a single statement that assigns the number of distributed 1 dollar bills to variable numOnes, given amountToChange. Hint: Use %. Sample...
  8. Dale

    Is the Current Density Operator Related to Classical Magnetic Moment?

    Is there a current density operator or something equivalent? If so, how does it relate to other operators like momentum and angular momentum? Basically, the classical picture of a magnetic moment is a little loop of current, I would like to understand the quantum analog.
  9. DavideGenoa

    Banach's inverse operator theorem

    Dear friends, I have been trying in vain for a long time to understand the proof given in Kolmogorov and Fomin's of Banach's theorem of the inverse operator. At p. 230 it is said that M_N is dense in P_0 because M_n is dense in P. I am only able to see the proof that (P\cap M_n)-y_0 \subset...
  10. S

    Applying Integration by Parts and Eikonal Equation to Fourier Integral Operators

    Hi! I have a question for you. At the end of the post there's a link. There's the homework which I have to do for an exam. I have to study the Fourier Integral Operator that there is at the begin of the paper. I did almost all the homework but I can't do a couple of things. First: at the point...
  11. M

    Simplifying Operator and Dirac Algebra for Kets

    Hi Guys, I am facing a problem playing around with some operators and Kets, would like some help! I have \langle \Psi | A+A^\dagger | \Psi \rangle .A Could someone simplify it? Especially is there a way to change the last operator A into A^\dagger? The way I thought about this is...
  12. P

    Hamiltonian Operator: Difference vs. E?

    Is there any difference between Hamiltonian operator and E? Or do we describe H as an operation that is performed over (psi) to give us E as a function of (psi)??
  13. J

    Can a Unitary Operator Be Expressed in Exponential Form?

    Homework Statement I'm working on this problem: Let \hat{U} an unitary operator defined by: \hat{U}=\frac{I+i\hat{G}}{I-i\hat{G}} with \hat{G} hermitian. Show that \hat{U} can be written as: \hat{U}=Exp[i\hat{K}] where \hat{K} is hermitian. Homework Equations...
  14. B

    Momentum operator of the quantized real Klein-Gordon field

    Homework Statement a+(k) creates particle with wave number vector k, a(k) annihilates the same; then the Klein-Gordon field operators are defined as ψ+(x) = ∑_k f(k) a(k) e^-ikx and ψ-(x) = ∑_k f(k) a+(k) e^ikx; the factor f contains constants and the ω(k). x is a Lorentz four vector, k is a...
  15. S

    Is the Operation Linear and Bijective?

    Could anyone help me solve this problem? Let A,B be two subspace of V, a \in A, b \in B. Show that the following operation is linear and bijective: (A + B)/B → A/(A \cap B): a + b + B → a + A \cap B I really couldn't understand how the oparation itself works, i.e, what F(v) really is in...
  16. Q

    Rotating eigenstates of J operator into each other?

    Homework Statement Consider the following set of eigenstates of a spin-J particle: |j,j > , ... , |j,m > , ... | j , -j > where \hbar^2 j(j+1) , \hbar m are the eigenvalues of J^2 and Jz, respectively. Is it always possible to rotate these states into each other? i.e. given |j,m> and...
  17. S

    How Can I Write an Operator in the Coupled and Uncoupled Basis?

    Given a system of two identical particles (let's say electrons), of (max) spin 1/2 (which means the magnetic quantum number of each of the electrons can be either 1/2 or -1/2), how can we write the operators (total angular momentum, z-component of the total angular momentum etc.) (a) in the...
  18. U

    What Does the Gradient Operator Really Mean?

    Homework Statement I need some help regarding the gradient operator. I recently came across this statement while reading Griffith's Electrodynamics "The gradient ∇T points in the direction of maximum increase of the function T." Wolfram Alpha also states that "The direction of ∇f is the...
  19. S

    NEW Proof that parity operator is hermitean

    If the parity operator ##\hat{P}## is hermitian, then: ##\langle \phi | \hat{P} | \psi \rangle = (\langle \psi | \hat{P} | \phi \rangle)^*## Let us see if the above equation is true. The left hand side of the above equation is: ## \langle \phi | \hat{P} | \psi \rangle =...
  20. JonnyMaddox

    Solve Hodge Star Operator Problem Step-by-Step

    Hi, I have real problems with the indices here, can someone give me a step by step explanation how to compute stuff with this formula? *\omega = \frac{\sqrt{|g|}}{r!(m-r)} \omega_{\mu_{1}\mu_{2}...\mu_{r}}\epsilon^{\mu_{1}\mu_{2}...\mu_{r}}_{v_{r+1}...v_{m}}dx^{v_{r+1}}\wedge...\wedge...
  21. M

    Make an operator to be hermitian

    I have an operator which isnot Hermitian is there any way to make it hermitian ?
  22. C

    Rigorous Definition of Infinitesimal Projection Operator?

