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. I

    How to integrate by parts when del operator is involved?

    i'm trying to integrate this: $$W=\frac{ε}{2}\int{\vec{∇}\cdot\vec{E})Vdτ}$$ where ε is a constant, E= -∇V, τ is a volume element how do i end up with the following via integration by parts? $$W=\frac{ε}{2}[-\int{\vec{E}\cdot(\vec{∇}V)dτ}+\oint{V\vec{E}\cdot d\vec{a}}$$] where the vector a...
  2. binbagsss

    Quantum Mechanics - Time evolution operator , bra ket states.

    The question is to calculate the time evoution of S_{x} wrt <\Psi(t)\pm l where <\Psi\pm (t) l= ( \frac{1}{\sqrt{2}}(exp(^{+iwt})< \uparrow l , \pm exp(^{-iwt})< \downarrow l ) [1] Sx=\frac{}{2}(^{0}_{1}^{1}_{0} ) Here is my attempt: - First of all from [1] I see that l \Psi\pm (t) > = (...
  3. S

    Quantum, matrix and momentum operator

    Homework Statement Write out matrix representation of P. Also, do P|ψ> Homework Equations ψ=ψ0 + 2ψ1 ψ0=(1/∏)1/4 exp(- u2/2) ψ1=(1/∏)1/4 √2 exp(- u2/2) P= 1/(i*∏) d/du The Attempt at a Solution I've no clue what to do. If I had a ψm ψn I would, but what do I do...
  4. N

    Find a basis for the null space of the transpose operator

    Homework Statement Let ##n## be a positive integer and let ##V = P_n## be the space of polynomials over ##R##. Let D be the differentiation operator on ##V## . Find a basis for the null space of the transpose operator ##D^t: V^*\to V^*##. Homework Equations Let ##T:V\to W## be a linear...
  5. nomadreid

    M-<M> for M operator: why not a mismatch?

    In "Quantum Computation and Quantum Information" by Nielsen & Chuang, on pp. 88-89, applying basic statistical definitions to operators, one of the intermediary steps uses the expression M-<M> where M is a Hermitian operator, and <M> is the expected value = <ψ|M|ψ> for a given vector ψ...
  6. S

    Is the Image of a Normal Operator the Same as Its Adjoint?

    Homework Statement Show that if T is a normal operator on a finite dimensional vector space than it has the same image as its adjoint. Homework Equations N/A The Attempt at a Solution I have been able to show that both T and T^{*} have the same kernel. Thus, by using the finite...
  7. L

    Inverse of the adjoint of the shift operator

    Hi there, Let S denote the shift operator on the Hardy space on the unit disc H^2, that is (Sf)(z)=zf(z). My question is to show the following identity (1-\lambda S^*)^{-1}S^*f (z)=\frac{f(z)-f(\lambda)}{z-\lambda}, where \lambda,z\in\mathbb{D} Thanks in advance
  8. X

    Time Evolution operator in Interaction Picture (Harmonic Oscillator)

    Homework Statement Consider a time-dependent harmonic oscillator with Hamiltonian \hat{H}(t)=\hat{H}_0+\hat{V}(t) \hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right) \hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right) (i)...
  9. C

    What is the correct formula for [AB,C] in terms of A, B, and C?

    Homework Statement I am solving a problem and I arrived near the end, and can't figure out what to do here: (1/(2m)) [P^2,X]+[P^2,X] m - mass P - Momentum operator X - Position operator Homework Equations P = -iħ(∂/∂x) [A,B]=AB-BA [AB,C]=A[B,C]+B[A,C] where A, B...
  10. H

    Is the Spin Exchange Operator for s=1/2 Particles Hermitian?

    Homework Statement Consider a system of two spin 1/2 particles, labeled 1 and 2. The Pauli spin matrices associated with each particle may then be written as \vec{\hat{\sigma _{1}}} , \vec{\hat{\sigma _{2}}} a)Prove that the operator \hat{A]}\equiv \vec{\hat{\sigma _{1}}}\cdot...
  11. H

    Momentum Operator: Comparing p_x and p

    This is not really a homework problem but rather a homework-related question.. When I came across my homework (and my textbook: Atkin's physical chemistry 9th Ed.), they defined the momentum operator as: p_x = - ( \hbar / i ) * d/dx... but i have seen in other sources that they define it...
  12. D

    Is the Momentum Operator Hermitian in Quantum Mechanics?

