# 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. ### Eigenvalues of Hamiltonian operator

Hello, I try to solve this problem, and I think a) wasn't too hard, I have the following solution: ##H = \lambda (\frac{\vec{S^2-(\vec{S_1}^2+\vec{S_2}^2)}{2})##. I struggle with 2. I find it very abstract. When I have H as a matrix I know how to calculate eigenvalues, but I don't know how...
2. ### A How to add higher dimensional operator at higher energy in SM?

Hi, I Learned that we can add higher dimensional operator but they are non-renormalizable - but effect of higher dimensional operator is vanishes at low energy - my question is than how can we add higher dimensional operator at higher energy - like dimension 5 operator ( weinberg operator)...
3. ### A Creation and annihilation operator

Hey, I have a short question. The quantized field in Schrödinger picture is given by: \hat{\phi} \left(\textbf{x}\right) =\int \frac{d^{3}p}{\left(2\pi\right)^3} \frac{1}{\sqrt{\omega_{2\textbf{p}}}}\left(\hat{a}_{\textbf{p}}e^{i\textbf{p} \cdot \textbf{x}} +...
4. ### Show that an operator is Hermitian

Hi, unfortunately, I have problems with the following task I tried the fast way, unfortunately I have problems with it I have already proved the following properties, ##\bigl< f,xg \bigr>=\bigl< xf,g \bigr>## and ##\bigl< f, \frac{d}{dx}g \bigr>=-\overline{f(0)} g(0)+\bigl< f,g...
5. ### I Can the Lieb-Robinson Bound be Intuited from the Taylor Series of an Operator?

Hi all, I was wondering if there was a way to intuit the Lieb-Robinson bound from simply looking at the taylor series for an operator ##A(t) = e^{-iHt}Ae^{iHt}## where ##H## is a k-local Hamiltonian and ##A(t)## initially starts off as a single-site operator. The generic idea is that at each...
6. ### I Spin operator and spin quantum number give different values, why?

Assume spin 1/2 particle So the spin operator gives +/- hbar/2 eg. S |n+> = +/- hbar/2 |n+> But S= s(s+1) hbar = sqrt(3)/2 hbar So I'm off by a factor of sqrt(3). I suspect I am missing something fundamental about my understanding of spin. My apologies and thanks in advance.

9. ### I How to add time variation to a Schrodinger operator?

I'm looking at Dirac's "Lectures on Quantum Field Theory" and I have a question about the basic mathematics of something that's part of ordinary quantum mechanics. On page 3, he says, The two pictures are connected in this way: any Schrodinger dynamical variable is connected with the...
10. ### A Relation between the density matrix and the annihilation operator

This question is related to equation (1),(3), and (4) in the [paper][1] [1]: https://arxiv.org/abs/2002.12252
11. ### A Momentum operator -- Why do we use the plane wave solution?

Why in order to derive the QM momentum operator we use the plane wave solution. Why later on in field theory and particle physics, the plane wave ansatz is so physically important?
12. ### POTW A Linear Operator with Trace Condition

Let ##V## be a finite dimensional vector space over a field ##F##. If ##L## is a linear operator on ##V## such that the trace of ##L\circ T## is zero for all linear operators ##T## on ##V##, show that ##L = 0##.
13. ### I Derivation of two-electron density operator

Hello, I am going over the derivation for two-electron density. I am having a hard time understanding how the second term in 2.11a seen below is derived. I know this term must eliminate the i=j products but can't seem to understand how. Thanks for the help.
14. ### A How to take a matrix outside the diagonal operator?

How to derive (proof) the following trace(A*Diag(B*B^T)*A^T) = norm(W,2), where W = vec(sqrt(diag(A^T*A))*B) & sqrt(diag(A^T*A)) is the square root of diag(A^T*A), B & A are matrix. Please see the equation 70 and 71 on page 2068 of the supporting matrial.
15. ### I If T is diagonalizable then is restriction operator diagonalizable?

