Operators Definition and 1000 Threads

This is a list of operators in the C and C++ programming languages. All the operators listed exist in C++; the fourth column "Included in C", states whether an operator is also present in C. Note that C does not support operator overloading.
When not overloaded, for the operators &&, ||, and , (the comma operator), there is a sequence point after the evaluation of the first operand.
C++ also contains the type conversion operators const_cast, static_cast, dynamic_cast, and reinterpret_cast. The formatting of these operators means that their precedence level is unimportant.
Most of the operators available in C and C++ are also available in other C-family languages such as C#, D, Java, Perl, and PHP with the same precedence, associativity, and semantics.

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  1. physicsxanime

    Operator algebra: Hermicity and Eigenstates

    A. I can show that A is either hermitian or antihermitian by $$(B^\dagger B=1-A^2)^\dagger$$ $$B^\dagger B=1-A^\dagger A^\dagger$$ comparing, we know that $$A^\dagger = \pm A$$ I don't know how I can make use of the communtation relation to get hermiticity of B. But I know that A and B must have...
  2. flyusx

    Operators On Multivariable Wave Functions

    I know the way to solve the first part is to find <ψ|Αψ> and compare it with <ψΑ|ψ>. This comparison can be done through an integral representation where we take ψ* and act A on ψ to be the integrand, or act A on ψ* and multiply by ψ for the integrand. If the integrals are the same, then the...
  3. D

    I Commutators, operators and eigenvalues

    Hi I just wanted to check my understanding of something which has come up when first studying path integrals in QM. If x and px are operators then [ x , px ] = iħ but if x and px operate on states to produce eigenvalues then the eigenvalues x and px commute because they are just numbers. Is...
  4. H

    I Can Operators Have Multiple Partial Derivatives in Quantum Mechanics?

    Suppose Q=2x+t and x=t2, then ∂Q/∂t=1. But Q can also be written as Q=x+t2+t, then ∂Q/∂t=2t+1. We now have 2 different answers. But I think there can only be one correct answer. In reference to the equation in the image, no matter we write Q=2x+t or Q=x+t2+t, <Q> should be the same, so the LHS...
  5. graviton_10

    I Showing that operators follow SU(2) algebra

    For two quantum oscillators, I have raising and lowering operators and , and the number operator . I need to check if operators below follow commutation relations. Now as far as I know, SU(2) algebra commutation relation is [T_1, T_2] = i ε^ijk T_3. So, should I just get T_1 and T_2 in...
  6. D

    I Dirac Notation for Operators: Ambiguity in Expectation Values?

    Hi If A is a linear operator but not Hermitian then the expectation value of A2 is written as < ψ | A2| ψ >. Now if i write A2 as AA then i have seen the expectation value written as < ψ | A+A| ψ > but if i only apply the operators to the ket , then could i not write it as < ψ | AA | ψ > ? In...
  7. T

    I Identity involving exponential of operators

    Hey all, I saw a formula in this paper: (https://arxiv.org/pdf/physics/0011069.pdf), specifically equation (22): and wanted to know if anyone knew how to derive it. It doesn't seem like a simple application of BCH to me. Thanks.
  8. E

    I Need help understanding Commuting Operators

    Here again with another question about the Quantum Sense video series. Thanks for all the useful feedback to my last question. My question concerns a very short chunk of about 20 seconds beginning at 4:25 of this link: At around 4:34, he says "B-alpha has to be the same eigenvector as alpha...
  9. O

    I Dot product of two vector operators in unusual coordinates

    Hi. I hope everyone is well. I'm just an old person struggling to make sense of something I've read and I would be very grateful for some assistance. This is one of my first posts and I'm not sure all the LaTeX encoding is working, sorry. Your help pages suggested I add as much detail as...
  10. Euge

    POTW Translation-Invariant Operators on Lebesgue Spaces

    Show that if there exists a nonzero, translation-invariant bounded linear operator ##T : L^p(\mathbb{R}^d) \to L^q(\mathbb{R}^d)## where ##1 \le p, q < \infty##, then necessarily ##q \ge p##.
  11. H

    If |a> is an eigenvector of A, is f(B)|a> an eigenvector of A?

