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. 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...
  2. 4

    I Creation and annihilation operator commutation confusion

    In Quantum Field Theory by Lancaster, equation 3.14 $$ [\hat{a_i},\hat{a_j}^\dagger]=\delta{ij}$$ is introduced as "we define". Yes, example 2.1, where the creation and annihilation operators applied to harmonic operator states, there is a nice simple proof that this is true (although...
  3. S

    Algebra of displacement operator

    Homework Statement Given an operator ##D(\alpha)=\exp\ (\alpha a^{\dagger}-\alpha^{*}a)## and a function ##g(a,a^{\dagger})##, where ##a## and ##a^{\dagger}## are operators and ##\alpha## and ##\alpha^{*}## are complex numbers, show that ##D^{-1}(\alpha)g(a,a^{\dagger})D(\alpha)=g(a+\alpha...
  4. W

    Linear Differential Equations and Linear Operator Problem

    Homework Statement I'm not sure how to approach this. The question involves linear operators and a non-homogenous differential equation. Here is the question: https://s15.postimg.org/cdmw80157/Capture.png Homework Equations They are given in the question The Attempt at a Solution I really...
  5. sa1988

    Verify that this kinetic energy operator is Hermitian

    Homework Statement Not actually a homework question but is an exercise in my lecture notes. Homework Equations I'm following this which demonstrates that the momentum operator is Hermitian: The Attempt at a Solution$$KE_{mn} = (\frac{-\hbar^2}{2m}) \int\Psi_{m}^{*} \Psi_{n}^{''} dx $$ $$...
  6. O

    Exponentiating Matrices: Representation of \exp{(iÔ)}

    Consider the operator Ô, choose a convenient base and obtain the representation of \ exp{(iÔ)} Ô = \bigl(\begin{smallmatrix} 1 & \sqrt{3} \\ \sqrt{3} & -1 \end{smallmatrix}\bigr) Attempt at solution: So, i read on Cohen-Tannjoudji's Q.M. book that if the matrix is diagonal you can just...
  7. Aswin Sasikumar 1729

    I Is there any operator for momentum in terms of t?

    Since there is an energy operator interms of t and a momentum operator interms of x as expected.For energy there is a hamiltanion operator interms of t which is unexpected for me.Similarly whether there is any operator interms of t for momentum also?
  8. Mayan Fung

    B Quantum Oscillator States: Is the Ladder Operator Enough?

    We learned that we can use the ladder operator to obtain the states of a quantum oscillator. However, I see no direct evidence to show that the solutions are complete. I mean, how can we know the energy state follows E is (E+hw). Why can't we have some more states in between? Does the derivation...
  9. Mayan Fung

    I Exploring the Kinetic Energy Operator: Why is it Differentiated Twice?

    When I learned about operators, I learned <x> = ∫ Ψ* x Ψ dx, <p> = ∫ Ψ* (ħ/i ∂/∂x) Ψ dx. The book then told me the kinetic energy operator T = p2/2m = -ħ2/2m (∂2/∂x2) I am just think that why isn't it -ħ2/2m (∂/∂x)2 Put in other words, why isn't it the square of the derivative, but...
  10. Guilherme Vieira

    How to Calculate Probability using Density Operator?

    Hello, I'm trying to understand how to calculate de probability of finding a system in a specific eigenstate using the density operator. In the book of Balian, Haar, Gregg I've found a good definition of it being the expectation value of the projector Pr in the orientation of the eingenstate...
  11. Kevin McHugh

    B Projection Operator: What Math Operation Does it Represent?

    What mathematical operation does this expression represent" |i><i| I know the i's are unit vectors, and I know <i|i> is the dot product, but what operation is the ket-bra?
  12. K

    I When is a given state an eigenstate of a given operator?