The usual theorem is talking about the linear operator being restricted to an invariant subspace: I had no problem understanding its proof, it appears here for example: https://math.stackexchange.com/questions/3386595/restriction-operator-t-w-is-diagonalizable-if-t-is-diagonalizable However, I...
16. ### Comp Sci Assigning a variable using a unary operator

int a[]={12,34,55,76,89,23}; int n=5; while(n>1){ a[n]=a[--n];} for(int i=0;i<6;i++){ printf("%d\t",a[i]); } Mentor note: Please use code tags in future posts. I've added them to this code. The above code should shift the element values to the right of each array cell, but after running...
17. ### I Does there exist momentum-shift operator?

As is well known there is translation operator in position space, such that., $$\exp(i\hat{p}a)x\exp(-i\hat{p}a)=x+a.$$ While in momentum space, can we have analog of the above mentioned translation operator? i.e., momentum shift operator? $$\exp(-i\hat{x}q)p\exp(i\hat{x}q)=p+q.$$ If so, why...
18. ### I What are the units of the squeezing parameter?

As I understand it, when the squeezing operator acts on an annihilation/creation operator, a function of sinh(r) and cosh(r) is produced, where r is the squeezing parameter. I've been reading some papers that say that up to '15 dB of squeezing' have been produced in a laboratory. Does this mean...
19. ### I Differential operator in multivariable fundamental theorem

I'm referring to this result: But I'm not sure what happens if I apply a linear differential operator to both sides (like a derivation ##D##) - more specifically I'm not sure at what point should each term be evaluated. Acting ##D## on both sides I'll get...
20. ### JavaScript Why use spread operator when sorting an array in JavaScript?

I Understand the basic theory behind spread operator. const girlNames = ['Jessica', 'Emma', 'Amandine'] const newGirlNames = [...girlNames] console.log(newGirlNames) // Output: ["Jessica", "Emma", "Amandine"] But I don't understand when it's applied. See here. function createList() {...
21. ### I Velocity operator, its expression and eigenvalues

Cohen Tannoudji pp 215 pp 225 pp 223 From above we can say that there exists a velocity operator ##\mathbf v=\frac{\mathbf p}{m}## ,whose eigenvalues are the observed values of velocity. 1. I've seen multiple times that we can't define velocity in quantum mechanics, but here I find that...
22. ### JavaScript Why use spread operator in this react program? What does it do?

Code for this is here: https://codesandbox.io/s/floral-frog-7zfon3 I got the general idea of program flow in this case. First user clicks a button, then function gets called, that function in turn updates the state from initial [] state to [{id:1,value:0.12232}]. Now the array map accesses...
23. ### B Operator that returns unique number of binary matrix

If we have an arbitrary square matrix populated randomly with 1s and 0s, is there an operator which will return a unique number for each configuration of 1s and 0s in the matrix? i.e. an operation on $$\begin{pmatrix} 1 &0 &0 \\ 1 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}$$ would return something...
24. ### Unitary Operators: Proving <Af,Ag>=<f,g>

Hello folks, I need to show that a unitary operator obeys <Af,Ag>=<f,g>, where A is a unitary operator. However, I am technically not yet given the fact, that the adjoint of A is equal to its inverse, and that is the problem. I have no clue how to prove the given task without using the...
25. ### I Unitary Operator: Can X Make L(A) Unitary?

Unitary operator If an operator L(A) = [X, A], are there matrix X for which the operator is unitary? Keep in mind it is in a complex matrix space with standard inner product (A, B) = tr(A*B)
26. ### Is an operator (integral) Hermitian?

Knowing that to be Hermitian an operator ##\hat{Q} = \hat{Q}^{\dagger}##. Thus, I'm trying to prove that ##<f|\hat{Q}|g> = <\hat{Q}f|g> ##. However, I don't really know what to do with this expression. ##<f|\hat{Q}g> = \int_{-\infty}^{\infty} [f(x)^* \int_{-\infty}^{\infty} |x> <x| dx f(x)] dx##...
27. ### I Why is K an anti-unitary operator in (26)?