    Hi, If ##|a\rangle## is an eigenvector of the operator ##A##, I know that for any scalar ##c \neq 0## , ##c|a\rangle## is also an eigenvector of ##A## Now, is the ket ##F(B)|a\rangle## an eigenvector of ##A##? Where ##B## is an operator and ##F(B)## a function of ##B##. Is there way to show...
  12. olgerm

    I Very basic questions about operators in QM

    Could you tell me if I have understood following about operators in QM correctly? Wavefunction takes all generalized coordinates of the system as arguments. for example if we have a system of proton and electron (in 3-dimensional space) then the wavefunction of this system has 7 arguments...
  13. K

    I Multiplying three vector operators

    Given vector operators as $$\mathbf{A} = (A_{1}, A_{2} ,A_{3}) $$ $$\mathbf{B} = (B_{1}, B_{2} ,B_{3}) $$ $$\mathbf{C} = (C_{1}, C_{2} ,C_{3}) $$ I know that for two vector operators $$\begin{equation} \mathbf{Q} \mathbf{P} = \sum_{\alpha = 1}^{3} Q_{\alpha} P_{\alpha} \end{equation}$$...
  14. P

    Unitary Operators: Proving <Af,Ag>=<f,g>

    Hello folks, I need to show that a unitary operator obeys <Af,Ag>=<f,g>, where A is a unitary operator. However, I am technically not yet given the fact, that the adjoint of A is equal to its inverse, and that is the problem. I have no clue how to prove the given task without using the...
  15. H

    Is an operator (integral) Hermitian?

    Knowing that to be Hermitian an operator ##\hat{Q} = \hat{Q}^{\dagger}##. Thus, I'm trying to prove that ##<f|\hat{Q}|g> = <\hat{Q}f|g> ##. However, I don't really know what to do with this expression. ##<f|\hat{Q}g> = \int_{-\infty}^{\infty} [f(x)^* \int_{-\infty}^{\infty} |x> <x| dx f(x)] dx##...
  16. J

    A Raising and lowering operators

    Suppose you take a Schroedinger-like equation $$-\psi''+F(x)\psi=0$$. (E.g. F(x)=V(x)-E, and not worried about factors of 2 etc.) This is positive definite if $$\int \left( \psi'^2+F(x)\psi^2 \right)dx>0$$. Is so, you can write this as the product $$(d/dx+g(x))(-d/dx+g(x))\psi=0$$, i.e. as...
  17. Addez123

    I don't understand simple Nabla operators

    Using the formula in 'relevant equations' I calculate $$div(fA) = \nabla(fA) = (\nabla f) \cdot A + f \nabla \cdot A$$ $$3r^2 \cdot (x^2, y^2, z^2) + r^3 \cdot (2x + 2y + 2z)$$But the answer is $$3r \cdot (x^3 + y^3 + z^3) + r^3 \cdot (2x + 2y + 2z)$$ I find no way of easily turning ##3r^2...
  18. H

    B Tensor product of operators and ladder operators

    Hi Pfs i have 2 matrix representations of SU(2) . each of them uses a up> and down basis (d> and u> If i take their tensor product i will get 4*4 matrices with this basis: d>d>,d>u>,u>d>,u>u> these representation is the sum equal to the sum of the 0-representation , a singlet represertation with...
  19. G

    I Why does the QFT Lagrangian not already use operators?

    I've learned that in canonical quantization you take a Lagrangian, transform to a Hamiltonian and then "put the hat on" the fields (make them an operator). Then you can derive the equations of motion of the Hamiltonian. What is the reason that you cannot already put hats in the QFT Lagrangian...
  20. S

    I Operators in finite dimension Hilbert space

    I have a question about operators in finite dimension Hilbert space. I will describe the context before asking the question. Assume we have two quantum states | \Psi_{1} \rangle and | \Psi_{2} \rangle . Both of the quantum states are elements of the Hilbert space, thus | \Psi_{1} \rangle , |...
  21. Dario56

    I Density Operators of Pure States

    Quantum states are most often described by the wavefunction ,##\Psi##. Variable ,##\Psi(x_1x_2\dots x_n) \Psi^*(x_1x_2\dots x_n)## defines probability density function of the system. Quantum states can also be described by the density matrices (operators). For a pure state, density matrix is...
  22. H

    Proving that ##T## is skew-symmetric, inner product is an integration.

    ##\langle T(f), g \rangle = \int_{0}^{1} \int_{0}^{x} f(t) dt ~ g(t) dt## As ##\int_{0}^{x} f(t) dt## will be a function in ##x##, therefore a constant w.r.t. ##dt##, we have ##\langle T(f), g \rangle = \int_{0}^{x} f(t) dt ~ \int_{0}^{1} g(t) dt## ##\langle f, T(g)\rangle = \int_{0}^{1} f(t)...
  23. P

    Quantum exam practice, operators and eigenstates

    I'm really not sure what the question expects me to do here but here is what I do know. If the state is an eigenstate it should satisfy the eigenvalue equation for example; $$\hat{H} f_m^l = \lambda f_m^l$$ but is the question asking me to use each operator on each state? How do I know if...
  24. P