    I've been reading Thomas Jordan's Linear Operators for Quantum Mechanics, and I am stalled out at the bottom of page 40. He has just defined the projection operator E(x) by E(x)(f(y)) = {f(y) if y≤x, or 0 if y>x.} Then he defines dE(x) as E(x)-E(x-ε) for ε>0 but smaller than the gap between...
  23. carllacan

    Matrix elements of position operator in infinite well basis

    Homework Statement Find the eigenfunctions of a particle in a infinite well and express the position operator in the basis of said functions.Homework Equations The Attempt at a Solution Tell me if I'm right so far (the |E> are the eigenkets) X_{ij}= \langle E_i \vert \hat{X} \vert E_j \rangle...
  24. D

    QFT - Commutator relations between P,X and the Field operator

    Hi all, I haven't been able to find an answer online but this seems like a pretty basic question to me. What are the commutator relations between the position/momentum operators and the field operator? I'm not even certain what the commutation relations between X/P and a single ladder operator...
  25. D

    Hermitian conjugate of the annihilation operator

    Hi I have been looking at the solutions to a past exam question. The question gives the annihilation operator for the harmonic oscillator as a= x + ip ( I have left out the constants ). The question then asks to calculate the Hermitian conjugate a(dagger). I thought to find the Hermitian...
  26. carllacan

    Conmutative Hermitian operator in degenerate perturbation theory

    Hi. In 2-fold degenerate perturbation theory we can find appropiate "unperturbate" wavefunctions by looking for simultaneous eigenvectors (with different eigenvalues) of and H° and another Hermitian operator A that conmutes with H° and H'. Suppose we have the eingenvalues of H° are ##E_n =...
  27. carllacan

    At most explicit time-dependent operator

    Hi. I have a little language problem. I'm studying in Germany, and my German is... nicht sehr gut, so I sometimes have problems understanding the exercises. The one I'm having issues right now has a part which says einen höchstens explizit zeitabhängigen Ope-rator I am Schrödingerbild. My...
  28. carllacan

    Does the Creation Operator Have Eigenvalues?

    Homework Statement Prove that the creation operator a_+ has no eigenvalues, for instance in the \vert n \rangle . Homework Equations Action of a_+ in a harmonic oscillator eigenket \vert n \rangle : a_+\vert n \rangle =\vert n +1\rangle The Attempt at a Solution Calling a the...
  29. carllacan

    Notation for the nabla operator arguments

    Hi. In this development (c ∇+ d A)(c ∇+dA)= c^{2} ∇^{2} + d^{2}A^{2} + cd A∇ + cd ∇A (c ∇+ d A)^{2}= c^{2} ∇^{2} + d^{2}A^{2} + cd A∇+ cd A∇+ cd (∇A) I feel like we have "two" different ∇ operators. At the end of the first line ∇ acts on A and the test function (not shown). At the...
  30. D

    Orthogonal operator and reflection

    Homework Statement Let ##n## be a unit vector in ##V## . Define a linear operator ##F_n## on ##V## such that $$F_n(u) = u-2\langle u, n \rangle n \; \mathrm{for} \; u \in V.$$ ##F_n## is called the reflection on ##V## along the direction of ##n##. Let ##S## be an orthogonal linear operator on...
  31. gfd43tg

    Conditional operator if-else-elseif-end with switch-case combined

    Homework Statement #1 in the attachment Homework Equations The Attempt at a Solution My code is working for all of the numeric, logical, character portions. I got 7/8 points if ischar(X_input) Y_output = upper(X_input); elseif isnumeric(X_input) switch...
  32. carllacan

    Squared gradient vs gradient of an operator

    Hi. This is driving me mad: \hat{\vec{\nabla}}(\hat{\vec{A}})f=(\vec{\nabla}\cdot\vec{A})f + \vec{A}\cdot(\vec{\nabla}f) for an arbitrary vector operator ##\hat{\vec{A}}## So if we set ##\vec{A} = \vec{\nabla}## this should be correct...
  33. Einj

    What is the Number of Quarks Operator in Quantum Field Theory?

    Hi everyone. In QFT one usually defines the "number of valence quarks" of a certain particle via the operator: $$ \hat N_{val}=\sum_f |\hat Q_f|,$$ where: $$ \hat Q_f=\int d^3x \bar \psi_f\gamma_0\psi_f.$$ According to this definition I expected, for example, for the J/\psi to have...
  34. R

    A Commutator of annihilation operator

    Hi, everybody: I encountered a problem when I am reading a book. It's about the atom-photon interaction. Let the Hamiltonian for the free photons be H_0=\hbar \omega(a^{\dagger}a+\frac{1}{2}). so the commutator of the annihilation operator and the Hamiltonian is [a,H_0]=\hbar\omega a and I...
  35. L

    Is A^k a Projection Operator if k is Even/Odd?