Hey all, I just wanted to double check my understanding of (26) in the following notes: https://arxiv.org/pdf/1512.08882.pdf. Is the reason that ##(U_{T}\cdot K) \cdot (U_{C}\cdot K) = U_{T}\cdot U_{C}^{*}## because ##K## is a unitary operators and thus ##(K\cdot U_{C}\cdot K) = U_{C}^{*}## as...
28. ### Using Parity operator for addition/subtraction

This is for a Quantum Mechanics class but part b of this question seemed like it relied more on math than physics so I think it appropriate to post here. If not, Mods please move to appropriate place. For the ##\Pi xf(\vec r)+x\Pi f(\vec r)=0## I have my answer circled in red on the first...
29. ### A Spectrum of the Liouville Operator

Context: Consider a classical system with Hamiltonian ##H##. The Liouville differential operator can be defined using the Poisson brackets as $$L=-i\left \{ ,H \right \}.$$ ##L## is Hermitian in the Hilbert space of square integrable wavefunctions over phase space. The spectrum of ##L## is easy...
30. ### Linear operator in 2x2 complex vector space

Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2? ____________________________________________________________ An ordered basis for C2x2 is: I don't...
31. ### I Proving U=eiA is Unitary: Exploring -1 as Exponent and Inverse

Hi If A is a Hermitian operator then U = eiA is a unitary operator. To prove this we take the Hermitian conjugate of U U+ = e-iA = (eiA)-1 (1) ⇒ U+ = U-1 (2) My question is - In line (1) , -1 is used as an exponent or power while in line (2) , -1 is used to refer to the inverse of a...
32. ### I Operators in finite dimension Hilbert space

I have a question about operators in finite dimension Hilbert space. I will describe the context before asking the question. Assume we have two quantum states | \Psi_{1} \rangle and | \Psi_{2} \rangle . Both of the quantum states are elements of the Hilbert space, thus | \Psi_{1} \rangle , |...
33. ### I Where does the QM creation operator get its energy?

We are all familiar with ladder operators, such as QM harmonic oscillators or in QFT to produce energy states which are interpreted as particles. But when a creation operator raises the energy level of a system, where does that energy come from?
34. ### Proving commutator relation between H and raising operator

I am going through my class notes and trying to prove the middle commutator relation, I am ending up with a negative sign in my work. It comes from [a†,a] being invoked during the commutation. I obviously need [a,a†] to appear instead. Why am I getting [a†,a] instead of [a,a†]?
35. ### I Vector operator acting on a ket gives a ket out of the state space

Definition of linear operator in quantum mechanics "A linear operator ##A## associates with every ket ##|\psi\rangle \in \mathcal{E}## another ket ##\left|\psi^{'}\right\rangle \in\mathcal{E}##, the correspondence being linear" We also have vector operators ##\hat{A}## (such as a position...
36. ### A Matrix representation of a unitary operator, change of basis

If ##U## is an unitary operator written as the bra ket of two complete basis vectors :##U=\sum_{k}\left|b^{(k)}\right\rangle\left\langle a^{(k)}\right|## ##U^\dagger=\sum_{k}\left|a^{(k)}\right\rangle\left\langle b^{(k)}\right|## And we've a general vector ##|\alpha\rangle## such that...
37. ### Tong QFT sheet 2, question 6: Normal ordering of the angular momentum operator

My attempt/questions: I use ##T^{0i} = \dot{\phi}\partial^i \phi##, ##\dot{\phi} = \pi##, and antisymmetry of ##Q_i## to get: ##Q_i = 2\epsilon_{ijk}\int d^3x [x^j \partial^k \phi(\vec{x})] \pi(\vec{x})##. I then plug in the expansions for ##\phi(\vec{x})## and ##\pi(\vec{x})## and multiply...
38. ### How can negative integers be used in deriving the Hamiltonian for open strings?