    Commutation relations between Ladder operators and Spherical Harmonics

    I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?
  25. K

    I A doubt regarding position representation of product of operators

    We've two operators ##\hat{a}##,##\hat{b}##. I know their position representation ##\langle r|\hat{b} \mid \psi\rangle=b## ##\langle r|\hat{a}| \psi\rangle=a ## Is it generally true that the position representation of the combined operator ##\hat{a}\hat{b}## is ##a b## where ##a, b## are the...
  26. H

    A Exploring Quantum Measurements and Unitary Operators

    Hi Pfs I read this answer in https://quantumcomputing.stackexchange.com/questions/136/if-all-quantum-gates-must-be-unitary-what-about-measurement Quantum measurements are special cases of quantum channels (CPTP cards). Stinespring dilation states that any quantum channel is realized by...
  27. H

    Bounded operators on Hilbert spaces

    I have to show that for two bounded operators on Hilbert spaces ##H,K##, i.e. ##T \in B(H)## and ##S \in B(K)## that the formula ##(T \bigoplus S) (\alpha, \gamma) = (T \alpha, S \gamma)##, defined by the linear map ##T \bigoplus S: H \bigoplus K \rightarrow H \bigoplus K ## is bounded...
  28. K

    Possible Results and Probabilities of a Measurement of Operator Q

    I first Normalise the wavefunction: $$ \Psi_N = A*\Psi, \textrm{ where } A = (\frac{1}{\sum {|a_n^{'}|^{2}}})^{1/2} $$ $$ \Psi_N = \frac{2}{7}\phi_1^Q+\frac{3}{7}\phi_2^Q+\frac{6}{7}\phi_3^Q $$ The Eigenstate Equation is: $$\hat{Q}\phi_n=q_n\phi_n$$ The eigenvalues are the set of possible...
  29. Hamiltonian

    I Solving Schrodinger's eqn using ladder operators for potential V

    The Schrodinger equation: $$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + \hat V\psi = E\psi$$ $$\frac{1}{2m}[\hat p^2 + 2m\hat V ]\psi = E\psi$$ The ladder operators: $$\hat a_\pm = \frac{1}{\sqrt{2m}}[\hat p \pm i\sqrt{2m\hat V}]$$ $$\hat a_\pm \hat a_\mp = \frac{1}{2m}[\hat p^2 + (2m\hat V) \mp...
  30. W

    A Define spin operators for numerical groundstate obtained by ED

    Hi, I want to measure spin components of a ground state of some models. These ground states are obtained by ED. The states for constructing the Hamiltonian are integers representing spins in binary. As the ground state (and the other eigenvectors) are now not anymore in a suitable representation...
  31. M

    I Rising and lowering operators in a molecule

    Hello! If I have a molecule in Hund case a, I can write its wavefunction (electronic + rotational) as ##|e,\Lambda,\Omega,S>|J,\Sigma,\Lambda,\Omega,M>##. I am not sure what happens if I apply, say, ##S_+## on this wavefunction (assuming I am not applying it to the top of the ladder), where this...
  32. Svend

    A How to derive the Momentum and Energy Operators from first principles?

    So we all know that the form of the momentum operator is: iħd/dx. And for energy it is iħd/dt. But how do we derive these operators? The only derivations of the i have seen is where the schrødinger equation was used, but that makes the logic circular, because the Schrødinger-Equation is derived...
  33. J

    Proof involving exponential of anticommuting operators

    For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that $$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$ which generalizes what I initially...
  34. L

    B Commutators of functions of operators

    I would like to ask whether if operators ##A## and ##B## commute also operators ##e^A## and ##e^B## commute? Also I have a question is it possible that ##e^A## is matrix where all elements are ##\infty## so that ##e^A \cdot e^B-e^B\cdot e^A## has all elements that are ##\infty##?
  35. A

    DMD is better without Ergodic Theory and Koopman Operators

    Hello everyone! This is my first post here. I am trying out an argument that I've been sculpting, and I thought this might be a good community where I can get some good feedback. My work is in data driven methods for dynamical systems, and in particular, I am an operator theorist. I have been...
  36. E

    A How can we show that the annihilation operators satisfy the given equation?

    I don't really know what I'm doing, I'd appreciate some nudges in the right direction. We defined ##\mathcal{S}## as the space of complex solutions to the Klein-Gordon equation, and for any ##\alpha, \beta \in \mathcal{S}## that ##(\alpha, \beta) =-\int_{\Sigma_0} d^3 x \sqrt{h} n_a j^a(\alpha...
  37. L

    A $S^+$ and $S^{-}$ operators formula

    For ##\hat{S}^+## and ##\hat{S}^{-}## operators for any given spin ##S## relation \hat{S}^+|S,m \rangle=\sqrt{S(S+1)-m(m+1)}\hbar|S,m+1 \rangle \hat{S}^-|S,m \rangle=\sqrt{S(S+1)-m(m-1)}\hbar|S,m-1 \rangle Can someone please explain how we get those factors ##\sqrt{S(S+1)-m(m+1)}\hbar## and...
  38. E

