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. P

    Matrix form of Density Operator

    Hi All, I have spent hours trying to understand the matrix form of Density Operator. But, I fail. Please see page 2 of the attached file. (from the book "Quantum Mechanics - The Theoretical Minimum" page 199). Most appreciated if someone could enlighten me this. Many thanks in advance. Peter Yu
  2. I

    Eigenvalue of lowering operator

    How to prove that eigenvalue of lowering operator is zero?
  3. B

    QM: Expectation value of raising and lowering operator

    Homework Statement Using J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle Derive that : \langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)] Homework Equations J_- = J_x - iJ_y J_+ = J_x + iJ_y The...
  4. I

    Quantum Mechanics - Lowering Operator

    Homework Statement let A be a lowering operator. Homework Equations Show that A is a derivative respects to raising operator, A†, A=d/dA† The Attempt at a Solution I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can...
  5. D

    Displacement Operator: Exponential of Parameter & Vector?

    Hi . I've just encountered something called the displacement operator which is the exponential of a parameter multiplied by a vector but I thought the argument of an exponential had to be a scalar. Is this not true ?
  6. S

    MHB Can Linear Integral Operators Be Combined to Prove a Trivial Inequality?

    I have a linear integral operator (related to integral equations) $(Ky)(x)=\int_{a}^{b} \,k(x,s) y(s) ds$ and another one $(Ly)(x)=\int_{a}^{b} \,l(x,s) y(s) ds$ both are continuous Before I proceed can I write: $Ky=\int_{a}^{b} \,k(.,s) y(s) ds$ ? (I saw...
  7. P

    Relationship between commutators and observables

    Homework Statement Suppose A^ and B^ are linear quantum operators representing two observables A and B of a physical system. What must be true of the commutator [A^,B^] so that the system can have definite values of A and B simultaneously? Homework Equations I will use the bra-ket notation for...
  8. J

    What Boundary Conditions Are Needed for Time-Dependent Hermitian Operators?

    Hello, could you please give me an insight on how to get through this proof involving operators? Proof: Given an eigenvalue-eigenvector equation, suppose that the vectorstate depends on an external parameter, e.g. time, and that over it acts an operator that is the fourth derivative w.r.t...
  9. L

    Unpacking the Physics Behind the Annihilator-Creator Operator Formula

    a In the formula above, on the left hand side, ρ(0) is a system's density operator in its initial state. a is the annihilator operator of the system, and a+ is the create operator of the system. ρss is the system's density operator in its steady state. But I don't understand why this formula...
  10. blue_leaf77

    Role of Angular Momentum in Defining Vector Operator ##\mathbf{V}##

    A vector operator ##\mathbf{V}## is defined as one satisfying the following property: ## [V_i,J_j] = i\hbar \epsilon_{ijk}V_k## where ##\mathbf{J}## is an angular momentum operator. My question is what is the role of ##\mathbf{J}##, does it have to be the total angular momentum from all...
  11. D

    KE operator and eigenfunctions

    I have just done a question and then looked at the solution which I don't get. The question gives a wavefunction as u = x - iy. It then asks if this function is an eigenfunction of the kinetic energy operator in 3-D. Applying this operator to u gives zero. I took this to mean that u is an...
  12. L

    Translation operator on ground state

    Homework Statement I am working through a time independent perturbation problem and I am calculating the first order correction to the energy, and I am stuck operating the perturbation : v = i b (Exp[i g x]-Exp[-i g x]) on the ground state |0>. Homework Equations <0| v |0> = 1st order...
  13. metapuff

    Easy Question About the Number Operator

    Suppose I have a system of fermions in the ground state ##\Psi_0##. If I operate on this state with the number operator, I get \langle \Psi_0 | c_k^{\dagger} c_k | \Psi_0 \rangle = \frac{1}{e^{(\epsilon_k - \mu)\beta} + 1} which is, of course, the fermi distribution. What if I operate with...
  14. S

