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. 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...
  2. 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...
  3. 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...
  4. tommyxu3

    I Can Normal Matrices Be Non-Self-Adjoint?

    Hello everyone, I have a question. I'm not sure if it is trivial. Does anyone have ideas of finding a matrix ##A\in M_n(\mathbb{C})##, where ##A## is normal but not self-adjoint, that is, ##A^*A=AA^*## but ##A\neq A^*?##
  5. J

    I Weinberg LN in QM (Section 3.5): Momentum operator

    Hi everyone, Weinberg uses spatial translation invariance to derive the momentum operator. But the way he does it puzzles me. Here is an excerpt of the book. Equation 3.5.1 is the definition of the unitary operator ##U(x)## for translation invariance: $$U^{-1}(x)XU(x) = X+x,$$ with...
  6. henry wang

    I Proving grad operator yields perpendicular vector to contour

    I saw a link on MIT open courseware proving grad operator yields perpendicular vector to contour, but I can't make sense of how dg/dt=0. Can someone explain to me please...
  7. E

    Displacement operator, squeeze operator - problem interpreting notation....

    Homework Statement Prove the following relations (for ##\zeta:=r e^{i\theta}##): \begin{align} D(\alpha)^\dagger a D(\alpha)&=a+\alpha\\ S(\zeta)^\dagger a S(\zeta)&= a \cosh r- a^ \dagger e^{i\theta} \sinh r \end{align} Homework Equations ##|\alpha\rangle## is the coherent state. ##a## and...
  8. S

    I Why does the kinetic operator depend on a second derivative?

    The formula T = -(ħ/2m)∇2 implies that T is proportional to the second spatial derivative of a wavefunction. What is the origin of this dependence? In classical mechanics, T = p2/2m. Is it also the case in classical mechanics that p2/2m is proportional to a second spatial derivative? I...
  9. Ernesto Paas

    I When can one clear the operator

    Hi all! I'm having problems understanding the operator algebra. Particularly in this case: Suppose I have this projection ## \langle \Phi_{1} | \hat{A} | \Phi_{2} \rangle ## where the ##\phi's ## have an orthonormal countable basis. If I do a state expansion on both sides then I suppose I'd get...
  10. kenyanchemist

    I Is Ψ2Px an Eigenfunction of L2 or Lz in Quantum Mechanics?

    hi, am major new on quantum mechanics. please help me understand. is the real wave function Ψ2Px= [Ψ2p+1 +Ψ2p-1]1/2 an eigen function of L2 or Lz? if so, how is it? and if so kindly explain the values of l and m thanks
  11. S

    I Angular momentum ladder operator derivation

    In the Griffiths textbook for Quantum Mechanics, It just gives the ladder operator to be L±≡Lx±iLy With reference to it being similar to QHO ladder operator. The book shows how that ladder operator is obtained, but it doesn't show how angular momentum operator is derived. Ive searched the...
  12. S

    I Why is the KE operator negative in QM?

    In the Hamilonian for an H2+, the kinetic energy of the electron (KE of nucleus ignored due to born-oppenheimer approximation) has a negative sign in front of it. I understand the signs for the potential energy operators but not for the KE apart from the strictly mathematical point of view. Can...
  13. P

    I Do All Physical States Satisfy the Hamiltonian Equation Hψ = Eψ?

    I came across a previous exam question which stated: Do all physical states, ψ, abide to Hψ = Eψ. I thought about it for a while, but I'm not really sure.
  14. S

    I What happens to the eigenvalue if an operator acts on a bra?

