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.

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  1. S

    Proof that HK is hermitian operator only if HK=KH

    Let H and K be hermitian operators on vector space U. Show that operator HK is hermitian if and only if HK=KH. I tried some things but I don't know if it is ok. Can somebody please check? I got a hint on this forum that statements type "if only if" require proof in both directions, so here...
  2. ShayanJ

    Understanding the Use of Controlled Not Gates in Quantum Computing

    In an article I'm reading, the author defines an operator as below: \hat{U}_{CNOT}(\theta)=\exp{(-i \theta \hat{U}_{CNOT})}=\hat{1} \cos{\theta}-i \hat{U}_{CNOT} \sin{\theta} Where \hat{U}_{CNOT} is the controlled not gate(http://en.wikipedia.org/wiki/Controlled_NOT_gate). Then the...
  3. F

    Measuring the physical quantity corresponding to an operator.

    Homework Statement Homework Equations N/A The Attempt at a Solution I am having trouble getting my head around these questions, the first part a) wasn't too tricky, I used the fact that eigenfunctions of a Hermitian operator \hat{O} are orthogonal and got my normalisation...
  4. Fernando Revilla

    MHB Amy's question at Yahoo Answers (Self adjoint operator)

    Here is the question: Here is a link to the question: Show a linear transformation is self adjoint? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  5. J.Hong

    Operator algebra of chiral quasi-primary fields

    Studying conformal field theory, I tried to derive general expression for the commutation relations of the modes of two chiral quasi-primary fields. At first, I expressed the modes \phi_{(i)m} and \phi_{(j)n} as contour integrals over each fields, and took commutation relation. I used...
  6. S

    Why do we consider higher dimensional operators in the search for new physics?

    I am confuse about the dimension of an operator? Why we need an operator of Dim six or greater for new physics?
  7. H

    Inverse of Operator: Is it True?

    is this true? (1/ηαβ∂α∂β)= ηαβ∂α∂β any help,pls!
  8. sunrah

    Statistical operator of hydrogen atom

    Homework Statement Individual hydrogen atoms have been prepared in the energy state n = 2. However, nothing is known about the remaining quantum numbers. Fine structure and all corrections can be ignored. What is the micro-canonical statistical operator. Homework Equations \hat{\rho_{mc}} =...
  9. sunrah

    What Is the Density Operator of an Unknown System?

    Homework Statement What is the density operator (statistical operator) of a system about which nothing is known? Homework Equations \hat{\rho} = \sum p_{i} |i\rangle\langle i| The Attempt at a Solution If nothing is known about a system we must assume something in order to make...
  10. J

    An integral equation Abel and L operator?

    an integral equation Abel and L operator?? 1. The Abel operator The general Abel integral equation \begin{gathered}\intop_{x}^{a}\dfrac{F(y)dy}{\left(y^{2}-x^{2}\right)^{\frac{1+u}{2}}}=f(x)\end{gathered} has the solution \begin{gathered}F(r)=-\dfrac{2\cos\frac{\pi...
  11. Y

    Rotation operator quantum mechanics

    Hi everyone, I'm stuck on the concept of the rotation operator in QM. From what I understand, one constructs a representation of SO(3) on a Hilbert space by mapping a rotation matrix R\in SO(3) specified by an angle \phi and a unit vector \vec{n} to D(R) = \exp[-\frac{i...
  12. M

    How to Estimate the Operator Norm ||A||_2 for a Difference Operator?

    Greetings everyone! I have a set of tasks I need to solve using using operator norms, inner product... and have some problems with the task in the attachment. I would really appreciate your help and advice. This is what I have been thinking about so far: I have to calculate a non trivial upper...
  13. BruceW

    Quantum Operator Derivation: Exploring Intuitive Approaches

    Hi all! I was reviewing some basic quantum mechanics, and I was trying to 'derive' the equation -i \hbar \frac{\partial}{\partial x} \psi_{(t,x)} = <x| \hat{P} | \psi > using the commutator relation, and the form of the identity operator. OK, I know that the proper, mathematical way to prove the...
  14. L

    Proof That Every Linear Operator L:ℝ→ℝ Has Form L(x)=cx

    Homework Statement Show that every linear operator L:ℝ→ℝ has the form L(x) = cx for some c in ℝ. Homework Equations A linear operator in vector space V is a linear transformation whose domain and codomain are both V. The Attempt at a Solution If L is a vector space of the real...
  15. S

