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.

View More On Wikipedia.org
Recent contents Top users
  1. W

    Solving Projection Operator Questions - QM Basics

    Hello, Suppose P is a projection operator. How can I show that I+P is inertible and find (I+P)^-1? And is there a phisical meaning to a projection operator? (Please be patient I have just started with QM). Thanks. Y.
  2. A

    Time Ordering Operator: Integrals & Step Function

    I really asked this question in another thread but it seems the original respondent gave up explaining me. My question is about the rewriting of the integrals from first to second line on the attached picture. The θ denotes the heaviside step function such that: θ(t1-t2) = {1 t1>t2 , 0 t1<t2}...
  3. FeDeX_LaTeX

    Treatment of Integral as an Operator?

    My tutor showed me something today, and I still can't completely wrap my head around why it makes sense. Consider the following integral equation: ##\int f(x) = f(x) - 1## Then: ##\int (f(x)) - f(x) = -1 \implies f(x) \left( \int (\text{Id}) - 1\right) = -1## so we get the geometric series...
  4. T

    Understanding the Normal Operator Equality in Proofs of Properties

    I'm having trouble seeing how this equality is possible which is seen in proofs of properties of normal operators. ||Tv||^2 = <T*Tv, v> = <TT*v, v> = ||T*v||^2 As far as I can get is ||Tv||^2 = <Tv, Tv>
  5. H

    Showing there are no eigenvectors of the annhilation operator

    Homework Statement Show there are no eigenvectors of a^{\dagger} assuming the ground state |0> is the lowest energy state of the system. Homework Equations Coherent states of the SHO satisfy: a|z> = z|z> The Attempt at a Solution Based on the hint that was given (assume there...
  6. A

    Derivative of Time Evolution Operator: Exp(-iHt)

    For the time evolution operator: exp(-iHt) How do I take the derivative of an operator like this keeping the order correct? I mean I of course know how to differentiate an exponential function, but this is the exponential of an operator.
  7. M

    Calculating the norm of linear, continuous operator

    Homework Statement . Let ##X=\{f \in C[0,1]: f(1)=0\}## with the ##\|x\|_{\infty}## norm. Let ##\phi \in X## and let ##T_{\phi}:X \to X## given by ##T_{\phi}f(x)=f(x)\phi(x)##. Prove that ##T## is a linear continuous operator and calculate its norm. The attempt at a...
  8. J

    Is there a way to eliminate the second derivative in calculus using integration?

    Given that D²f(x) = g(x), one form that eliminate the second derivate is integrating the equation: ∫∫D²f(x)dx² = ∫∫g(x)dx². But, and if I try so: \\ \sqrt{D^2f(x)}=\sqrt{g(x)} \\ D\sqrt{f(x)}=\sqrt{g(x)} \\ PD\sqrt{f(x)}=P\sqrt{g(x)} \\ \sqrt{f(x)}=P\sqrt{g(x)} \\ f(x)=[P\sqrt{g(x)}]^2 \\...
  9. A

    Operator which is written in k space

    I have an operator which is written in k space as something like: H = Ʃkckak where a_k and c_k are operators. So it is a sum of operators of different k but there are no crossterms as you can see. Being no crossterms does this mean that the operator is diagonalized in the language of linear...
  10. B

    Does a Group Action Always Use the Group's Original Operation?

    A group ##G## is said to act on a set ##X## when there is a map ##\phi:G×X \rightarrow X## such that the following conditions hold for any element ##x \in X##. 1. ##\phi(e,x)=x## where ##e## is the identity element of ##G##. 2. ##\phi(g,\phi(h,x))=\phi(gh,x) \ \ \forall g,h \in G##. My...
  11. C

    Expectation value of the time evolution operator

    This problem pertains to the perturbative expansion of correlation functions in QFT. Homework Statement Show that \langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle = \left(\langle0|T\left[exp\left(-i\int_{-t}^{t}dt'...
  12. J

    The angular momentum operator acting on a wave function

    Hi guys, I need help on interpreting this solution. Let me have two wave functions: \phi_1 = N_1(r) (x+iy) \phi_2 = N_2(r) (x-iy) If the angular momentum acts on both of them, the result will be: L_z \phi_1 = \hbar \phi_1 L_z \phi_2 = -\hbar \phi_2 My concern is, \phi_1 and \phi_2...
  13. K

    Multigrid : Restriction operator

    Hello everybody, I have a small question related to multigrid. I am trying to solve a Poisson equation in 3D with periodic boundary conditions with cell-centered multigrid. I have programmed a quite fast serial code which performs V-cycles with a maximum of 7 grids. For the interpolation...
  14. M

    Particle Number Operator (Hermitian?)

