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. S

    I Klein-Gordon Operator: Creating Particles at Position x

    Hello! I read that the Klein-Gordon field can be viewed as an operator that in position space, when acted upon vacuum at position x creates a particle at position x: ##\phi(x) |0 \rangle \propto |x \rangle##. It make sense intuitively and the mathematical derivation is fine too, but I was...
  2. S

    I Propagator operator in Heinsenberg picture

    Hello! I read that in Heisenberg picture the propagator from x to y is given by ##<0|\phi(x)\phi(y)|0>##, where ##\phi## is the Klein-Gordon field. I am not sure I understand why. I tried to prove it like this: ##|x>=\phi(x,0)|0>## and after applying the time evolution operator we have...
  3. B

    B Non-Hermitian operator for superposition

    It is said that no Hermitian operator gives a time evolution where "I observed the spin to be both up and down" is a possible result. If you use non-Hermitian operator.. then it's possible.. and what operator is that where it is possible in principle where "I observed the spin to be both up and...
  4. Vitani11

    Show that if H is a hermitian operator, U is unitary

    Homework Statement Show that if H is a hermitian operator, then U = eiH is unitary. Homework Equations UU† = I for a unitary matrix A†=A for hermitian operator I = identity matrix The Attempt at a Solution Here is what I have. U = eiH multiplying both by U† gives UU† = eiHU† then replacing U†...
  5. ChrisVer

    What is the best way to avoid copying objects in an assignment operation?

    I have written the following class: class Rectangle{ private: double length; double height; const char* name; public: Rectangle (){}; Rectangle (double l, double h, const char* n) : length(l), height(h), name(n) {}...
  6. V

    I Is the Fermion number operator squared equal to itself?

    What the title says. Acting on a fermionic state with the number operator to a power is like acting with the fermionic operator itself. Does this allow us to define ## \hat{n}^k=\hat{n} ##? Or is there any picky mathematical reason not to do so?
  7. J

    Operator in three level system -- Eigenvalues/Eigenvectors

    There is an operator in a three-state system given by: 2 0 0 A_hat = 0 0 i 0 -i 0 a) Find the eigenvalues and Eigenvectors of the operator b) Find the Matrix elements of A_hat in the basis of the eigenvectors of B_hat c) Find the Matrix Elements of A_hat...
  8. Sharkey4123

    A Reduced Density Operator and Entanglement

    I'm having a little bit of trouble getting my head around the idea of the reduced density operator being used to tell us about the entanglement of a state. I understand that if you take the reduced density operator of any of the Bell states, you get a reduced density operator proportional to...
  9. S

    Differential operator acting on scalar fields

    Homework Statement I really cannot seem to be able to follow the logic of how you would use the product rule when using 4 vector differential operator. ∂μ is the differential operator, Aμ is a scalar field and φ and φ* is it's complex conjugate scalar field. I have the answer, I'd just really...
  10. M

    Assignment Operator Overloading Question

    Hey everyone. First I am new to programming (so my vocabulary and skills are not very proficient with C++), and I'm learning operator overloading. I have questions about line 1 and lines 5-8 of this code. Mod note: changed quote tags to code tags to preserve indentation. NumberArray&...
  11. S

    Are A and B Densely Defined if A+B is Densely Defined?

    Homework Statement Let A and B be two unbounded operators, is what if A+B is densely defined then A and B are also densely defined? Homework Equations D(A+B)=D(A)∩D(B) The Attempt at a Solution Since A+B is densely defined then A and B are also densely defined because D(A+B)=D(A)∩D(B)
  12. U

    Hamiltonian operator affecting observable

    I'm working on this problem "Consider an experiment on a system that can be described using two basis functions. In this experiment, you begin in the ground state of Hamiltonian H0 at time t1. You have an apparatus that can change the Hamiltonian suddenly from H0 to H1. You turn this apparatus...
  13. P

    Density Operator to Matrix Form

    Homework Statement Write the density operator $$\rho=\frac{1}{3}|u><u|+\frac{2}{3}|v><v|+\frac{\sqrt{2}}{3}(|u><v|+|v><u|, \quad where <u|v>=0$$ In matrix form Homework Equations $$\rho=\sum_i p_i |\psi><\psi|$$ The Attempt at a Solution [/B] The two first factors ##\frac{1}{3}|u><u|##...
  14. D

    What is the x operator in momentum space?