    How do I know if some given state is and eigenstate of some given operator?
  13. mertcan

    I Quantum mechanics getting position operator from momentum

    hi, initially I want to put into words that I looked up the link (http://physics.stackexchange.com/questions/86824/how-to-get-the-position-operator-in-the-momentum-representation-from-knowing-the), and I saw that $$\langle p|[\hat x,\hat p]|\psi \rangle = \langle p|\hat x\hat p|\psi \rangle -...
  14. S

    MHB Operator form of integro-differential equation

    Hi For brevity one usually writes Fredholm integral equation of the 2nd kind $\psi(x)=f(x)+\int_{a}^{b} \,k(x,s)\psi(s) ds$ into the form $\psi=f+K \psi$ where $K$ is the operator kernel My question can one write an integro differential equation $\d{\psi(x)}{x}=f(x)+\int_{a}^{b}...
  15. D

    I Complex conjugate and time reversal operator

    Hi. I'm confused about the action of the complex conjugate operator and time reversal operator on kets. I know K(a |α > ) = a* K | α > but what is the action of K on | α > where K is the complex conjugation operator ? What is the action of the time reversal operator Θ on a ket , ie. what is Θ...
  16. Romanopoulos Stelios

    Differentiation of unitary operator U(t,t') in Peskin and Schroeder

    How the authors came to the conclusion (eq. 4.25) that $$ U(t,t')=e^{iH_0(t-t_0)} e^{-iH(t-t')} e^{-iH_0(t'-t_0)} $$
  17. N

    B Position Operator: Mathematically Defined

    Well i am noobie to quantum physics so i matbe totally incorrect so please bear with me. I had question how is position operator defined mathematically. I was reading the momentum position commutator from...
  18. binbagsss

    QM Bra & Ket Linear Algebra Hermitian operator proof -- quick question

    Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...
  19. L

    Finding Eigenvalues and Wave Function in a Basis of Orthonormalized Vectors

    Homework Statement Eigenvalues of the Hamiltonian with corresponding energies are: Iv1>=(I1>+I2>+I3>)/31/2 E1=α + 2β Iv2>=(I1>-I3>) /21/2 E2=α-β Iv3>= (2I2> - I1> I3>)/61/2 E3=α-β Write the matrix of the Hamiltonian in the basis of...
  20. F

    A Action of an anti-difference operator

    hi, i want to ask you that how to perform an anti-difference operator on the product of two functions? i.e. D^{-1}(f_{n} g_{n})
  21. A

    Proof for the Hermitian operator

    1. Homework Statement prove the following statement: Hello, can someone help me prove this statement A is hermitian and {|Ψi>} is a full set of functions Homework Equations Σ<r|A|s> <s|B|c>[/B]The Attempt at a Solution Since the right term of the equation reminds of the standard deviation, I...
  22. A

    Matrix representation of certain Operator

    Homework Statement Vectors I1> and I2> create the orthonormal basis. Operator O is: O=a(I1><1I-I2><2I+iI1><2I-iI2><1I), where a is a real number. Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors are...
  23. olgerm

    B An equation from terms of operator del to terms of sums

    https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fd3adddbdfb95797d11ef6167ecda4efe3e0b9 https://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_in_terms_of_potentials How to write this formula in terms of sums and vector components? What is ##v\cdot\nabla## ? I think it is some...
  24. E

    A Commutator of field operator with arbitrary functions

    In QFT, the commutation relation for the field operator \hat{\phi} and conjugate momentum is [\phi(x,t),\pi(y,t)] = i\delta(x-y) Maybe this is obvious, but what would the commutator of \phi or \pi and, say, e^{i k\cdot x} be?
  25. M

    A How Does the Dirac Spin Exchange Operator Work in Quantum Mechanics?

    The spin exchange operator would have the property $$\begin{align*}P\mid \chi_{\uparrow\downarrow} \rangle = \mid\chi_{\downarrow\uparrow} \rangle & &P\mid \chi_{\downarrow\uparrow} \rangle =\mid \chi_{\uparrow\downarrow} \rangle \end{align*}$$ This also implies ##P\mid \chi_{\text{sym.}}...
  26. F

    Operator, eigenstate, small calculation

    Hello :-) I have a small question for you :-) 1. Homework Statement The Operator e^{A} is definded bei the Taylor expanion e^{A} = \sum\nolimits_{n=0}^\infty \frac{A^n}{n!} . Prove that if |a \rangle is an eigenstate of A, that is if A|a\rangle = a|a\rangle, then |a\rangle is an...
  27. M