    I'm going through a derivation and it shows: (dirac notation) <una|VA-AV|unb>=(anb-ana)<una|V|unb> V and A are operators that are hermition and commute with each other and ana and anb are the eigenvalues of the operator A. I imagine it is trivial and possibly doesn't even matter but why does...
  15. carllacan

    Eigenvectors of "squeezed" amplitude operator

    Homework Statement Prove that the states $$|z, \alpha \rangle = \hat S(z)\hat D(\alpha) | 0 \rangle $$ $$|\alpha, z \rangle = \hat D(\alpha) \hat S(z)| 0 \rangle $$ are eigenvectors of the squeezed amplitude operator $$ \hat b = \hat S(z) \hat a \hat S ^\dagger (z) = \mu \hat a + \nu \hat a...
  16. DrPapper

    I Expression for Uncertainty of Arbitrary Operator

    Hello all, as far as I can see this question is not posted already, my apologies if it is and please provide a link. But I'm watching this video on youtube: And at 22:38 there's an expression given for the uncertainty of an arbitrary operator Q, however I'm concerned the expression is incorrect...
  17. K

    Show that Momentum Operator is Hermitian: Q&A

    Homework Statement Hi, my task is to show that the momentum operator is hermitian. I found a link, which shows how to solve the problem: http://www.colby.edu/chemistry/PChem/notes/MomentumHermitian.pdf But there are two steps that I don't understand: 1. Why does the wave function approach...
  18. G

    MHB Proving $(T^2-I)(T-3I) = 0$ for Linear Operator $T$

    Problem: Let $T$ be the linear operator on $\mathbb{R}^3$ defined by $$T(x_1, x_2, x_3)= (3x_1, x_1-x_2, 2x_1+x_2+x_3)$$ Is $T$ invertible? If so, find a rule for $T^{-1}$ like the one which defines $T$. Prove that $(T^2-I)(T-3I) = 0.$ Attempt: $(T|I)=\left[\begin{array}{ccc|ccc} 3 &...
  19. J

    How do I find eigenstates and eigenvalues from a spin operator?

    Homework Statement I have a spin operator and have to find the eigenstates from it and then calculate the eigenvalues. I think I managed to get the eigenvalues but am not sure how to get the eigenstates.Homework Equations The Attempt at a Solution I think I managed to get the eigenvalues out...
  20. Joshua L

    B What is the Hamiltonian operator for a decaying Carbon-14 atom?

    Hey, here's a quick question: What is the Hamiltonian operator corresponding to a decaying Carbon-14 atom. Any insight is quite appreciated!
  21. M

    Definition of an operator in a vector space

    In the book that I read, an operator is defined to be a linear map which maps from a vector space into itself. For example, if ##T## is an operator in a vector space ##V##, then ##T:V\rightarrow V##. Now, what if I have an operator ##O## such that ##T:V\rightarrow U## where ##U## is a subspace...
  22. Z

    Is the differential in the momentum operator commutative?

    As it says; I was looking over some provided solutions to a problem set I was given and noticed that, in finding the expectation value for the momentum operator of a given wavefunction, the following (constants/irrelevant stuff taken out) happened in the integrand...
  23. M

    Confusion about eigenvalues of an operator

    Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This...
  24. E

    Verify Ψ is solution of quantum oscillator using H operator

    Homework Statement verify that Ψ(x) = ( 1/a√π)½ exp(-(x2/2a2)) is a solution to the TISE for linear harmonic oscillator. Where a = √(hbar/mw). and V(x) = ½ mw2x2. Homework Equations HΨ=EΨ E_n = (n+½)hbar*wThe Attempt at a Solution I've started by differentiating the wave function twice to...
  25. Z

    Relativistic vs. Nonrelativistic KE Operator question

    Hey, folks. I, on a whim today, started taking a MOOC quantum mechanics course that I have the functional math skills necessary to do but have virtually no background knowledge of quantum to start with and am incredibly rusty on stuff like PDE's; Quite frankly I'm out of my league, but the...
  26. H

    Converting operator matrix (Quantum Chemistry question)

    Dear all, I want to know how to convert operator matrix when using Dirac Bra-Ket notation when it must be converted into a new dimension. I am currently working on transition dipole moment operator matrix D which I am going to use the following one: D = er Where e is charge of electron, r is...
  27. amjad-sh

    Hamiltonian and momentum operator acting on a momentum eigenstate

    suppose that the momentum operator \hat p is acting on a momentum eigenstate | p \rangle such that we have the eigenvalue equation \hat p | p \rangle = p| p \rangle Now let's project \langle x | on the equation above and use the completeness relation \int | x\rangle \langle x | dx =\hat I we...
  28. J

    Laplace-Beltrami Operator non-curvilinear coordinates

    Homework Statement I have to find the Laplace operator asociated to the next quasi-spherical curvilinear coordinates, for z>0. Homework Equations \begin{align} x&=\rho \cos\phi\nonumber\\ y&=\rho \sin \phi\nonumber\\ z&=\sqrt{r^2-\rho^2}, \end{align} The Attempt at a Solution I computed...
  29. C

    How do I evaluate <x> with the k-space representation?