    The Hilbert space for a free relativistic particle has inner product (in the momentum representation) $$ \langle \chi | \phi \rangle = \int \frac {d \vec k^3} {(2 \pi)^3 2 \sqrt{\vec k^2 + m^2}} \chi (\vec k) * \phi (\vec k)$$ States undergo time evolution $$i∂t|ψ \rangle = H_0 | ψ \rangle$$...
  15. PedroBittar

    I Extended hamiltonian operator for the Hydrogen atom

    I am familiar with the usual solution of the hydrogen atom using the associated legendre functions and spherical harmonics, but my question is: is it possible to extend the hamiltonian of the hydrogen atom to naturally encompass half integer spin? My guess is that spin only pops in naturally in...
  16. davidge

    I Position Operator Action on Wave Function: $\psi(x)$

    Would the action of the position operator on a wave function ##\psi(x)## look like this? $$\psi(x) \ =\ <x|\psi>$$ $${\bf \hat x}<x|\psi>$$ Question 2: the position operator can act only on the wave function?
  17. LarryS

    I Why no position operator for photon?

    Apparently, in QM, the photon does not have a position operator. Why is this so? As usual, thanks in advance.
  18. redtree

    A Deriving the Lagrangian from the Hamiltonian operator

    In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian...
  19. M

    Complex conjugation operator

    Homework Statement Hi, so I have been given the following operator in terms of 3 orthonormal states |Φi> A = |Φ2><Φ2| + |Φ3><Φ3| - i|Φ1><Φ2| - |Φ1><Φ3| + i|Φ2><Φ1| - |Φ3><Φ1| So I need to determine whether A is unitary and/or Hermitian and/or a projector and then calculate the eigenvalues and...
  20. KennethK

    Calculate the spectrum of a linear operator

    <mod note: moved to homework> Calculate the spectrum of the linear operator ##T## on ##B(\ell^1)##. $$T(x_1,x_2,x_3,\dots)=(\sum_{n=2}^\infty x_n, x_1, x_2, x_3, \dots)$$I think the way to do it is to find the point spectrums of ##T## and its adjoint ##T^*##. But I don't know how to calculate...
  21. davidge

    I Extending an infinitesimal operator

    I notice that in Quantum Mechanics when extending an infinitesimal operation to a finite one, we should end with the exponential. For example: (rf. Sakurai, Modern Quantum Mechanics) $$ D(\boldsymbol{\hat n}, d\phi) = 1 - \frac{i}{\hbar} ( \boldsymbol {J \cdot \hat n})d\phi $$ This is the...
  22. K

    I The fractional derivative operator

    I've been thinking about it since yesterday and have noticed this pattern: We have, the first order derivative of a function ##f(x)## is: $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} ...(1)$$ The second order derivative of the same function is: $$f''(x)=\lim_{h\rightarrow...
  23. D

    I Proof that parity operator is Hermitian in 3-D

    Hi. I have been looking at the proof that the parity operator is hermitian in 3-D in the QM book by Zettili and I am confused by the following step ∫ d3r φ*(r) ψ(-r) = ∫ d3r φ*(-r) ψ(r) I realize that the variable has been changed from r to -r. In 3-D x,y,z this is achieved by taking the...
  24. S

    Given operator, show the Hamiltonian

    Homework Statement Given \hat{P}_r\psi=-i\hbar\frac{1}{r}\frac{\partial}{\partial{r}}(r\psi), show \hat{H}=\frac{1}{2m}(\hat{P}^2_r+\frac{\hat{L}^2}{r^2}) Homework EquationsThe Attempt at a Solution The solution starts out with...
  25. N

    I Eigenvalues of Fermionic field operator

    Hello, I'm a bit confused about the eigenvalues of the second quantized fermionic field operators \psi(x)_a. Since these operators satisfy the condition \{\psi(x)_a, \psi(y)_b\} = 0 the eigenvalues should also anti-commute? Does this mean that the eigenvalues of \psi(x)_a are...
  26. DoobleD

    I No r dependence in L operator?