    If ##A## is not projection operator. Could ##A^k## be a projection operator where ##k## is even or odd degree. Thanks for the answer.
  36. C

    Unitary operator acting on state

    In the operation $$U(\Lambda)|{\bf p}\rangle=|{\Lambda\bf p}\rangle,$$ if we define the state covariantly, $$|{\bf p}\rangle=\sqrt{2E_{\bf p}}a_{\bf p}^\dagger|0\rangle,$$ then does the unitary operator U(\Lambda) affect the factor of \sqrt{2E_{\bf p}}? In other words, can we write...
  37. P

    Proof that the linear momentum operator is hermitian

    hello i have to proof that Px (linear momentum operator ) is hermitian or not i have added my solution in attachments please look at my solution and tell me if its correct thank you all
  38. Ravi Mohan

    Expectation values of unbounded operator

    I am reading an intriguing article on rigged Hilbert space http://arxiv.org/abs/quant-ph/0502053 On page 8, the author describes the need for rigged Hilbert space. For that, he considers an unbounded operator A, corresponding to some observable in space of square integrable functions...
  39. P

    The operator of momentum (layman question)

    I found two "forms" of it: p=\frac{\hbar}{i}\frac{d}{dx} p=-i\hbar\frac{d}{dx} how could they be the same??
  40. K

    Questions about tensor operator

    Hi. Before question, sorry about my bad english. It's not my mother tongue. My QM textbook(Schiff) adopt J x J = i(h bar)J. as the defining equations for the rotation group generators in the general case. My question is, then tensor J must have one index which has three component? (e.g...
  41. laramman2

    Why does the Laplacian operator still maintain its unit vectors i, j, k?

    When two vectors are dotted, the result is a scalar. But why here http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/mathbas/vectors.htm , the del-squared still maintains its unit vectors i, j, k? Isn't it this way ∇2 = (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2) and not (i∂2/∂x2 + j∂2/∂y2 +...
  42. E

    Quantum operator hermiticity. Show that S is Hermitian

    Homework Statement Spin Operator S has eigenvectors |R> and |L>, S|R> = |R> S|L> =-|L> eigenvectors are orthonormal Homework Equations Operator A is Hermitian if <ψ|A|Θ> = <Θ|A|ψ>* The Attempt at a Solution <ψ|S|L> = <L|S|ψ>* // Has to be true if S is Hermitian LHS...
  43. B

    Is L a Self-Adjoint Operator with Non-Negative Eigenvalues?

    Homework Statement We have a linear differential operator ##Ly=-y^{''}## working on all ##y## that can be derived at least twice on ##[-\pi ,\pi ]## and also note that ##y(-\pi )=y(\pi )## and ##y^{'}(-\pi )=y^{'}(\pi )##. a) Is ##0## eigenvalue for ##L##? b) Is ##L## symmetric? (I think the...
  44. S

    Expectation value of an operator

    When we say expectation value of an operator like the pauli Z=[1 0; 0 -1], like when <Z> = 0.6 what does it mean? What is difference between calculating expectation value of Z and its POVM elements{E1,E2}? thanks
  45. K

    Translation operator on a sphere

    I'm considering a system where an electron is subjected to magnetic field which is produced by dirac monopole. Here I'm interested in looking for a translation operator. Now how can I get a translation operator in presence of field and in absence of field.?? I need both the operators. Can...
  46. B

    Geometric Derivation of the Complex D-Bar Operator

    This picture from https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20 is all you need to derive the Cauchy-Riemann equations, i.e. from the picture we see i \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} should hold so we have i \frac{\partial f}{\partial x} = i...
  47. U

    Perturbation Theory, exchange operator

    Homework Statement Part (a): Find eigenvalues of X, show general relation of X and show X commutes with KE. Part (b): Give conditions on V1, V2 and VI for X to commute with them. Part (c): Write symmetric and antisymmetric wavefunctions. Find energies JD and JE. Part (d): How are...
  48. O

    Introduction to Liouvillian Operator inStatistical Mechanics

    Hello, who can suggest me a book, or a PDF where i can find an introduction to Liouvillian operator in statistical mechanics? I understand that it's correlated to time evolution of density of an Hamiltonian system but i don't know anything else thank you sorry for my wrong english.. :(
  49. J

    Shift operator is useful for what?

    Definition: ##f(x+k) = \exp(k \frac{d}{dx}) f(x)## So I thought, how take advantage this definition? Maybe it be usefull in integration like is the laplace transform. So I tried to integrate the expression ##\int f(x+k) dx = \int \exp(k \frac{d}{dx}) f(x) dx ## that is an integration by parts...
  50. S

    Differential of exponential operator

    If \hat{U}(r) = e^{\hat{A}(r)}, can we say \frac{d\hat{U}}{dr} = \frac{d\hat{A}}{dr}e^{\hat{A}(r)}?
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