    Square Root of Positive Operator

    Homework Statement Suppose U = T^2 + \alpha T + \beta I is a positive operator on a real inner product space V with \alpha^2 < 4 \beta . Find the square root operator S of U.Homework Equations The Attempt at a Solution Isn't this just the operator S \in L(V) such that S e_k = \sqrt{...
  16. Ikaros

    Show a real, smooth function of Hermitian operator is Hermitian

    Homework Statement If B is Hermitian, show that BN and the real, smooth function f(B) is as well. Homework Equations The operator B is Hermitian if \int { { f }^{ * }(x)Bg(x)dx= } { \left[ \int { { g }^{ * }(x)Bf(x) } \right] }^{ * } The Attempt at a Solution Below is my...
  17. S

    QM operator and double slit experiment doubt basics

    well i am a starter in QM, i have 2 big doubts ! let me first tell what i understood , there is phsi which defines a state of a system, phsi times x is a position operator and phsi 's derivative of x multiplied by i h is its momentum operator ... well then i operator these in phsi and what do...
  18. T

    Quantum mechanics operator manipulation

    Homework Statement consider operator defined as \hat{O_A} = \hat{A} -<\hat{A}> show that (ΔA)^2=<\hat{O_A}^2> Homework Equations (ΔA)^2=<\hat{A}^2>-<\hat{A}>^2 The Attempt at a Solution (ΔA)^2=<\hat{A}^2>-<\hat{A}>^2 = <\hat{A}^2> - (\hat{A} -\hat{O_A})^2 = <\hat{A}^2> -...
  19. T

    Prove angular momentum operator identity

    Homework Statement Using the operator identity: \hat{L}^2=\hat{L}_-\hat{L}_+ +\hat{L}_z^2 + \hbar\hat{L}_z show explicitly: \hat{L}^2 = -\hbar^2 \left[ \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2} + \frac{1}{\sin\theta} \frac{\partial}{\partial\theta}...
  20. B

    Complex Numbers: Equation involving the argument operator.

    Homework Statement Question: Homework Equations Any relevant to complex numbers. The Attempt at a Solution Given, Arg(\frac{z}{w})= Arg(z)-Arg(w) z=x+yi z1 = -1-2i z2 = 2+3i Arg(z-z1)=Arg(z2-z1) LHS: Arg(x+yi+1+2i) Arg((x+1) + i(y+2)) tan(\theta)=\frac{y+2}{x+1}...
  21. B

    Quantum Harmonic Oscillator ladder operator

    Homework Statement What is the effect of the sequence of ladder operators acting on the ground eigenfunction \psi_0 Homework Equations \hat{A}^\dagger\hat{A}\hat{A}\hat{A}^\dagger\psi_0The Attempt at a Solution I'm not sure if I'm right but wouldn't this sequence of opperators on the ground...
  22. J

    Shouldn't the operator be applied to both the wf anD its modulus?

    Why is it: $$ \langle A \rangle = \int dV ~ \psi^* (\hat{A} \psi) $$ As opposed to: $$ \langle A \rangle = \int dV ~ (\hat{A} \psi^*) (\hat{A} \psi) $$ The op can substantially change the wf, so it would seem to make more mathematical sense (at least from a linear algebra pov) to...
  23. T

    How to understand current-density operator

    Dear all, There is a definition of current-density operator, and the form is as follows: j(r)=1/2iƩ[∇lδ(r-rl)+δ(r-rl)∇l] I cannot understand this form, because i think ∇ operator and δ operator are commutable. Another form of current-density operator can be found from this website...
  24. V

    Operator in a real vector space has an upper block triangular matrix

    Hello All, I was trying to prove that an operator T in a real vector space V has an upper block triangular matrix with each block being 1 X 1 or 2 X 2 and without using induction. The procedure which i followed was : We already know that an operator in a real vector space has either a one...
  25. atyy

    Time operator in string theory

    In covariant quantization of the string, say as in David Tong's http://arxiv.org/abs/0908.0333 (p28), time is an operator. Is the time operator Hermitian, and does it correspond to an observable?
  26. fluidistic

    Why can't we define a time related operator?