    Fluctuation operator and partial wave

    Can someone please explain to me why the expression ##[-\Box + U''(\Phi(r))]## is called the fluctuation operator? I was also wondering how to derive the following for the ##l^{th}## partial wave of the above operator: ##-\frac{d^2}{dr^2}-\frac{3}{r}\frac{d}{dr} + \frac{l(l+2)}{r}+...
  15. gfd43tg

    Triplets/Singlets and applying lowering S operator

    Hello, I am going through this and I am totally confused. Where do they come up with ##\mid 1 \hspace {0.02 in} 0 \rangle = \frac {1}{\sqrt{2}}(\uparrow \downarrow + \downarrow \uparrow)##? They just use the lowering operator, but I'm wondering if the switch in order from equation 4.177 and the...
  16. Safinaz

    Helicity Op: Commuting Dirac Hamiltonian

    Hi there, The question about the helicity operator ## h= S . \bf{p} ## ( where S is 2 by 2 matrix, with ##\sigma^i ## on the diagonal ), that as mentioned in a reference as [arXiv:1006.1718], it commutes with the Dirac Hamiltonian ## H = \gamma^0 ( \gamma^i p^i + m ) ## equ. (3.3), due to...
  17. gfd43tg

    Spin angular momentum operator queries

    Hello, For the spin angular momentum operator, the eigenvalue problem can be formed into matrix form. I will use ##S_{z}## as my example $$S_{z} | \uparrow \rangle = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac {\hbar}{2} \begin{pmatrix} 1 \\ 0...
  18. ognik

    Derive infinitesimal rotation operator

    Homework Statement Derive the infinitesimal rotation operator around the z-axis. Homework Equations My book gives this equation (which I follow) with epsilon the infinitesimal rotation angle: $$ \hat{R}(\epsilon) \psi(r,\theta, \phi) = \psi(r,\theta, \phi - \epsilon) $$ but I just don't get...
  19. F

    Unitary translation operator and taylor expansion

    Homework Statement I have quite a straightforward question on the taylor expansion however I will try to provide as much context to the problem as possible: ##T(a)## is unitary such that ##T(-a) = T(a)^{-1} = T(a)^{\dagger}## and operates on states in the position basis as ##T(a)|x\rangle =...
  20. gfd43tg

    How Do You Approach Angular Momentum Operator Algebra in Quantum Mechanics?

    Homework Statement Homework EquationsThe Attempt at a Solution This whole thing about angular momentum has me totally confused and stumped, but I am trying this problem given in a youtube video lecture I watched. I know of this equation ##L^{2} = L_{\pm}L_{\mp} + L_{z}^{2} \mp \hbar L_{z}##...
  21. 1

    Eigenvalues of operator in dirac not* (measurement outcomes)

    Homework Statement A measurement is described by the operator: |0⟩⟨1| + |1⟩⟨0| where, |0⟩ and |1⟩ represent orthonormal states. What are the possible measurement outcomes? Homework Equations [/B] Eigenvalue Equation: A|Ψ> = a|Ψ> The Attempt at a Solution Apologies for the basic...
  22. J

    2 particle exchange operator P

    Trying to derive two functions which are eigenfunctions of the hamiltonian of 2 identical and indistinguishable particles and also eigenfunctions of the 2-particle exchange operator P. Need some help with my workings I think.Have particle '1' and particle '2' in a hamiltonian given as...
  23. S

    Momentum operator eigenstates/eigenvalues

    As far as I know, the momentum operator is as follows: -iħ(∂/∂x) Now let's say that I enact this operator on the famous solution to the 1-D particle in a box example: Ψ= squrt(2/L) sin(πnx/L) If the momentum operator operates on the above wave function, it yields: -iħ * squrt(2/L) * (πn/L)...
  24. Matternot

    Using the D operator without true understanding.