    In classical mechanics, angular momentum, L = r x p, depends of r. For a given momentum p, the bigger r is, the bigger is the angular momentum. Event in spherical coordinates, r still appears in the classical angular momentum. However, the angular momentum operator in QM has no r dependence...
  27. F

    A On the equivalence of operator vs path integral in QFT

    I have read many textbooks and googled google times for a clear explanation, but I could not find one. How does raising and lowering -annihilation/ creation-(is that energy or particle number?) translate to transition probabilities of path integral.
  28. Konte

    I Internal rotation kinetic energy operator

    Hello everybody, My question today is: Given a molecule that has an internal degree of liberty ( let's take the ethane molecule with its internal rotation as an example), how to write the kinetic energy operator by means of the corresponding internal coordinate? Thank you guys. Konte
  29. F

    I Expectation of an operator (observable) how to calculate it

    Hello Forum, I understand that in order to calculate the average of a certain operator (observable), whatever that observable may be that we are interested in, we need to prepare many many many identical copies of the same state and apply the operator of interest to those identical state. By...
  30. C

    Relation between operator and experimental action

    Hi I have some questions about operator and experimental action 1, For each experimental action(no matter how trivial or complex), can they ALWAYS be described by some corresponded operator? how to proof? For example, adding some energy to excit a particle can be described in operator language...
  31. B

    I Generalizing the translation operator

    If I have the operator, ##e^{a\partial_p}## acting on ##f(p)##, I know that $$e^{a\partial_p}f(p)=f(p+a)\,.$$ If I have ##e^{a\partial_p^2}f(p)##, this is just the Weierstrass transform of ##f(p)##. However, what happens if I have a general operator, ##e^{g(p)\partial_p}## or...
  32. Crush1986

    Position Operator in Momentum Space?

    Homework Statement So, I'm doing this problem from Townsend's QM book 6.2[/B] Show that <p|\hat{x}|\psi> = i\hbar \frac{\partial}{\partial p}<p|\psi> Homework Equations |\psi(p)> = \int_\infty^{-\infty} dp |p><p|\psi> The Attempt at a Solution So, <p|\hat{x}|\psi> = <p|\hat{x}...
  33. ReidMerrill

    Physical chemistry: Energy operator and eigenfunction

    Homework Statement The energy operator for a time-dependent system is iħ d/dt. A possible eigenfunction for the system is Ψ(x,y,z,t)=ψ(x,y,z)e-2πiEt/h Show that the probability density is independent of time Homework Equations ĤΨn(x) = EnΨn The Attempt at a Solution [/B] I understand the...
  34. N

    I Symmetric, self-adjoint operators and the spectral theorem

    Hi Guys, at the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be self-adjoint (or in many cases hermitian what maybe means symmetric or maybe self-adjoint). Which condition do we need for our observables...
  35. C

    Pauli Spin Matrices - Lowering Operator - Eigenstates

    This is not part of my coursework but a question from a past paper (that we don't have solutions to). 1. Homework Statement Construct the matrix ##\sigma_{-} = \sigma_{x} - i\sigma_{y}## and show that the states resulting from ##\sigma_{-}## acting on the eigenstates of ##\sigma_{z} ## are...
  36. maxhersch

    I Kronecker Delta and Gradient Operator

    I am looking at an explanation of the gradient operator acting on a scalar function ## \phi ##. This is what is written: In the steps 1.112 and 1.113 it is written that ## \frac {\partial x'_k} {\partial x'_i} ## is equivalent to the Kronecker delta. It makes sense to me that if i=k, then...
  37. ShayanJ

    A Field operator eigenstates & Fock states(Hatfield's Sch rep)

    In chapter 10 of his book "Quantum Field Theory of Point Particles and Strings", Hatfield treats what he calls the Schrodinger representation of QFT. He starts with a free scalar field and introduces field operators ## \hat \varphi(\vec x) ## and its eigenstates ## \hat \varphi(\vec...
  38. Muthumanimaran

    A Kraus Operator in Fock basis

    The Kraus operator is defined as, $$A_{k}(t)={\sum_{\{k_i\}}^{k}}'\langle\{k_i\}|U(t)|\{0\}\rangle$$ is given in eqn(5) in the [Arxiv link](https://arxiv.org/pdf/quant-ph/0407263.pdf) the matrix representation of $A_k(t)$ is given in eqn (7) as...
  39. ShayanJ

    A What is the Role of Twist Operator in Complex Analysis?

    Is there any concept in mathematics called twist operator or twist field? It should somehow be related to branch points and Riemann sheets in complex analysis. Thanks
  40. Konte

    I Will the kinetic energy operator change with a variable shift?