    A Hermitian operator A is defined by A=A(dagger) which is the transpose and complex conjugate of A. In 1-D the momentum operator is -i(h bar)d/dx. How can this be Hermitian as the conjugate has the opposite sign ? Thanks
  13. A

    Nuclear force tensor operator expectation value.

    Homework Statement I have a question asking me to find the expectation value of S_{12} for a system of two nucleons in a state with total spin S = 1 and M_s = +1 , where S_{12} is the tensor operator inside the one-pion exchange nuclear potential operator, equal to S_{12} =...
  14. S

    Expectation value for momentum operator using Dirac Notation

    Question and symbols: Consider a state|ε> that is in a quantum superposition of two particle-in-a-box energy eigenstates corresponding to n=2,3, i.e.: |ε> = ,[1/(2^.5)][|2> + |3>], or equivalently: ε(x) = [1/(2^.5)][ψ2(x) + ψ3. Compute the expectation value of momentum: <p> = <ε|\widehat{}p|ε>...
  15. K

    Quantum States and ladder operator

    In any textbooks I have seen, vacuum states are defined as: a |0>= 0 What is the difference between |0> and 0? Again, what happens when a+ act on |0> and 0? and Number Operator a+a act on |0> and 0?
  16. Z

    Why Does <n',l',m'|\hat{z}|n,l,m> Equal Zero Unless m=m'?

    Homework Statement I want to show that <n',l',m'|\hat{z}|n,l,m> = 0 unless m=m', using the form of the spherical harmonics. Homework Equations Equations for spherical harmonics The Attempt at a Solution Not sure how to begin here since there aren't any simple eigenvalues for...
  17. D

    The uncertainty operator and Heisenberg

    In deriving the Heisenberg uncertainty relation for 2 general Hermitian operators A and B , the uncertainty operators ΔA and ΔB are introduced defined by ΔA=A - (expectation value of A) and similarly for B. My question is this - how can you subtract(or add) an expectation value , which is just...
  18. J

    What is the Definition of the Shape Operator and How is it Calculated?

    In wolframpage there is follows definition for shape operator in a given point by vector v: I think that this equation means: S(\vec{v})=-\frac{d\hat{n}}{d \vec{v}} correct, or not? If yes, of according with the matrix calculus...
  19. D

    Exponential projection operator in Dirac formalism

    Homework Statement Hey guys. So here's the situation: Consider the Hilbert space H_{\frac{1}{2}}, which is spanned by the orthonormal kets |j,m_{j}> with j=\frac{1}{2}, m_{j}=(\frac{1}{2},-\frac{1}{2}). Let |+> = |\frac{1}{2}, \frac{1}{2}> and |->=|\frac{1}{2},-\frac{1}{2}>. Define the...
  20. P

    Solving Operator Nabla Example Problem

    Homework Statement So I have this rather komplex example and I am looking for help. ∇(3(r*a)r)/R5 -a/R5) r=xex+yey+zez a-constant vector R=r1/2 Homework Equations The Attempt at a Solution So the nabla " works" on every member individualy,and i have to careful here:(r*∇a),because...
  21. G

    Joint Number state switching operator?

    Hi, I was just wondering if anyone know of an operator, which has some realistic analogue, that would perform the following action: A|N,0> = A|0,N> Where the ket's represent two joint fock states (i.e. two joint cavities) and A is the opeator I desire. I thought that the beam...
  22. M

    Vector calculus: angular momentum operator in spherical coordinates

    Note: physics conventions, \theta is measured from z-axis We have a vector operator \vec{L} = -i \vec{r} \times \vec{\nabla} = -i\left(\hat{\phi} \frac{\partial}{\partial \theta} - \hat{\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial \phi} \right) And apparently \vec{L}\cdot\vec{L}=...
  23. R

    When can we move the del operator under an integral sign?