    Particle Number Operator (Hermitian??) Hey guys, I'm studying the quantic harmonic oscillator and I'm using "Cohen-Tannoudji Quantum Mechanics Volume 1". At some point he introduced the particle number operator, N, such that: N=a+.a , where a+ is the conjugate operator of a. The...
  15. Sudharaka

    MHB Uniquely Determined Linear Operator

    Hi everyone, :) Here's a problem that I want to confirm my answer. Note that for the second part of the question it states, "prove that \(T\) is bonded by the above claim". I used a different method and couldn't find a method that relates the first part to prove the second. Problem: Suppose...
  16. K

    MHB Eigenvector and eigenvalue for differential operator

    My friends and I have been struggling with the following problem, and don't understand how to do it. We have gotten several different answers, but none of them make sense. Can you help us? **Problem statement:** Let $V$ be the vector space of real-coefficient polynomials of degree at most $3$...
  17. W

    Can an annihilation operator be found for this Hamiltonian?

    Homework Statement Given the Hamiltonian H(t) = \frac{P^2}{2m} + \frac{1}{2}mw^2X^2 + b(XP+PX) from some b>0. Find an annihilation operator a_b s.t. [a_b,a_b^{\dagger}]=1 and H = \hbar k (a_b^{\dagger}a_b+\frac{1}{2}) for some constant k. Hint: [P + aX,X]=[P,X], \forall a.Homework Equations...
  18. O

    How do operators combine in quantum mechanics?

    Hello everyone I've got these things buzzing in my head and not exactly knowing how to solve them. Homework Statement Operator Ahat = (d/dx + x) and Bhat = (d/dx - x) a. Chat = AhatAha b. Chat = AhatBhat What do the position and momentum operator Xhat = x and Phat = -i*hbar*d/dx, give when...
  19. S

    Finding the eigenvalues of the spin operator

    1. What are the possible eigenvalues of the spin operator \vec{S} for a spin 1/2 particle? Homework Equations I think these are correct: \vec{S} = \frac{\hbar}{2} ( \sigma_x + \sigma_y + \sigma_z ) \sigma_x = \left(\begin{array}{cc}0 & 1\\1 & 0\end{array}\right),\quad...
  20. D

    Trace of operator with continuous spectrum

    Greetings, I must be missing something obvious but how is Tr{} defined exactly in case of contunuous spectrum operators? Everywhere I look I see it defined as a sum of [possibly infinite sequence of] eigenvalues. Is the following correct: Given Q = \int f(q) \left| q\right\rangle...
  21. D

    Expectation value of a hermitian operator prepared in an eigenstate

    Hey guys, So this question is sort of a fundamental one but I'm a bit confused for some reason. Basically, say I have a Hermitian operator \hat{A}. If I have a system that is prepared in an eigenstate of \hat{A}, that basically means that \hat{A}\psi = \lambda \psi, where \lambda is real...
  22. T

    What is the relationship between creation and annihilation operators in k-space?

    Hi, Could anyone tell if there exists an identity a_k^\dagger = a_{-k} because intuitively there should be no difference between creating a particle with momentum k and destroying a particle with momentum -k. If true is it possible to show that from the definition a_k = \frac{1}{√V}∫e^{ikx}...
  23. L

    On SUSY irreps: eigenvalues of Pauli-Lubanski operator?

    Hi everyone, Just an easy question that came to my mind while studying basics of SUSY. Consider in N=1, D=4 a massive clifford vacuum |m,s,s_3\rangle, and for cconcreteness take its spin to be s=1/2. Now, acting with the four supercharges on both the |m,1/2,1/2\rangle and |m,1/2,-1/2\rangle...
  24. D

    Checking that a coherent state is an eigenfunction of an operator

    Homework Statement Hey guys, I'll type this thing up in Word. http://imageshack.com/a/img716/8219/wycz.jpg
  25. R

    Why Doesn't My Del Operator Conversion Work for Vector Quantities?

