Questions tagged [operators]

In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as observables and as time evolution operators in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!

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$[(\hat{a}^{\dagger})^2, \hat{a}] = -2\hat{a}^{\dagger}$?

I'm confused by a line in the following wikipedia article on the squeeze operator in deriving the action of the squeeze operator on Heisenberg basis, the article seems to imply that $$[(\hat{a}^{\...
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What is wrong with the dependence of result with the order of the operation in my computation of Higgs sector derivative term?

In Standard Model Higgs sector, there are terms that introduce : $\left(D_{\mu}\Phi\right)= \left( \begin{array}{cc} \partial_{\mu}+i\frac{g}{2}W_{\mu}^3+i\frac{g'}{2}B_{\mu} &i\frac{g}{2}W_{\mu}^...
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Normal ordering of number operators $n$th power

In resources I keep seeing the normally ordered form of the number operator to the $n$th power, $${(a^\dagger a)}^n=\sum_{k=1}^n S(n,k){(a^\dagger)}^ka^k.$$ Why are we interested in the number ...
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Wick's theorem: From operators to fields

I understand Wick's Theorem when operators are involved to be, $$\mathcal{N}(f(a,a^\dagger) = :\!\sum\textbf{All contractions}\!:$$ But I'm getting slightly confused when this is expanded to fields, I'...
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What is the role of Hermitian Hamiltonians in relativistic QFT?

In single-particle quantum mechanics, the probability of finding the particle in all space is conserved due to the hermiticity of the Hamiltonians (and remains equal to unity for all times, if ...
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Polar decomposition of a complex scalar field theory

In the text I am referring to, the field was substituted in terms of a number density and phase: $$\psi(x) = \sqrt(ρ(x))e^{iθ(x)}.$$ While quantizing the field, a commutation relation was imposed: $$[\...
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Factorization of the wavefunction in a central Hamiltonian problem

I am trying to understand the topic of the title. If I consider a central Hamiltonian, so an Hamiltonian of the form $H=\frac{p^2}{2m}+V(r)$ what are the logical steps that lead me to the known result?...
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What does the exponentiated generator of scale transformation do when it acts on a function? [duplicate]

We know that $d/dx$ is the generator of translation in the sense that $$e^{ad/dx}f(x)=f(x+a)\tag{1}$$ which can be easily be proved from the Taylor series of $f(x+a)$. Studying the very basics of ...
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Different values for the Normal ordering

I've come across 2 examples approaching the ordering of $a^2({a^\dagger})^2$, each reach different results: $a^2({a^\dagger})^2=\;:\!\sum\text{all contractions}\!:\;=\;:\!aaa^\dagger a^\dagger\!:+\;4:...
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Resolution of the identity of operator with mixed spectrum

In most quantum mechanics text books, the resolution of the identity or completeness relation is stated in the following (or similar) form $$ \mathbb I_\mathcal H = \sum\limits_n |\lambda_n\rangle \...
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Why does applying the kinetic energy operator to a free particle result in a divergent integral?

The wavefunction of a free particle is just $$\psi = Ae^{i(kx-\omega t)}$$ and when you plug this into the Schrodinger equation you get the dispersion relation $$E = \frac{\hbar^2 k^2}{2m}$$ However, ...
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Do we care about the distance between quantum observables?

In quantum mechanics, we care about the spectrum of quantum observable, the eigenvalues of observables, as they give the measurement results. I wonder if we have to care about the "distance&...
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Using Wick's Theorem in an example with the harmonic oscillator

I understand Wick's theorem to be, $$T(x)=\mathcal{N}(x)=\sum:\textbf{all contractions}:$$ And I'm researching combinatorics and quantum theory in general. How would one connect Wicks theorem to the ...
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What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?

In this post, I claimed in my answer that $\partial_{\mu}^{\dagger}=\partial_{\mu}$. The reason I claimed that is because $D^{\dagger}_{\mu}=\partial_{\mu}-iqA_{\mu}$ and I assumed that $$(\partial_{\...
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4 answers
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Precise definitions for higher spin operators

I am trying to understand the matrices and vectors presented in this section https://en.wikipedia.org/wiki/Spin_(physics)#Spin_projection_quantum_number_and_multiplicity I am looking for a reference ...
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Product rule for bras and kets

For the time evolution of expectation value of an operator $\Omega$, we can write $$\frac{d}{dt}\langle\psi|\Omega |\psi\rangle=\langle\dot\psi|\Omega|\psi\rangle+\langle\psi|\dot\Omega|\psi\rangle+\...
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Derivation of the path integral in QM [duplicate]

So I'm going through the path integral derivation for a general quantum system where the generalized coordinates and momentum are given as $q^i$ and $p^i$ respectively. The transition amplitude we ...
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2 answers
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How to write a matrix $\mathcal{M}$ such that $\mathcal{M} \boldsymbol{x}=\boldsymbol{\omega}\times\boldsymbol{x}$? [duplicate]

As is well known, it is possible to use the $\nabla$ operator as if it were a vector.  Someone consider it an abuse of notation but surely something that works well and is very useful. Well, how is it ...
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How to get the weight of an eigenstate inside the state of the system without knowing the state?

Let us suppose we have a system in a state $\Psi$, with: $\Psi = \sum_m c_m \psi_m$ Let us further suppose that we don't know what $\Psi$ or the $c_m$ are, but that we know what the $\psi_m$ are since ...
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What is the physical meaning of the eigenstates of an operator in quantum mechanics?