    I was introduced to the d operator to help solve constant coefficient differential equations for the particular integral without using trial solutions: http://en.wikipedia.org/wiki/Differential_operator http://en.wikipedia.org/wiki/Shift_theorem The results generally seem sensible, but there...
  25. S

    MHB Norm of a linear integral operator

    i have a problem concerning the norm of linear integral operator. ii found the answer in a book called unbounded linear operators theory and applications by Dover books author seymour Goldberg. the proof runs as follows ||T|| is less than max over x in [a,b] of integral (|k(x,y)|dy) Then he...
  26. G

    Improved energy-momentum tensor changing dilation operator

    I'm trying to show that \int d^3x \,x^\mu \left(\partial_\mu \partial_0-g_{\mu 0} \partial^2 \right)\phi^2(x)=0 . This term represents an addition to a component of the energy-momentum tensor \theta_{\mu 0} of a scalar field and I want to show that this does not change the dilation operator...
  27. gfd43tg

    Finding relationships of inner products with operator

    Homework Statement Homework EquationsThe Attempt at a Solution a) I am having some trouble understanding the notation. I'm uncertain whether it should be $$ \langle {f} | \hat {O_{2}} | g \rangle = \int_{-\infty}^{\infty} f^{*}g \frac {dg}{dx} dx $$ or $$ \langle {f} | \hat {O_{2}} | g...
  28. T

    Material Derivative (Convective Derivative Operator)

    Hi, I've learned that material derivative is equal to local derivative + convective derivative, but can't seem to find out which way the convective derivative acts, like for example in velocity fields: The equation my teacher gave us was (with a and v all/both vectors): Acceleration = material...
  29. L

    Exploring Quantum State Changes with Competitive Operators

    In a quantum state, if you use the position operator, it gives you position, momentum operator, momentum, Hamiltonian, energy.. can you give an example or all experiments done where a quantum state has 3 competitive operators acting on it.. I want to see the quantum state changing in between...
  30. Muthumanimaran

    Classical Differences Between DAlembertian & Laplacian Operators

    what are the differences between D'Alembertian and Laplacian operator? while reading Electrodynamics by Griffiths I learned that Laplacian operator is used in non-relativistic cases and D'Alembertian operator is used in relativistic cases, But i don't what is the difference between these operators
  31. C

    Divergence Operator on the Incompressible N-S Equation

    Hello All, If I apply the Divergence Operator on the incompressible Navier-Stokes equation, I get this equation: $$\nabla ^2P = -\rho \nabla \cdot \left [ V \cdot \nabla V \right ]$$ In 2D cartesian coordinates (x and y), I am supposed to get: $$\nabla ^2P = -\rho \left[ \left( \frac...
  32. TrickyDicky

    Projection operator and measurement

    I'm aware there have been plenty of discussions about Copenhagen interpretation vs ensemble interpretations (myself I have always been more fond of the latter) but I intend to explore new perspectives and stick as much as possible to what QM practitioners do in practice as opposed to obscure...
  33. R

    Green's Function - modified operator

    Hi, I'm stuck with a question from one of my examples sheets from uni. The question is as follows: If G(x,x') is a greens function for the linear operator L, then what is the corresponding greens function for the linear operator L'=f(x)L, where f(x) =/=0? So I've started by writing...
  34. lfqm

    Raising operator for s in |s,m> states

    Are there any known (collective spin) operators to raise or lower the quantum number s in \left|{s,m}\right> spin states? I'm trying to construct coherent states varying the quantum number s instead of the well known spin coherent states varying m. I found a coherent-like state similar to the...
  35. G

    Expectation value of operator derivation

    Where one can find a proof of the expectation value of operator expression. <A> = < Ψ | A | Ψ > or <A> = integral( Ψ* A Ψ dx ) Thanks.
  36. teroenza

    Creation/Anhilation Operator Commutation Relation

    Homework Statement Simplify the following commutator involving the creation and annihilation operators. [a^{\dagger}a,a \sqrt{a^\dagger a} ] Homework Equations I know that [a,a^\dagger] = 1. The Attempt at a Solution I think I should be trying to put the creation operators to the left...
  37. C

    Operator Fields: Introduction for Quantum Field Theory

    Hi, are there any math texts out there that are good introductions to operator fields as used in quantum field theory ("fields" in the physics, not mathematical sense, in this case.)?
  38. Einj

    Gaussian functional integral with constant operator

    Hello everyone. What it the result for a Gaussian functional integral when the "matrix" is nothing but a number? Mathematically speaking is the following true? $$ \int \mathcal{D}\phi e^{-\int d^3k f(k) |\phi(k)|^2}\propto \left(f(k)\right)^{-1/2} $$ Here ##f(k)## is just a function of k, not...
  39. L

    Are Unitary Operator Eigenvalues Always Modulus 1 and Eigenvectors Orthogonal?