    I have been trying to convert the Del operator from Cartesian to Cylindrical coords since like 5 days. but still i can't see why my way doesn't work. It worked for the 3D heat equation and 3D wave equation but for vector quantities no :( ... This is the way i followed \nabla P =...
  26. L

    <p> Operator on Probability Density in X-Space

    Homework Statement Consider a particle whose wave function is: \Psi(x)=\left\{\begin{array}{ccc} 2\alpha^{3/2}xe^{-\alpha x} & \text{if} & x> 0\\ 0 & \text{if} & x\leq 0 \end{array}\right. Calculate <p> using the \hat{p} operator on probability density in x space. Homework...
  27. H

    Hamiltonian function vs. operator

    I've dealt with both the Hamiltonian function for Hamiltonian mechanics, and the Hamiltonian operator for quantum mechanics. I have a kind of qualitative understanding of how they're similar, especially when the Hamiltonian function is just the total energy of the system, but I was wondering if...
  28. D

    Bra-kets and operator formalism in QM - Expectation values of momentum

    Homework Statement sup guys! I think I've solved this set of problems, but I was just wondering if I've done it right - I have no way to tell. I'll put all the questions and answers here - plus the stuff I used. So could you please tell me if there's any mistakes? Here it is - using Word...
  29. L

    Problem understanding operator algebra

    "It is left as a problem for the reader to show that if [S,T] commutes with S and T, then [e^{tT}, S] = -t[S,T]e^{tT} I'm not sure if I'm missing something here, but i don't even see how it is possible to arrive at this answer. I get: [e^{tT}, S] = e^{tT}S - Se^{tT} Then using the fact...
  30. T

    What exaclty is a differential operator?

    Homework Statement I have fallen behind on my Numerical Methods course and am starting to fail it. I need to know how to make a differential approximation and I'm reading through the materials but I have too little time and it doesn't even explain what a differential operator is. At first it is...
  31. ajayguhan

    Linear Operator vs Linear Function: Technical Difference

    What is the exact technical difference between a linear operator and linear function?
  32. J

    Figuring symmetries of a differential operator from its eigenfunctions

    So, I understand that the derivative operator, D=\frac{d}{dx} has translational invariance, that is: x \rightarrow x - x_0, and its eigenfunctions are e^{\lambda t}. Analogously, the theta operator \theta=x\frac{d}{dx} is invariant under scalings, that is x \rightarrow \alpha x, and its...
  33. P

    Expectation value for non commuting operator

    if 2 hermitian operator A, B is commute, then AB=BA, the expectation value <.|AB|.>=<.|BA|.>. how about if A and B is non commute operator? so we can not calculate the exp value <.|AB|.> or <.|BA|.>?
  34. R

    How Can Hermitian Operators Prove Key Quantum Mechanics Equations?

    Prove the equation A\left|\psi\right\rangle = \left\langle A\right\rangle\left|\psi\right\rangle + \Delta A\left|\psi\bot\right\rangle where A is a Hermitian operator and \left\langle\psi |\psi\bot\right\rangle = 0 \left\langle A\right\rangle = The expectation value of A. \Delta A...
  35. D

    Expectation value of an operator in matrix quantum mechanics

    Homework Statement Hey everyone. Imma type this up in Word as usual: http://imageshack.com/a/img577/3654/q9ey.jpg Homework Equations http://imageshack.com/a/img22/3185/pfre.jpg The Attempt at a Solution http://imageshack.com/a/img703/8571/xogb.jpg
  36. jk22

    Rotation of Spin Operator and Vector in 3D Space

    If we consider a spin 1/2 particle, then, the rotation of the spinor for each direction is given by a rotation matrix of half the angle let say theta Rspin=\left(\begin{array}{cc} cos(\theta/2) & -sin(\theta/2)\\sin(\theta/2) & cos(\theta/2)\end{array}\right) and the new component of the spin...
  37. D

    Finding normalized eigenfunctions of a linear operator in Matrix QM

    Homework Statement Hey everyone! The question is this: Consider a two-state system with normalized energy eigenstates \psi_{1}(x) and \psi_{2}(x), and corresponding energy eigenvalues E_{1} and E_{2} = E_{1}+\Delta E; \Delta E>0 (a) There is another linear operator \hat{S} that acts by...
  38. M

    Why Is Vector Notation Essential in Cross Product Calculations?

    Why do we use the coordinates of r in terms of x,y,z?Why don't we express coordinates of A in x,y,z?
  39. O

    Ehrenfest theorem, is there any condition for the operator Q?

