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. Mayan Fung

    I How do we know that the raising operator only raises the state by one step?

    In the simple harmonic oscillator, I was told to use the raising and lowering operator to generate the excited states from the ground state. However, I am just thinking that how do we confirm that the raising operator doesn't miss some states in between. For example, I can define a raising...
  2. L

    A Is the Frechet Derivative of a Bounded Linear Operator Always the Same Operator?

    I understand the Frechet derivative of a bounded linear operator is a bounded linear operator if the Frechet derivative exists, but is the result always the same exact linear operator you started with? Or, is it just "a" bounded linear operator that may or may not be known in the most general case?
  3. F

    I Divergence & Curl -- Is multiplication by a partial derivative operator allowed?

    Divergence & curl are written as the dot/cross product of a gradient. If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator. is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation
  4. JD_PM

    Evaluating a momentum operator

    I think I get the approach. We first need to evaluate the term ##\dot A_{\mu} \nabla A^{\mu}## and then evaluate the 3D space integral; we may need to take the limit ##V \rightarrow \infty## (i.e ##\sum_{\vec k} (2 \pi)^3/V \rightarrow \int d^3 \vec k##) at some point. The mode expansions of...
  5. koroshii

    Finding a unitary operator for quantum non-locality.

    My trouble might be from how I interpret the problem. Alice and Bob are entangled. After Alice makes the measurement both of their states should collapse to one of these states with a certain probability. (Unless my understand of how entanglement is wrong.) The way I am understand the question...
  6. LCSphysicist

    I Hermitian Operators and Non-Orthogonal Bases: Exploring Infinite Spaces

    The basis he is talking about: {1,x,x²,x³,...} I don't know how to answer this question, the only difference i can see between this hermitians and the others we normally see, it is that X is acting on an infinite space, and, since one of the rules involving Hermitian fell into decline in the...
  7. Filip Larsen

    I Rotational invariance of cross product matrix operator

    Given that the normal vector cross product is rotational invariant, that is $$\mathbf R(a\times b) = (\mathbf R a)\times(\mathbf R b),$$ where ##a, b \in \mathbb{R}^3## are two arbitrary (column) vectors and ##\mathbf R## is a 3x3 rotation matrix, and given the cross product matrix operator...
  8. DuckAmuck

    I Momentum operator acting to the left

    Is the following true if the momentum operator changes the direction in which it acts? \langle \phi | p_\mu | \psi \rangle = -\langle \phi |\overleftarrow{p}_\mu| \psi \rangle My reasoning: \langle \phi | p_\mu | \psi \rangle = -i\hbar \langle \phi | \partial_\mu | \psi \rangle \langle...
  9. LCSphysicist

    Normalize the eigenfunction of the momentum operator

    I am just solving the equation $$\frac{h}{2\pi i}\frac{\partial F}{\partial x} = pF$$, finding $$F = e^{\frac{ipx2\pi }{h}}C_{1}$$, and$$ \int_{-\infty }^{\infty }C_{1}^2 = 1$$, which gives me $$C_{1} = \frac{1}{(2\pi)^{1/2} }$$, so i am getting the answer without the h- in the denominator...
  10. chocopanda

    Harmonic oscillator with ladder operators - proof using the Sum Rule

    I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator: $$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$ The exercise explicitly says to use laddle operators and to express $p$ with $$b=\sqrt{\frac {mw}{2 \hbar}}-\frac...
  11. BenKlesc

    How to become a commercial nuclear plant operator?

    I have a question for anyone on here that has pursued the nuclear energy field. I'm interested in becoming an equipment technician or operator at a nuclear power plant, but I am already 26 years old. I understand that many of the operators and technicians were former Navy nukes. I looked into...
  12. Mayan Fung

    I Energy operator in Quantum Mechanics

    I learned that the energy operator is ##\hat{E} = i\hbar \frac{\partial}{\partial t} ## and the Hamiltonian is ##\hat{H} = \frac{-\hbar^2}{2m}\nabla^2+V(r,t)## If the Hamiltonian represents the total energy of the system. I expect the two should be the same. Did I misunderstand the concept of...
  13. K

    Calculating Curve Integrals with the Del Operator: A Pain in the Brain?