    Homework Statement I know that Unitary operators act similar to hermitean operators. I want to prove that the eigenvalues of unitary operators are complex numbers of modulus 1, and that Unitary operators produce orthogonal eigenvectors. Homework Equations U†U = I U-1=U† λ = eiΦ{/SUP]...
  40. B

    Calculating eigenstates of an operator

    Homework Statement Consider a two-dimensional space spanned by two orthonormal state vectors \mid \alpha \rangle and \mid \beta \rangle . An operator is expressed in terms of these vectors as A = \mid \alpha \rangle \langle \alpha \mid + \lambda \mid \beta \rangle \langle \alpha \mid +...
  41. C

    Need some help understanding boundary operator on simplicies

    I am currently reading up on some algebraic topology\differential geometry and have reached the section on de Rham theory. This is my first encounter with such notions and I am a little confused by what is meant when one applies a boundary operator to a simplex. Conceptually, I know that it...
  42. A

    Angular momentum operator justification

    One can represent the mean of the angular momentum operator as a vector. But what is the (mathematical) justification to represent the operator by a vector which has a direction that the operator has not. Yet worse, l(l+1) h2 is the proper value of operator L^2 and from such result it is assumed...
  43. D

    Represent Action of Stern Gerlach Operator as a Matrix

    Homework Statement Given the series of three Stern-Gerlach devices: Represent the action of the last two SG devices as matrices ##\hat{A}## and ##\hat{B}## in the ##|+z\rangle, |-z\rangle## basis. Homework Equations ##|+n\rangle = cos(\frac{\theta}{2})|+z\rangle +...
  44. D

    Understanding the Parity-Flipping Nature of the Momentum Operator

    What does it mean when it is said that the momentum operator flips the parity of the function on which it operates ?
  45. ShayanJ

    Momentum operator in curvilinear coordinates

    This paper is about momentum operator in curvilinear coordinates. The author says that using \vec p=\frac{\hbar}{i} \vec \nabla is wrong and this form is only limited to Cartesian coordinates. Then he tries to find expressions for momentum operator in curvilinear coordinates. He's starting...
  46. B

    How to Solve the Creation Operator Problem in Problem 3a?

    Anyone having an idea of how to solve problem 3a) file:///C:/Users/Administrator/Downloads/handin1%20(2).pdf ? I've been stuck for a great while but have not idea.
  47. A

    Parity operator commutes with second derivative?

    How do I prove that the parity operator Af(x) = f(-x) commutes with the second derivative operator. I am tempted to write: A∂^2f(x)/∂x^2 = ∂^2f(-x)/∂(-x)^2 = ∂^2f(-x)/∂x^2 = ∂^2Af(x)/∂x^2 But that looks to be abuse of notation..
  48. P

    Derivation of momentum operator

    hello, i am trying to learn the derivation of the momentum operator and i found 2 ways of deriving it. one is using Fourier transform and the other is taking the time derivative of the expectation value of x. i just want to know what is the physical interpretation of the time rate of change...
  49. W

    Eigenvalues for a bounded operator

    Homework Statement Let C be the composition operator on the Hilbert space L_{2}(\mathbb{R}) with the usual inner product. Let f\in L_{2}(\mathbb{R}), then C is defined by (Cf)(x) = f(2x-1), \hspace{9pt}x\in\mathbb{R} give a demonstration, which shows that C does not have any eigenvalues...
  50. teroenza

    Creation/Anhilation Operator Exponential Commutator Relation

    Homework Statement Given that the function f can be expanded in a power series of a and a^\dagger, show that: [a,f(a,a^{\dagger})]=\frac{\partial f }{\partial a^\dagger} and that [a,e^{-\alpha a^\dagger a}] = (e^{-\alpha}-1)e^{-\alpha a^{\dagger} a}aThe Attempt at a Solution I've tied using...
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