    I asked my professor if we could define time as an operator and he said no. I've read on the web that time isn't an observable because "you don't measure the time of a particle". Yet, to me at least, the sentence "you don't measure the time of a particle" is similar to "you don't measure the...
  27. S

    Quick Matrix Element Question using Hermitian Operator

    Hi there, This should be very simple... If I have a state <1|AB|2> where A and B are Hermitian operators, can I rewrite this as <2|BA|1> ? That would be, taking the complex conjugate of the matrix element and saying that A*=A and B*=B. Thank you!
  28. fluidistic

    Eigenvalues of the position operator

    I'm new to QM, but I've had a linear algebra course before. However I've never dealt with vector spaces having infinite dimension (as far as I remember). My QM professor said "the eigenvalues of the position operator don't exist". I've googled "eigenvalues of position operator", checked into...
  29. 7

    Some weird integration by parts to derive momentum operator

    In a Griffith's book (page 15-16) an author derives a momentum operator. In the derivation he states that he used a integration by parts two times. He starts with this equation which i do understand how to get to. $$ \begin{split} \frac{d \langle x \rangle}{dt} = -\frac{i\hbar}{2m}...
  30. M

    What are the effects of a translation on a vector in R2?

    Homework Statement Let a be a fixed nonzero vector in R2. A mapping of the form L(x) = x + a is called a translation. Show that a translation is not a linear transformation. Illustrate geometrically the effect of a translation. My work is in the photo below, can you check and see if I'm...
  31. R

    Is the Displacement Operator Tψ(x)=ψ(x+a) Hermitian?

    Consider the displacement operator Tψ(x)=ψ(x+a). Is T Hermitian?
  32. T

    Operator Transformation under Change of Basis

    Homework Statement Consider the three operators defined by $$\left(S_i\right)_{jk} = -i\epsilon_{ijk}$$ in the x-y-z space and the basis vectors given in x-y-z space as $$e^{\left(1\right)} = -\frac{1}{\sqrt{2}}\left(e_x + ie_y\right), e^{\left(0\right)} = e_z, e^{\left(-1\right)} =...
  33. M

    Compact linear operator in simple terms?

    Hi, I'm struggling to understand this concept. I think the term probably comes from functional analysis and I don't know any of the terms in that field so I'm having trouble understanding the meaning of what a compact linear operator is. I posted this in linear algebra because I'm reading...
  34. C

    Differential Operator to prove identity

    Homework Statement Use ##D = \frac{d}{dx}##as a differential operator and the following $$(D - a)(D -b)[f(x)e^{\lambda x}] = e^{\lambda x} (D + \lambda -a)(D + \lambda -b)f(x)$$ to obtain $$(D^2 + D +1)[(Ax^2 + Bx + C)e^{ix}] = (iAx^2 + [iB + (4i + 2)A]x + 2A + (2i + 1)B + iC)e^{ix}$$ The...
  35. V

    Inverse Weyl quantization of the projection operator.

    I am trying to solve the following problem on an old Quantum Mechanics exam as an exercise. Homework Statement Homework Equations I know that the trace of an operator is the integral of its kernel. \begin{equation} Tr[K(x,y)] = \int K(x,x) dx \end{equation} The Attempt at a...
  36. J

    What is the mathematical intuition behind operator embedding?

    Can someone explain to me the mathematical intuition that motivates the embedding of quantum operators between the conjugate wave function and the (non-conjugated) wave function? That is, we write: \Psi^{*}\hat{H}\Psi, that is: \Psi^{*}(\hat{H}\Psi), so that \hat{H} operates on \Psi (not...
  37. R

    MHB Is the Exponential of a Linear Operator Defined?

    In class we recently learned that for a linear operator T: V \rightarrow V and function g(t) = a_0 + a_1t + \dots + a_nt^n one can define the operator g(T) = a_0I + a_1T + \dots + a_nT^n (where I is the identity transformation). We also recently learned about the exponential of a matrix. My...
  38. L

    Translation operator. Infinite potential well.