    A What Is the Spectrum of a Linear Operator in L2 Spaces?

    http://<img src="https://latex.codecogs.com/gif.latex?L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left&space;|t-s&space;\right&space;|f(s)ds" title="L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left |t-s \right |f(s)ds" />[/PLAIN] I have many doubts on linear operator. How I can find a spectrum of a linear...
  28. J

    I Box normalization for the free particle-Position operator

    Hello, When we normalize the free particle by putting it in a box with periodic boundary conditions, we avoid the "pathological" nature of the momentum representation that take place in the normal problem of a particle in a box with the usual boundary conditions of Ψ=0 at the two borders. Thus...
  29. D

    I Confused about operator action

    Hi. I have come across the following equation < x | p | ψ > = - iħ d/dx < x | ψ > where p is the momentum operator. I'm confused as to how p which acts to the right on | ψ >can then be taken outside the inner product so it now also acts on the bra < x | ? Thanks
  30. J

    A How to understand the electric-field operator?

    I know the positive field operator E+ is actually an annihilation operator a while the negative field E- is a creation operator a+. I also learned that the absorption process can be represented as E-E+, which should be the number of photons n accroding to the principle of ladder operator. Also...
  31. B

    A Interpreting the K-G Annihilation Operator Expression

    I can derive $$a(k) = \int d^3 x e^{ik_{\mu} x^{\mu}} (\omega_{\vec{k}} \psi + i \pi)$$ for a free real scalar Klein-Gordon field in three ways mathematically: the usual Fourier transform way in Peskin/Srednicki, an awesome direct a = ½(2a) = ... way (exercise!), and as a by-product of a clever...
  32. weezy

    I What is a time-dependent operator?

    While studying Ehrenfest's theorem I came across this formula for time-derivatives of expectation values. What I can't understand is why is position/momentum operator time-independent? What does it mean to be a time-dependent operator? Since position/momentum of a particle may change...
  33. D

    Question about the del operator under a translation

    Homework Statement This isn't really a problem. I am just re-reading some section "Classical Mechanics" by John Taylor. I think this belongs in the math section, since my question is mainly about the del operator. There is just one fragment of one sentence that I want to make sure I am...
  34. otaKu

    I Regarding the eigenvalues of the translation operator

    I don't understand what the eigenvalue of a translation operator means physically. The eigenvalues of other operators like momentum and hamiltonian give us the physically measurable values I suppose. Then what exactly do we obtain by the translation eigenvalues? I am new to the field of quantum...
  35. munirah

    Reduced Density operator in matrix form

    Homework Statement I already read book of Quantum Computation and Quantum Information by Nielsen and Chuang according to reduced density operator and I already understand how to do the reduced density using Dirac notation, Ket Bra.Homework Equations [/B] My problem here I want to know the...
  36. S

    B P → q with NOR operator and Constant F (false)

    How can I show (without using truth table) that p → q is equivalent to F ↓ ((F ↓ p) ↓ q) where F is constant "false" and p and q are propositions? Is it possible to have a similar kind of expression with T (true) instead of F? Thanks in advance!
  37. bananabandana

    How Does Time Evolution Affect Quantum Operator Matrix Elements?

    Homework Statement [/B] For a general operator ## \hat{O}##, let ##\hat{O}_{mn}(t)## be defined as: $$ \hat{O_{mn}}(t) = \int u^{*}_{m}(x,t) \hat{O} u_{n}(x,t) $$ and $$ \hat{O_{mn}} = \int u^{*}_{m}(x) \hat{O} u_{n}(x) $$ ##u_{m}## and ##u_{n}## are energy eigenstates with corresponding...
  38. G

    I Raising index on covariant derivative operator?

    In Carroll, the author states: \nabla^{\mu}R_{\rho\mu}=\frac{1}{2} \nabla_{\rho}R and he says "notice that, unlike the partial derivative, it makes sense to raise an index on the covariant derivative, due to metric compatibility." I'm not seeing this very clearly :s What's the reasoning...
  39. ChrisVer

    What Is the Use of the Bitwise OR Operator in Python?