    For commutator, HQ-QH = 0 . But for this case as shown below, complex ψQHψ - HcomplexψQψ= 0? If the operator Q is in term of (∂/∂t) and (∂/∂x) ,then the HQ-QH may not be zero. Is there any restriction for Q operator?
  40. H

    Understand Operator Dispersion in Sakurai's "Modern Quantum Physics

    I'm trying to get my head around quantum mechanics with the help of Sakurai "Modern Quantum Physics". It's been good so far, but I came across a formula I don't really understand. When discussing uncertainty relation (in 1.4) the author begins with defining an "operator": \Delta A \equiv A -...
  41. J

    Confusion about this min() operator

    I'm reading about planning algorithms and I'm having some difficulty understanding a bit of notation here. The pdf I'm reading is "planning.cs.uiuc.edu/ch2.pdf" and the equation in question is on page 11. I'm not sure I understand what the min operator with all the subscripts actually means...
  42. G

    What Are the Probabilities of Measuring Each Spin State for a Spin-1 Particle?

    The S_{z} operator for a spin-1 particle is S_{z}=\frac{h}{2\pi}[1 0 0//0 0 0//0 0 -1] I'm given the particle state |\phi>=[1 // i // -2] What are the probabilities of getting each one of the possible results? Now... we can say the possible measure results will be 1,0,-1 and the...
  43. L

    Interchanging Linear Operator and Infinite Sum

    Suppose that x\in H, where H is a Hilbert space. Then x has an orthogonal decomposition x = \sum_{i=0}^\infty x_i. I have a linear operator P (more specifically a projection operator), and I want to write: P(x) = \sum_{i=0}^\infty P(x_i). How can I justify taking the operator inside the...
  44. N

    Del operator - order of operations

    Hey! Is it true that when you dot the del-operator on another vector, the differentiation has priority over the dot-product? That's why you get all those weird formulas for the divergence in circular and cylindrical coordinates (which are very different to the Cartesian ones)? So in the case of...
  45. K

    Spectrum of Momentum operator in the Hilbert Space L^2([-L,L])

    Homework Statement Find the spectrum of the Momentum operator in the Hilbert Space defined by L^2([-L,L]), consisting of all square integrable functions ψ(x) in the range -L, to L Homework Equations We can get the resolvent set containting all λ in ℂ such that you can always find a...
  46. L

    Commutator Relations; Conjugate Product of a Dimensionless Operator

    Consider the following commutator for the product of the creation/annihilation operators; [A*,A] = (2m(h/2∏)ω)^1 [mωx - ip, mωx + ip] = (2m(h/2∏)ω)^1 {m^2ω^2 [x,x] + imω ([x,p] - [p,x]) + [p,p]} Since we have the identity; [x,p] = -[p,x] can one assume that.. [x,p] - [p,x] =...
  47. D

    Proof of a linear operator acting on an inverse of a group element

    Hey guys! Basically, I was wondering how to prove the following statement. I've seen it in the Hamermesh textbook without proof, so I wanted to know how you go about doing it. Let's say you have a group element g_{1}, which has a corresponding inverse g_{1}^{-1}. Let's also define a linear...
  48. J

    Show that a unitary operator maps one ON-basis to another

    Homework Statement Given an inner product space V, a unitary operator U and a set \left\{\epsilon_i\right\}_{i=1,2,\dots} which is an orthonormal basis of V, show that the image of \left\{\epsilon_i\right\} under U is also an orthonormal basis of V Homework Equations The Attempt at a...
  49. V

    Nabla Operator in Spherical Coordinates

    Homework Statement Exercise 1.3 on uploaded Problem Sheet. Homework Equations Shown in Exercise 1.3 on Problem Sheet The Attempt at a Solution Uploaded working: I have found the inverse of the Transformation Matrix from Cartesian to Spherical Coordinates by transposing...
  50. B

    Getting Eigenvalues Into a Differential Operator

    Following Butkov, a second order ode A(x)y'' + B(x)y' + C(x)y = D(x) can always be brought into Sturm-Liouville form \tfrac{d}{dx}[p(x)y'] - s(x)y = f(x) after multiplying across by H(x) = - \tfrac{1}{A(x)}e^{\int^x \tfrac{B(t)}{A(t)}dt}. He then says the function s(x) can...
Back
Top