On ***page 38*** of Becker Becker Schwarz, we're given ***equation 2.69*** which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2})$$ Considering the open string we have...
39. ### B Muons: How Particles "Pop In and Out of Existence

This a quote from Nature of the 30 of june 2022 'Like the electron, the muon has a magnetic field that makes it act like a tiny bar magnet. As muons travel, they generate various particles that briefly pop in and out of existence." Now I would like to know how 'various particles' pop in and out...
40. ### MHB Forward Unit Push Operator Equation

Hopefully the symbols I am using are standard. I will define them upon request. I have a theorem that says, given a difference equation \left ( \sum_{j = 0}^m a_j E^j \right ) y_n = \alpha ^n F(n), we can define a polynomial function \phi (E) = \sum_{j = 0}^m a_j E^j such that \phi (E) y_n =...
41. ### I What is the Role of the Radial Position Operator in Quantum Mechanics?

While trying to find the expectation value of the radial distance ##r## of an electron in hydrogen atom in ground state the expression is : ##\begin{aligned}\langle r\rangle &=\langle n \ell m|r| n \ell m\rangle=\langle 100|r| 100\rangle \\ &=\int r\left|\psi_{n \ell m}(r, \theta...
42. ### I Deriving the Commutator of Exchange Operator and Hamiltonian

In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange...
43. ### I Operator acting on ket state n

I tried playing with the number's operator eigenvalues equation but couldn't get anywhere, can s/b help me out?
44. ### I Position representation of angular momentum operator

One of the component of angular momentum operator is ##\hat{L}_{x}=\hat{y} \hat{P}_{z}-\hat{z} \hat{P}_{y}## I want it's position representation. My attempt : I'll find the representation of the first term ##\hat{y} \hat{P}_{z}##. The total representation is the sum of two terms. The...
45. ### I Form of potential operator of two interacting particles

Considering two interacting particles in 3d, the corresponding Hilbert space ##H## is the tensor product of the two individual Hilbert spaces of the two particles. If the particle interaction is given by a potential ##V(\mathbf r_1 -\mathbf r_2)## ,what is the corresponding potential operator...
46. ### B Operator as the outcome of a measurement?

for any set of POVM outcomes it is possible to construct a setup with say one incoming photon and possible outcomes that will click differently. so this is not only mathématics. but what is physically an operator valued measurement?
47. ### I Basic question on meaning of momentum operator

How do we apply the momentum operator on a wavefunction? Wikipedia says > the momentum operator can be written in the position basis as: ##{ }^{[2]}## ## \hat{\mathbf{p}}=-i \hbar \nabla ## where ##\nabla## is the gradient operator, ##\hbar## is the reduced Planck constant, and ##i## is the...
48. ### I How to apply potential operator ##V(\hat{x})##

I want some clarification on the potential operator ##V(\hat{x})##. Can you please help me ------------------------------ Is the action of ##V(\hat{x})## defined by its action on the position kets as ##\hat{V}(x)|x\rangle=V(x)|x\rangle##? Then we'd have for any ket ##|\psi\rangle## that...
49. ### I Adjoint of position operator

I'm trying to find the adjoint of position operator. I've done this: The eigenvalue equation of position operator is ##\hat{x}|x\rangle=x|x\rangle## The adjoint of position operator acts as ##\left\langle x\left|\hat{x}^{\dagger}=x<x\right|\right.## Then using above equation we've...
50. ### Total Momentum Operator for Klein Gordon Field

As $$\hat{P_i} = \int d^3x T^0_i,$$ and $$T_i^0=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi-\delta_i^0\mathcal{L}=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi=\pi\partial_i\phi.$$ Therefore, $$\hat{P_i} = \int d^3x \pi\partial_i\phi.$$ However...