    With an example, can someone explain me what the following code would do? In fact I am not certain I understand how the |= operator is used. (I have checked for operator overloading within the classes but there is none, so it supposedly does the default operation) objectA = ClassA(arguments)...
  40. A

    I How do you derive Slater determinant from creation operator?

    Hello, Could someone provide me with a good proof or explain me here how we can derive Slater determinant for N fermions by starting with the vacuum state and the creation operators with anticommutation equations. I see that the idea of both these structures is similar but I cannot work it out...
  41. S

    I Proving the Linearity of the Curl Operator in Electromagnetic Theory

    Hi, I stumbled upon thinking that "Is curl operator a linear operator" ? I was reading EM Theory and studied that the electromagnetic field satisfies the curl relations of E and B. But if the operator was not linear then how can a non linear operator give rise to a linear solution. Thus it...
  42. K

    Pauli Equation - Simple operator algebra question

    Homework Statement I am watching a course on Relativistic Quantum Mechanics to freshen up, and I have found to have some issues regarding simple operator algebra. This particular issue on the Pauli Equation (generalization of the Schrodinger equation that includes spin corrections) in an...
  43. J

    I QM - Ladder Operator QHO - factorization

    Hi, quick question with A being the lowering operator and A† the raising operator for a QHO (A A† - 1 + 1/2) ħω [Aψ] = A (A† A - 1 + 1/2) ħω ψ By taking out a factor of A. Why has the ordering of A A† swapped around? I would have thought taking out a factor of A would leave it as A (A† - 1 +...
  44. ShayanJ

    A Non-negativity of the eigenvalues of the Dirac operator

    How can I prove that the eigenvalues of the operator ## i\gamma^\mu \partial_\mu ## are non-negative? I've tried using the ansatz ## \psi=u(p) e^{ip_\nu x^\nu} ## but it didn't help. I've also tried playing with the equation using the properties of gamma matrices but that doesn't seem to lead...
  45. AwesomeTrains

    Ladder operator commutator with arbitary function

    Hey there! 1. Homework Statement I've been given the operators a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}} and a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}} without the constants and definition of the momentum operator: a=x+\partial_x and a^\dagger=x-\partial_x with...
  46. J

    I What is the complex conjugate of the momentum operator?

    Hello, i am kind of confused about something. What is the complex conjugate of the momentum operator? I don't mean the Hermitian adjoint, because i know that the Hermitian adjoint of the momentum operator is the momentum operator. Thanks!
  47. DarkMatter5

    Calculating the momentum operator in a quantum state

    Homework Statement A gaussian wave packet is given by the formula: Ψ(x)=(1/(π1/4d1/2))eikx-(x2/2d2) Calculate the expectation value in this quantum state of the momentum squared. Homework Equations <p2>=-ħ∫Ψ*(X) (d2Ψ(x)/dx2) dx ∫e(-x2/d2) dx= d√π ∫xe(-x2/d2) dx =0 ∫x2e(-x2/d2) dx = (d3√π)/2...
  48. Smalde

    QM: Time development of the probability of an Eigenvalue

    The problem is actually of an introductory leven in Quantum Mechanics. I am doing a course on atomic and molecular physics and they wanted us to practice again some of the basics. I want to know where I went conceptually wrong because my answer doesn't give a total probability of one, which of...
  49. P

    I When should one eigenvector be split into two (same span)?

    This question was inspired by 3c) on https://people.phys.ethz.ch/~muellrom/qm1_2012/Solutions4.pdf Given the operator \hat{B} = \left(\matrix{b&0&0\\0&0&-ib\\0&ib&0}\right) I find correctly that the eigenvalues are \lambda = b, \pm b. To find the eigenvectors for b, I do the following...
  50. D

    I Momentum operator on positon/momentum representation

    Hi. I have come across the following step in a derivation of the harmonic oscillator groundstate wavefunction using ladder operators ∫ <x | p | p><p | o > dp = ∫ p<x | p><p | o > dp = -iħ d/dx ∫ <x | p><p | 0>dp I am confused about how the -iħ d/dx arises. I thought the p produced when the p...
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