    Homework Statement Hi, it's me again. I'm new to vector calculus so this might sound like a stupid question, but in relation to a specific problem, I was wondering when we could move the del operator under the integration sign - in relation to a specific problem, which is: A(r) = integral...
  24. B

    MHB If an operator commutes, its inverse commutes

    Prove that if operator on a hilbert space $T$ commutes with an operator $S$ and $T$ is invertible, then $T^{-1}$ commutes with $S$. $T^{-1}S$=$T^{-1}T^{-1}TS$=$T^{-1}T^{-1}ST$
  25. B

    MHB Limit of Inverse Operators: Proving Convergence for Bounded Linear Sequences

    Let $T_{n}$ be a sequence of invertible bounded linear operators with limit $T$ Prove that $(T_{n})^{-1}$ tends to $T^{-1}$
  26. S

    Parity Operator and odd potential function.

    Homework Statement This is a university annual exam question: Show that for a potential V (-r)=-V (r) the wave function is either even or odd parity. Homework Equations The Attempt at a Solution We can determine whether a wavefunctions' parity is time independent based on if the...
  27. F

    MHB Ker(I-L) Finite-Dimensional: Proof

    Let L be an compact operator on a compact space K , and Let I be the identity on K. Show that Ker(I-L) is finite-dimensional. My attempt: Let $x_{n}$ be a sequence in the unit ball. K is compact so $(I(x_{n}))=(x_{n})$ has a convergent subsequence and L is compact operator so $L(x_{n})$ has a...
  28. A

    Condition for expectation value of an operator to depend on time

    Homework Statement A particle is in a 1D harmonic oscillator potential. Under what conditions will the expectation value of an operator Q (no explicit time dependence) depend on time if (i) the particle is initially in a momentum eigenstate? (ii) the particle is initially in an energy...
  29. H

    What is the Time Evolution of the Angular Momentum Operator?

    Hi guys, this might be a stupid question but if I wanted a general expression for the time evolution of the angular momentum operator is it just the same as Hamiltonian? i.e ih ∂/∂t ψ = L2 ψ Solving this partial differential gives the time evolution of the angular momentum operator...
  30. M

    Is the energy operator (time derivative) a linear one?

    Typically in mathematics time derivative is linear in the sense that constants are pulled out the operator which then operates on a time dependent function. But in quantum mechanics we say linear to mean that the operator passes over the coefficients of the kets (which themselves might be time...
  31. CrimsonFlash

    What are the matrix elements of the angular momentum operator?

    What are the "matrix elements" of the angular momentum operator? Hello, I just recently learned about angular momentum operator. So far, I liked expressing my operators in this way: http://upload.wikimedia.org/math/8/2/6/826d794e3ca9681934aea7588961cafe.png I like it this way because it...
  32. ChrisVer

    Second Quantization-Kinetic operator

    I am feeling a little stupid tonight... So let me build the problem... For a single particle operator O, we have in the basis |i> we have that: O= \sum_{ij} o_{ij} |i><j| with o_{ij}=<i|O|j> Then for N particles we have that: T=\sum_{a}O_{a}= \sum_{ij} o_{ij} \sum_{a} |i>_{a}<j|_{a} with...
  33. F

    Find eigenfunctions and eigenvalues of an operator

    Homework Statement \hat T = \frac{{\hat L_z^2}}{{2I}} = - \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} Homework Equations Find eigenfunctions and eigenvalues of this operatorThe Attempt at a Solution It leads to the differential eqn - \frac{{{\hbar...
  34. B

    Why can momentum be expressed as an operator

    Homework Statement Im having a hard time figuring out how in quantum mechanics things such as momentum can be expressed as an operator. I know the simple algebra to get the relation. Starting with the 1D solution to wave equation\Psi=e^{i\omega x} then differentiating that with respect to x...
  35. L

    Eigenvalue of angular momentum operator

    Homework Statement I'm running through practice papers for my 3rd year physics exam on atomic and nuclear physics: This is the operator we found in the previous part of the question L = -i*(hbar)*d/dθ Next, we need to find the eigenvalues and normalised wavefunctions of L The...
  36. A

    Continuity of the inverse of a linear operator

    If g(a) \neq 0 and both f and g are continuous at a, then we know the quotient function f/g is continuous at a. Now, suppose we have a linear operator A(t) on a Hilbert space such that the function \phi(t) = \| A(t) \|, \phi: \mathbb R \to [0,\infty), is continuous at a. Do we then know that...
  37. A

    Commutate my hamiltonian H with a fermionic anihillation operator

    Homework Statement I have problem where I need to commutate my hamiltonian H with a fermionic anihillation operator. Had H been written in terms of fermionic operators I would know how to do this, but the problem is that it describes phonon oscillations, i.e. is written in terms of bosonic...
  38. A

    How Is the Particle Density Operator Defined?