    My attempt is below. Could somebody please check if everything is correct? Thanks in advance!
  14. Haorong Wu

    I Variational operator in the least action principle

    Hello. Since I learned the least action principle several years ago, I cannot figure out the difference between the variational operator ##\delta## in ##\delta S=0## and the differential operator ##d## in, say ##dS##. Everytime I encountered the variational operator, I just treated it as a...
  15. Haorong Wu

    Could an operator act on a bra vector?

    I am confused about the problem. I thought operators do not act on bra vectors, and the problem is equivalent to ##a^{\dagger} \left | \alpha \right > = \left ( \alpha ^{*} + \frac {\partial} {\partial \alpha} \right ) \left | \alpha \right > ##. Then, strangely, ##\left < \alpha \right |##...
  16. Haorong Wu

    How to calculate an operator in the Heisenberg picture?

    I have some problems when calculating the operators in Heisenberg picture. First, ##\frac {dx} {dt} = \frac {1} {i \hbar} \left [ x, H \right ] = \frac {p} {m}##. Similarly, ##\frac {dp} {dt} = \frac {1} {i \hbar} \left [ p, H \right ] = - m \omega ^ 2 x##. These are coupled equations. I...
  17. N

    I A question about operator power series

    Hi All, I've been going through Shankar's 'Principles of Quantum Mechanics' and I don't quite understand the point the author is trying to make in this exercise. I get that this wavefunction is not a solution to the Schrodinger equation as it is not continuous at the boundaries and neither is...
  18. troglodyte

    I Struggling with one step to show quantum operator equality

    Hello guys, I struggle with one step in a calculation to show a quantum operator equality .It would be nice to get some help from you.The problematic step is red marked.I make a photo of my whiteboard activities.The main problem is the step where two infinite sums pops although I work...
  19. abivz

    I Obtaining the Dirac function from field operator commutation

    Hi everyone, I'm new to PF and this is my second post, I'm taking a QFT course this semester and my teacher asked us to obtain: $$[\Phi(x,t), \dot{\Phi}(y,t) = iZ\delta^3(x-y)]$$ We're using the Otto Nachtman: Elementary Particle Physics but I've seen other books use this notation: $$[\Phi(x,t)...
  20. abivz

    I QFT - Field operator commutation

    Hi everyone, I'm taking a QFT course this semester and we're studying from the Otto Nachtman: Texts and Monographs in Physics textbook, today our teacher asked us to get to the equation: [Φ(x,t),∂/∂tΦ(y,t)]=iZ∂3(x-y) But I am unsure of how to get to this, does anyone have any advice or any...
  21. F

    Does operator L^2 commute with spherical harmonics?

    My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions. I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero...
  22. Replusz

    I Commutator with gradient operator (nabla)

  23. C

    A Partial differential equation containing the Inverse Laplacian Operator

    I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...
  24. Q

    Action of the time reversal operator on the QM wave equation

    Applying the time reversal operator to the plane wave equation: Ψ = exp [i (kx - Et)] T[Ψ ] = T{exp [i (kx - Et)]} = exp [i (kx + Et)] This looks straightforward as I have simply applied the 'relevant equation' however my doubt is in relation to the possible action of operator T on the i...
  25. A

    I When does the exchange operator commute with the Hamiltonian

    I am attaching an image from David J. Griffith's "Introduction to Quantum Mechanics; Second Edition" page 205. In the scenario described (the Hamiltonian treats the two particles identically) it follows that $$PH = H, HP = H$$ and so $$HP=PH.$$ My question is: what are the necessary and...
  26. O