    Homework Statement Given the following k-space representation of the wave function: Ψ(k,t) = Ψ(k)e-iħk2t/2m use the wave number representation to show the following: <x>t=<x>0 + <p>0t/m <p>t=<p>0 Homework Equations <x>=∫Ψ*(x,t)xΨ(x,t)dx <p>=∫Ψ*(x,t)(-iħ ∂/∂x)Ψ(x,t)dx The Attempt at a...
  30. sa1988

    QM: "What are the possible results of measuring Operator A?"

    Homework Statement Homework EquationsThe Attempt at a Solution I'm fine with parts a) and b) However I don't understand what part c) is asking me to do. How do I 'measure' an operator? There are only two things I can think to do: 1. Find the expectation values of A for <Φ1|A|Φ1> and...
  31. ShayanJ

    Coulomb potential as an operator

    I want to calculate the commutator ##{\Large [p_i,\frac{x_j}{r}]}## but I have no idea how I should work with the operator ##{\Large\frac{x_j}{r} }##. Is it ## x_j \frac 1 r ## or ## \frac 1 r x_j ##? Or these two are equal? How can I calculate ##{\Large [p_i,\frac 1 r]}##? Thanks
  32. Jianphys17

    Is there a generalized curl operator for dimensions higher than 3?

    Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions! On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you...
  33. Alain De Vos

    Einstein Tensors and Energy-Momentum Tensors as Operators

    Can these tensor be seen as operators on two elements. So given two elements of something they produce something, for instance a scalar ?
  34. H

    Operator r is a diagonal matrix in position representation

    What does it mean by "In the position representation -- in which r is diagonal" in the paragraph below? How can we show that? Does it mean equation (3) in http://scienceworld.wolfram.com/physics/PositionOperator.html? (where I believe the matrix is in the ##|E_n>## basis)
  35. S

    Propagation amplitude and time-evolution operator

    I know that the time-evolution operator in quantum mechanics is ##e^{-iHt}##. Is this also called the Schrodinger time-evolution operator? Also, can you guys explain why the amplitude ##U(x_{a},x_{b};T)## for a particle to travel from one point ##(x_{a})## to another ##(x_{b})## in a given...
  36. G

    Zettili QM Problem on Trace of an Operator

    Homework Statement In Zettili's QM textbook, we are asked to find the trace of an operator |\psi><\chi| . Where the kets |\psi> and |\chi> are equal to some (irrelevant, for the purposes of this question) linear combinations of two orthonormal basis kets. Homework Equations...
  37. E

    Integrate Laplacian operator by parts

    This is the key step to transform from position space Schrodinger equation to its counterpart in momentum space. How is the first equation transformed into 3.21? To be more specific, how to integral Laplacian term by parts?
  38. G

    Deriving hamiltonian operator for rotational kinetic energy.

    Homework Statement I am trying to get the hamiltonain operator equality for a rigid rotor. But I don't get it. Please see the red text in the bottom for my direct problem. The rest is just the derivation I used from classical mechanics. Homework Equations By using algebra we obtain: By...
  39. Safinaz

    Effective operator and allowed loop level interactions

    Hi all, Some processes can not happen at the tree level, but it happen via loops, like for Higgs decay to pair of glouns or pair of photons, (h -> gg), (h -> y y) . For instance, effectively h -> gg written as ##~ h~ G^a_{\mu\nu} G_a^{\mu\nu}~ ## which is Lorentz and gauge invariant .. Now if...
  40. Matt atkinson

    Entanglement, projection operator and partial trace

    Homework Statement Consider the following experiment: Alice and Bob each blindly draw a marble from a vase that contains one black and one white marble. Let’s call the state of the write marble |0〉 and the state of the black marble |1〉. Consider what the state of Bob’s marble is when Alice...
  41. H