    There is something I do not understand. One way to define the current density operator is through the particle density operator Ï(r). From the fundamental interpretation of the wavefunction we have: Ï(r)= lψ(r)l2 But my book takes this a step further by rewriting the equality above...
  39. A

    Understanding Time Ordering Operator in Imaginary Time Formalism

    I am studying how to express Greens functions in imaginary time formalism. I have however big problems understanding the attached derivation. In equation 10.15 and 10.16 in...
  40. V

    Quantum physics - Symmetrizer operator

    Hi everyone. I am studying 'identical particles' in quantum mechanics, and I have a problem with the properties of the Symmetrizer (S) and Antisymmetrizer (A) operators. S and A are hermitian operators. Therefore, for what I know, their set of eigenkets must constitute a basis of the space...
  41. H

    Momentum operator as differentiation of position vector

    Is it possible to take momentum operator as dr/dt (r is position operator)? If not, why?
  42. N

    Does position operator have eigen wave function?

    I am learning quantum mechics. The hypothesis is: In the quantum mechanics, all operators representing observables are Hermitian, and their eigen functions constitute complete systems. For a system in a state described by wave function ψ(x,t), a measurement of observable F is certain to...
  43. Seydlitz

    Question on checking the linearity of a differential operator

    Suppose I have this operator: ##D^2+2D+1##. Is the ##1## there, when applied to a function, considered as identity operator? Say: ##f(x)=x##. Applying the operator results in: ##D^2(x)+2D(x)+(x)## or ##D^2(x)+2D(x)+1##? If ##1## here is considered as an identity operator then the...
  44. D

    Is the Sturm Liouville Operator Symmetric?

    Hello, I'm solving the previous exams and I have a problem with an exercise: Homework Statement q(x) a real function defined in [0,1] and continuous L a sturm Liouville operator : Lf(x)=f''(x)+q(x)*f(x) f ∈ C²([0,1]) with f(0)=0 and f'(1)=0. Is L a symetric operator relative to the...
  45. A

    Defining the square root of an unbounded linear operator

    I have started coming across square roots (H+kI)^{\frac 12} of slight modifications of Schrodinger operators H on L^2(\mathbb R^d); that is, operators that look like this: H = -\Delta + V(x), where \Delta is the d-dimensional Laplacian and V corresponds to multiplication by some function. But...
  46. A

    Fermionic Operator Equation Derivation Troubleshooting

    Attached is a derivation of the equation of motion for the fundamental fermionic anihillation opeator but I am having a bit of trouble with the notation. Does the notation Vv2-v and the other V_ simply mean that all terms in the sum of q have canceled except for when q=v2-v or v-v1? And second...
  47. T

    MHB Find T cyclic operator that has exactly N distinct T-invariant subspaces

    Let T be a cyclic operator on $R^3$, and let N be the number of distinct T-invariant subspaces. Prove that either N = 4 or N = 6 or N = 8. For each possible value of N, give (with proof) an example of a cyclic operator T which has exactly N distinct T-invariant subspaces. Am I supposed to...
  48. TrickyDicky

    What is the impact of nonperturbative effects on time-dependent quantum theory?

    If I haven't understood this tricky stuff very badly when the Hamiltonian is time independent, then Schrödinger’s equation implies that the time evolution of the quantum system is unitary, but for the time-dependent Hamiltonian one must add some mathematically "put by hand" assumptions (although...
  49. C

    Srednicki Ch5 creation operator time dependence

    Hi folks, originally I read Peskin & Schroeder but then I realized it was too concise for me. So I switched to Srednicki and am reading up to Chapter 5. (referring to the textbook online edition on Srednicki's website) Two questions: 1. In the free real scalar field theory, the creation...
  50. T

    Prove that a linear operator is indecomposable

    Homework Statement Let V be a fi nite-dimensional vector space over F, and let T : V -> V be a linear operator. Prove that T is indecomposable if and only if there is a unique maximal T-invariant proper subspace of V. Homework Equations The Attempt at a Solution I tried using the...
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