    A Help with the Proof of an Operator Identity

    I'm trying to come up with a proof of the operator identity typically used in the Mori projector operator formalism for Generalized Langevin Equations, e^{tL} = e^{t(1-P)L}+\int_{0}^{t}dse^{(t-s)L}PLe^{s(1-P)L}, where L is the Liouville operator and P is a projection operator that projects...
  27. P

    I Proof of Commutator Operator Identity

    Hi All, I try to prove the following commutator operator Identity used in Harmonic Oscillator of Quantum Mechanics. In the process, I do not know how to proceed forward. I need help to complete my proof. Many Thanks.
  28. sophiatev

    I Significance of the Exchange Operator commuting with the Hamiltonian

    In an Introduction to Quantum Mechanics by Griffiths (pg. 180), he claims that "P and H are compatible observables, and hence we can find a complete set of functions that are simultaneous eigenstates of both. That is to say, we can find solutions to the Schrodinger equation that are either...
  29. JD_PM

    Finding the eigenfunctions and eigenvalues associated with an operator

    The eigenvalue equation is $$\frac{d^2}{d \phi^2} f(\phi) = q f(\phi)$$ This is a second order linear homogeneous differential equation. The second order polynomial associated to it is $$\lambda ^2 - q = 0 \rightarrow \lambda = \pm \sqrt{q}$$ As both roots are real and distinct, the...
  30. JD_PM

    Show that the Hamiltonian operator is Hermitian

    $$<f|\hat H g> = \int_{-\infty}^{\infty} f^*\Big(-\frac{\hbar}{2m} \frac{d^2}{dx^2} + V(x) \Big) g dx$$ Integrating (twice) by parts and assuming the potential term is real (AKA ##V(x) = V^*(x)##) we get $$<f|\hat H g> = -\frac{\hbar}{2m} \Big( f^* \frac{dg}{dx}|_{-\infty}^{\infty} -...
  31. T

    Density operator and natural log/trace

    I feel like I'm going around in circles trying to do something with the expression ## tr( \rho *log(\rho)) ##. I thought about a Taylor expansion, but I don't think there's a useful one here because of the logarithm. We learned the Jacobi's formula in class, but I don't think I want a derivative...
  32. BeyondBelief96

    I Use of the Beam Splitter Operator

    Hello, I am a senior undergrad doing research in quantum optics, and I am trying to work out at the moment the output state of sending a coherent state through one input port and a squeezed vacuum state through the other, just to see what happens tbh. The problem I have constantly been running...
  33. H

    What is the inverse of the covariance operator in Brownian motion?

    in fact the answer is given in the book (written by philippe Martin). we have $$ (\tau_1| A^{-1} | \tau_2) = 2D \ min(\tau_1 ,\tau_2) = 2D(\tau_1 \theta (\tau_2 -\tau_1)+\tau_2 \theta (\tau_1 -\tau_2))$$ So $$-1/2D \frac{d^2}{d\tau_1^2} (\tau_1| A^{-1} | \tau_2) = \delta( \tau_1 - \tau_2) $$...
  34. B

    I Isospin Operator: Act on |ud> State

    How does the isospin operator I_3 act on a state |ud>, where u ist an up- and d a Down quark?
  35. George Keeling

    I What is a good formula for the Laplace operator?

    I have found various formulations for the Laplacian and I want to check that they are all really the same. Two are from Wikipedia and the third is from Sean Carroll. They are: A Wikipedia formula in ##n## dimensions: \begin{align} \nabla^2=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial...
  36. The black vegetable

    I Charge conjugation operator

    I have in my notes the charge conjugation operator converts the spinnor into its complex conjugate , ## C\begin{pmatrix} \varepsilon \\ \eta \end{pmatrix}=\begin{pmatrix} \varepsilon^{*}{} \\ \eta ^{*} \end{pmatrix}##when applied to gamma matrix from dirac equation does it do the same...
  37. Math Amateur