    For potential well problem for well with potential which is zero in the interval ##[0,a]## and infinite outside we get ##\psi_n(x)=\sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a}##. If I want to get this result for well with potential which is zero in the interval ##[-\frac{a}{2},\frac{a}{2}]## and...
  39. V

    Operator acting on the function

    Homework Statement Calculate the action of the operator on the function f(x) Homework Equations Operator - exp(a*x^2*(d/dx)) The Attempt at a Solution
  40. S

    Prove that the angular momentum operator is hermitian

    Greetings, My task is to prove that the angular momentum operator is hermitian. I started out as follows: \vec{L}=\vec{r}\times\vec{p} Where the above quantities are vector operators. Taking the hermitian conjugate yields \vec{L''}=\vec{p''}\times\vec{r''} Here I have used double...
  41. W

    Why the Casimir operator is proportional to the unit matrix ?

    Hello, now I'm reading Peskin Shroeder. I have a question about the Casimir operator on page 500 in Chapter 15. From the following eq, ## \ \ \ [T^b , T^a T^a ] = 0 \ \ \ \ \ \ \ (15.91) ## ## T^2(=T^a T^a) ## is an invariant of the algebra. Thus the author concludes that ##T^2## is...
  42. M

    Understanding the Translational Operator and Its Applications

    e^{\alpha\frac{d}{dx}}=1+\alpha\frac{d}{dx}+\frac{\alpha^2}{2!}\frac{d^2}{dx^2}+...=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^n}{dx^n} Why this is translational operator? ##e^{\alpha\frac{d}{dx}}f(x)=f(x+\alpha)##
  43. C

    Help with vector operator Del.

    Homework Statement In the Pauli theory of the electron, one encounters the expresion: (p - eA)X(p - eA)ψ where ψ is a scalar function, and A is the magnetic vector potential related to the magnetic induction B by B = ∇XA. Given that p = -i∇, show that this expression reduces to ieBψ...
  44. E

    Unexpected operator for 'plotinequality'

    When I try to operate this command plot(plot::Inequality(x^2 + y^2 < 1,x = -1.5..1.5, y = -1.5..1.5)) It failed, and displayed Error: Unexpected MATLAB operator. How can I fix it?
  45. B

    Fourier Transfrom and expectation value of momemtum operator

    Homework Statement Using <\hat{p}n> = ∫dxψ*(x)(\hat{p})nψ(x) and \hat{p} = -ihbar∂x and the definition of the Fourier transform show that <\hat{p}> = ∫dk|\tilde{ψ}(k)|2hbar*k 2. The attempt at a solution Let n = 1 and substitute the expression for the momentum operator. Transform the...
  46. P

    Fermionic Number Operator Help

    Hi can anyone tell me why in the fermionic number operator case: <0|N/V|0>= \sum_{\pm r}\int d^3 k a^{\dagger}(t,r)a(t,r) because if: N=a^{\dagger}(t,k)a(t,k) then after Fourier decomposition surely one gets: \int d^3 r d^3 r \frac{1}{(2Pi)^{3}} a^{\dagger}(t,r)a(t,rk) and when...
  47. C

    Using Ternary Operator in C for If Else & Else If

    I was wondering if anyone knew if it is possible to construct an if else if with the ternary operator in C. I know that we can use it for if else, but what if you wanted multiple conditions for else if in your statement? printf("%d",(a>5)?1:(a<5)?0:10); //Just a silly example perhaps?
  48. L

    Quick question: Momentum operator in QM

    Homework Statement There are two ways to write the momentum operator, p = (-i hbar d/dx) and p = (hbar / i)d/dx. How do you go from one to the other? Homework Equations The two I gave above. The Attempt at a Solution I tried to see if -ih = h/i by squaring both sides, but one came out...
  49. P

    Composite system, rigged Hilbert space, bounded unbounded operator, CSCO, domain

    Is something wrong in my assertions below? Suppose we have two quantum systems N and X. Let N is described by discrete observable \hat{n} (bounded s.a. operator with discrete infinite spectrum) with eigenvectors |n\rangle. Let X is described by continuous observable \hat{x} (unbounded s.a...
  50. dexterdev

    Derivation of Del Operator in Spherical & Cylindrical Coordinates

    Hi all, Del = i ∂/∂x + j ∂/∂y + k ∂/∂z in x y z cordinate similarly I require to see the derivation of del in other coordinates too. Please give me a link for the derivation.
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