    A The vertex factors in QCD penguin operators

    Have a look at O5 & O6 in Eqtns(5.4) . Why is there a (V+A) ? (V+A) contains the projection operator which projects out the right Weyl from a Dirac spinor. As per the Feynman rules of electroweak theory, there is a (V-A) assigned to each (Dirac) spinor-W boson vertex because W only couple to...
  39. berkeman

    Finding expectation value of two operators in a 3 state QM system

    A recent thread by @coolcantalope was accidentally deleted by a Mentor (I won't say which one...), so to restore it we had to use the cached version from Yahoo.com. Below are the posts and replies from that thread. The cached 2-page thread can be found by searching on the thread title, and is...
  40. J

    Prove that Casimir operators commute with the elements of Lie algebra

    I want to show that ##[C, a_{r}] = 0##. This means that: $$ Ca_{r} - a_{r}C = \sum_{i,j} g_{ij}a_{i}a_{j}a_{r} - a_{r}\sum_{i,j} g_{ij}a_{i}a_{j} = 0$$ I don't understand what manipulating I can do here. I have tried to rewrite ##g_{ij}## in terms of the structure...
  41. H

    A How can we measure these Hermitian operators?

    Hi Pf, I am reading this article about generalization of Pauli matrices https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices#Generalized_Gell-Mann_matrices_%28Hermitian%29 When i receive a qubit in a given density matrix , i can measure the mean values of the Pauli matrices by...
  42. chocopanda

    Harmonic oscillator with ladder operators - proof using the Sum Rule

    I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator: $$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$ The exercise explicitly says to use laddle operators and to express $p$ with $$b=\sqrt{\frac {mw}{2 \hbar}}-\frac...
  43. chocopanda

    Quantum Mechanics: creation and annihilation operators

    Hello everyone, I'm new here and I'm struggling with the mathematical formalities in quantum mechanics. $$\langle n+1|b^\dagger bb^\dagger + \frac 12 |n \rangle = \langle n+1|b^\dagger bb^\dagger |n \rangle + \langle n+1| \frac 12 |n \rangle $$ $$ = \langle n+1|b^\dagger b \sqrt{n+1} |n+1...
  44. Q

    A Invariance of discrete Spectrum with respect a Darboux transformation

    According to this this the Darboux transformation preserves the discrete spectrum of the Haniltonian in quantum mechanics. Is there a proof for this? My best guess is that it has to do with the fact that $$Q^{\pm}$$ are ladder operators but I'm not sure.
  45. E

    Normalisation constants with ladder operators

    The previous part was to show that ##a_+ \psi_n = i\sqrt{(n+1)\hbar \omega} \psi_{n+1}##, which I just did by looking at$$\int |a_+ \psi_n|^2 dx = \int \psi_n^* (a_{-} a_+ \psi_n) dx = E+\frac{1}{2}\hbar \omega = \hbar \omega(n+1)$$so the constant of proportionality between ##a_+ \psi_n## and...
  46. pellman

    I Anti-unitary operators and the Hermitian conjugate

    The definition of the hermitian conjugate of an anti-linear operator B in physics QM notation is \langle \phi | (B^{\dagger} | \psi \rangle ) = \langle \psi | (B | \phi \rangle ) where the operators act to the right, since for anti-linear operators ( \langle \psi |B) | \phi \rangle \neq...
  47. VVS2000

    I Proof of Commutative Operators for Simultaneous Measurement

    I was seeing a lecture and the professor told that i) you can measure an operator if it's hermitian, therefore observables are hermitian operators ii) if you can measure two observables simultaneously, then those two observables(operators) Is there any proof for this or is it some kind of rule...
  48. A

    QFT question about using momentum raising and lowering operators

    How did you find PF?: Google I know how to express Hamiltonian for scalar field written in field operators through the raising and lowering momentum operators, but I can't figure out how to do the same for the number of particles written in field operators: the 1/2E coefficient within the...
  49. PeroK

    How Does Wick's Theorem Apply to Time-Independent Bose Operators?

    This is problem 18.3 from QFT for the gifted amateur. I must admit I'm struggling to interpret what this question is asking. Chapter 18 has applied Wick's theorem to calculate vacuum expectation values etc. But, there is nothing to suggest how it applies to a product of operators. Does the...
  50. Yellotherephysics

    A Functional Determinant of a system of differential operators?

    So in particular, how could the determinant of some general "operator" like $$ \begin{pmatrix} f(x) & \frac{d}{dx} \\ \frac{d}{dx} & g(x) \end{pmatrix} $$ with appropriate boundary conditions (especially fixed BC), be computed? And assuming that it diverges, would it be valid in a stationary...
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