    MHB Understanding Andrew Browder's Prop 8.7: Operator Norm and Sequences

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need yet further help in fully understanding the proof of Proposition 8.7 ...Proposition...
  38. Math Amateur

    MHB Understanding Proposition 8.7: Operator Norm and Sequences

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some further help in fully understanding the proof of Proposition 8.7 ...Proposition...
  39. Math Amateur

    MHB Operator Norm and Cauchy Sequence .... Browder, Proposition 8.7 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the proof of Proposition 8.7 ...Proposition 8.7 and...
  40. Math Amateur

    MHB Operator Norm and Distance Function .... Browder, Proposition 8.6 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the concepts in Proposition 8.6 ...Proposition 8.6...
  41. Math Amateur

    MHB Operator norm .... Field, Theorem 9.2.9 ....

    I am reading Michael Field's book: "Essential Real Analysis" ... ... I am currently reading Chapter 9: Differential Calculus in \mathbb{R}^m and am specifically focused on Section 9.2.1 Normed Vector Spaces of Linear Maps ... I need some help in fully understanding Theorem 9.2.9 (3) ...
  42. M

    Show that the position operator does not preserve H

    The attempt ##\int_{-\infty}^{\infty} |ψ^*(x)\, \hat x\,\psi(x)|\, dxˆ## Using ˆxψ(x) ≡ xψ(x) =##\int_{-\infty}^{\infty} |ψ^*(x)\,x\,\psi(x)|\, dxˆ## =##\int_{-\infty}^{\infty} |ψ^*(x)\,\psi(x)\,x|\, dxˆ## =##\int_{-\infty}^{\infty} |x\,ψ^2(x)|\, dxˆ## I'm pretty sure this is not the...
  43. Math Amateur

    MHB Operator norm --- Remarks by Browder After Lemma 8.4 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding some remarks by Browder after Lemma 8.4 pertaining to...
  44. S

    I Operator for the local average of a growing oscillating function

    First some background, then the actual question... Background: (a) Very simple example: if we take ##Asin(x+ϕ)+0.1##, the average is obviously 0.1, which we can express as the integral over one period of the sine function. (assume that we know the period, but don't know the phase or other...
  45. Math Amateur

    MHB Operator Norm .... differences between Browder and Field ....

    I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the differences between Andrew Browder and Michael...
  46. Riemann9471

    Operator that commutes with the Hamiltonian

    Homework Statement: In the case of the quantum harmonic oscillator in 3D , does the z-component of the angular momentum of a particle commute with the Hamiltonian? Does the fundamental state has a well defined value of L_z (variance = 0) ? If you said no , why? If you said yes , what is the...
  47. binbagsss

    Exponential of ladder operator

    I'm just trying to follow the below And I understand all, I think, except what's happened to the term when A hits 1: [A,1] ? If I'm correct basically we're just hitting on the first operator so reducing the power by one each time of the operator in the right hand bracket thanks
  48. Haynes Kwon

    Hermiticity of AB where A and B are Hermitian operator?

    Trying to prove Hermiticity of the operator AB is not guaranteed with Hermitian operators A and B and this is what I got: $$<\Psi|AB|\Phi> = <\Psi|AB\Phi> = ab<\Psi|\Phi>=<B^+A^+\Psi|\Phi>=<BA\Psi|\Phi>=b^*a^*<\Psi|\Phi>$$ but since A and B are Hermitian eigenvalues a and b are real, Therefore...
  49. H

    I Matrix Representation of the Angular Momentum Raising Operator

    In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem. This should be value...
  50. Q

    A Eigenvalues for a non self adjoint operator

    Hi all- I am trying to obtain eigenvalues for an equation that has a very simple second order linear differential operator L acting on function y - so it looks like : L[y(n)] = Lambda (n) * y(n) Where y(n) can be written as a sum of terms in powers of x up to x^n but I find L is non self...
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