    Hello everybody, A special problem constrain me to make a variable change in my Hamiltonian operator, so with the kinetic energy operator, I have a doubt. The variable change is: ## \theta \longrightarrow (\theta + k) ## (where ##k## is a constant). And the kinetic energy operator...
  41. J

    Argue, why given Operators are compact or not.

    Which of the operators T:C[0,1]\rightarrow C[0,1] are compact? $$(i)\qquad Tx(t)=\sum^\infty_{k=1}x\left(\frac{1}{k}\right)\frac{t^k}{k!}$$ and $$(ii)\qquad Tx(t)=\sum^\infty_{k=0}\frac{x(t^k)}{k!}$$ ideas for compactness of the operator: - the image of the closed unit ball is relatively...
  42. P

    Normalization of the Angular Momentum Ladder Operator

    Homework Statement Obtain the matrix representation of the ladder operators ##J_{\pm}##. Homework Equations Remark that ##J_{\pm} | jm \rangle = N_{\pm}| jm \pm 1 \rangle## The Attempt at a Solution [/B] The textbook states ##|N_{\pm}|^2=\langle jm | J_{\pm}^\dagger J_{\pm} | jm \rangle##...
  43. D

    I Commutator of Parity operator and angular momentum

    Hi I have seen an example of commutator of the Parity operator of the x-coordinate , Px and angular momentum in the z-direction Lz calculated as [ Px , Lz ] ψ(x , y) = -2Lz ψ (-x , y) I have tried to calculate the commutator without operating on a wavefunction and just by expanding...
  44. P

    I Determining if an operator is degenerate

    Hi, I was wondering how you can formally determine whether a given operator is degenerate. I undertand you can produce the 'usual equation' det(Q-##\lambda ##)=0 and solve for ##\lambda ##, where Q is our operator. But if Q is a differential (for example ##\frac{p^2}{2m}= - \bar{h}...
  45. RJLiberator

    ODE: System of Linear Equations usuing Diff. Operator

    Homework Statement This is an ordinary differential equation using the differential operator. Given the system: d^2x/dt - x + d^2y/dt^2 + y = 0 and dx/dt + 2x + dy/dt + 2y = 0 find x and y equation Answer: x = 5ce^(-2t) y = -3ce^(-2t) Homework EquationsThe Attempt at a Solution We change...
  46. Tspirit

    I Can the expectation of an operator be imaginary?

    Assume ##\varPsi## is an arbitrary quantum state, and ##\hat{O}## is an arbitrary quantum operator, can the expectation $$\int\varPsi^{*}\hat{O}\varPsi$$ be imaginary?
  47. S

    A Transformation of the spinor indices of the Weyl operator under the Lorentz group

    The left-handed Weyl operator is defined by the ##2\times 2## matrix $$p_{\mu}\bar{\sigma}_{\dot{\beta}\alpha}^{\mu} = \begin{pmatrix} p^0 +p^3 & p^1 - i p^2\\ p^1 + ip^2 & p^0 - p^3 \end{pmatrix},$$ where ##\bar{\sigma}^{\mu}=(1,-\vec{\sigma})## are sigma matrices.One can use the sigma...
  48. P

    Momentum and Position Operator Commutator Levi Civita Form

    Homework Statement Prove that ##[L_i,x_j]=i\hbar \epsilon_{ijk}x_k \quad (i, j, k = 1, 2, 3)## where ##L_1=L_x##, ##L_2=L_y## and ##L_3=L_z## and ##x_1=x##, ##x_2=y## and ##x_3=z##. Homework Equations There aren't any given except those in the problem, however I assume we use...
  49. T

    I Permutation operator and Hamiltonian

    The permutation operator commutes with the Hamiltonian when considering identical particles, which implies: $$ [\hat{P}_{21}, \hat{H}] = 0 \tag{1}$$ Now given a general eigenvector ##{\lvert} {\psi} {\rangle}##, where $$ \hat{P}_{21} (\hat{H}{\lvert}{\psi}){\rangle} = (\hat{P}_{21} \hat{H})...
  50. D

    Understanding Hermitian Operators: Exploring Their Properties and Applications

    Basically I've seen some expressions involving Hermitian Operators that I can't seem to justify, that others on the internet throw around like axiomatic starting points. (AB+BA)+ = (AB)++(BA)+? Why does this work? Assuming A&B are hermitian, I get why we can assume A+B is hermitian, but does...
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