    Prove the time evolution operator is unitary

    How is (5.240b) derived? I get {U^{-1}}^\dagger(t, t_0)\,U^{-1}(t, t_0)=I instead. My steps: \begin{align}<\psi(t_0)\,|\,\psi(t_0)>&=\,<U(t_0, t)\,\psi(t)\,|\,U(t_0, t)\,\psi(t)>\\ &=\,<U^{-1}(t, t_0)\,\psi(t)\,|\,U^{-1}(t, t_0)\,\psi(t)>\\ &=\,<\psi(t)\,|\,{U^{-1}}^\dagger(t, t_0)\,U^{-1}(t...
  42. H

    Matrix representation of an operator with a change of basis

    Why isn't the second line in (5.185) ##\sum_k\sum_l<\phi_m\,|\,A\,|\,\psi_k><\psi_k\,|\,\psi_l><\psi_l\,|\,\phi_n>##? My steps are as follows: ##<\phi_m\,|\,A\,|\,\phi_n>## ##=\int\phi_m^*(r)\,A\,\phi_n(r)\,dr## ##=\int\phi_m^*(r)\,A\,\int\delta(r-r')\phi_n(r')\,dr'dr## By the closure...
  43. S

    How to Derive Raising and Lowering Operators from Ladder Operator Definitions?

    Homework Statement Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L. Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½. with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length. Show that a=(1/2)ñ+∂n and a†=(1/2)n -∂ñ a and a† are the lowering and raising operators of quantum...
  44. G

    Prove that linear operator is invertible

    Homework Statement Let \mathcal{A}: \mathbb{R^3}\rightarrow \mathbb{R^3} is a linear operator defined as \mathcal{A}(x_1,x_2,x_3)=(x_1+x_2-x_3, x_2+7x_3, -x_3) Prove that \mathcal{A} is invertible and find matrix of \mathcal{A},A^{-1} in terms of canonical basis of \mathbb{R^3}. Homework...
  45. S

    Unitary and linear operator in quantum mechanics

    Given a transformation ##U## such that ##|\psi'>=U|\psi>##, the invariance ##<\psi'|\psi'>=<\psi|\psi>## of the scalar product under the transformation ##U## means that ##U## is either linear and unitary, or antilinear and antiunitary. How do I prove this? ##<\psi'|\psi'>## ##= <U\psi|U\psi>##...
  46. S

    Commutation relations for angular momentum operator

    I would like to prove that the angular momentum operators ##\vec{J} = \vec{x} \times \vec{p} = \vec{x} \times (-i\vec{\nabla})## can be used to obtain the commutation relations ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##. Something's gone wrong with my proof below. Can you point out the mistake...
  47. H

    Derivation of the momentum-to-the-power-of-n operator

    Homework Statement Homework Equations The Attempt at a Solution First substitute ##\Phi(p,t)## in terms of ##\Psi(r,t)## and similarly for ##\Phi^*(p,t)##, and substitute ##p_x^n## in terms of the differentiation operator ##< p_x^n>\,=(2\pi\hbar)^{-3}\int\int...
  48. G

    How Does the Linear Operator \(\phi\) Transform Matrices to Polynomials?

    Homework Statement Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where B= \begin{bmatrix} 3 & -2 \\ 2 & -2 \\ \end{bmatrix} Find rank,defect and one basis of an image and kernel of linear operator...
  49. A

    Operator works on a quantum state yields another state?

    The well-known eigen value expression A(a)=a(a) assuming the operator which represents a physical phenomena acts on a quantum state which is represented by an eigen vector, (a) corresponds to an observed value a. But I am wondering if the same operator A can act on (a) and produce another eigen...
  50. snoopies622

    Seeking a phase angle operator for the QHO

    According to Daniel Gillespie in A Quantum Mechanics Primer (1970), " . . . any observable which in classical mechanics is some well behaved function of position and momentum, f(x,p), is represented in quantum mechanics by the operator f ( \hat{x} , \hat {p} ) . That is, a = f (x,p) . . ...
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