Let us suppose that we have an Hamiltonian that describes a quantum system. If one would like to know all of the possible values that the energy of the system described by that hamiltonian, one has to ...
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Radial position operator

While trying to find the expectation value of the radial distance $r$ of an electron in hydrogen atom in ground state the expression is: $$\begin{aligned}\langle r\rangle &=\langle n \ell m|r| n \...
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QFT field being operators

Why is it necessary for fields to be operator valued in quantum field theory instead of just having scalar amplitudes?
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Diagonalizing a given Hamiltonian

The following Hamiltonian, which has to be diagonalized, is given: $H = \epsilon(f^{\dagger}_1f_1 + f_2^{\dagger}f_2)+\lambda(f_1^{\dagger}f_2^{\dagger}+f_1f_2)$ $f_i^{\dagger}$ and $f_i$ represent ...
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Why is Dirac's Phase Operator Non-Hermitian?

I'm self-studying Gerry and Knight. To prove Dirac's phase operator is non-existent, the book makes the following argument. The conventions used are as follows: $\hat{n}$ is the number operator, $\hat{...
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Unitarity and Vector Space relation [migrated]

Can an operator be a unitary operator in a vector space and not be a unitary operator in another? If so, can I have a simple example or can anyone tell me when does that happen?
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Linearity of exponentials VS formalism [closed]

$\hat p = \frac{\hbar}{i} \frac{\partial}{\partial x} $ acting on $e^{\frac{ipx}{\hbar}}$ gives $p e^{\frac{ipx}{\hbar}}.$ By linearity, we have $\hat p \frac{1}{\sqrt{2\pi \hbar}} \int dp \Phi(p)e^{\...
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Steady-State solution of a driven damped quantum harmonic oscillator using QUTIP [duplicate]

I have a standard QHO Hamiltonian and a harmonic driving force $f(t) = \epsilon e^{-i\omega_{d}t} + \epsilon^* e^{i\omega_{d}t}$, where $\epsilon$ is a constant. $$H = \hbar\omega a^{\dagger}a + f(t)(...
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2 votes
1 answer
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Unitary transform using displacement operator to get time-independent Hamiltonian?

I am considering a driven cavity field with Hamiltonian $$H = \hbar\omega a^{\dagger}a + f(t)(a + a^{\dagger})$$ where $f(t) = \epsilon e^{-i\omega_{d}t} + \epsilon^* e^{i\omega_{d}t}$ is a classical ...
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2 answers
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A squared quantum operator is the same as using the same operator twice? So can't we find $L^2$ operator by using the $\mathbf L$ operator twice?

I understand that we can use the ladder formula and dervie the $L^2$ operator but is it wrong to use the L operator formula directly and apply it twice to observe the $L^2$ instead? \begin{align} L^2 &...
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2 answers
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Do operators always give a number after operating?

I am having some doubts regarding operators. In QM, when operators work on a wave function, will it always give a number times the wave function? Suppose I applied it on any normal function of x. Will ...
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A spectrum of the Liouvillian after unitary transformation of the operators

Let us consider the following GKSL equation $$ \dot{\rho}=-i[{H}, \rho]+\sum\limits_{k}\gamma_{k} \mathcal{D}\left[L_{k}\right] \rho = \mathcal{L} \rho, $$ here $H$ -- hermitian Hamiltonian, $\gamma_k ...
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3 votes
1 answer
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Is there a minimum value different from zero for the uncertainty in momentum for a particle in a box?

Lately, I have been studying QM more deeply and I just discovered how many important subtleties the 'well-known' particle in an infinite potential well hides, which are precious for extending ...
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1 answer
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Deriving an identity with rotation generators [closed]

I am trying to justify the following identity on page 68 of Osborn's notes on group theory: $$e^{-i\pi J_{3}}J_{2}e^{i\pi J_{3}} = -J_{2}.$$ Here, the $J_{i}$ are the typical angular momentum ...
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How to define the position operator in higher dimensions?

I am familiar with the traditional definition of the position and of the momentum operator in several dimensions. I do not like it, as it uses from the very beginning a concrete Cartesian basis. What ...
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Formula for the n-th iteration of rotational operator

Hello i was wondering since we have a formula for $$\operatorname{rot}(\operatorname{rot}(A))=\operatorname{grad}(\operatorname{div}(A))-\Delta(A)$$ Is there any nice generalisation of it ? Thanks
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5 votes
1 answer
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What does Haag's theorem say about the Schrodinger picture?

Suppose there are two interacting fields $\phi _1 $ and $\phi_2 $. Let $\psi [\phi_1, \phi_2]$ be a functional with the two fields as the input functions and complex numbers as the output, such that ...
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"Precession" of a free electron

I'd like to gain as deep of an understanding as possible of the following diagram from Introductory Quantum Mechanics, 4th ed., by Richard Liboff: I'm not looking for help with solving the easy ...
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1 vote
1 answer
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Under what conditions is a POVM a von Neumann measurement?

I want to know a definition of a von Neumann measurement. Because I can't find this concept referenced correctly in internet, and what differentiates it from a POVM, that by definition is the ...
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Commutator of $[\hat{x}, \hat{k}]$

We define the operators $$\hat{x} |x\rangle = x |x\rangle\tag{1}$$ and $$\hat{k} |k\rangle = k |k\rangle\tag{2}$$ where $$\sqrt{2\pi} \Psi(k)=\int dxe^{-ikx}\Psi(x)\tag{3}$$ and $$\langle x|k\rangle = ...
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1 vote
1 answer
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Commutator of $V(\hat{\vec{r}})$ and $\hat{L_z}$

Can someone explain to me why this is true? to me I see that $$x[\hat{p_y},V(r)]- y[\hat{p_x},V(r)]$$ $$=x(\hat{p_y}V(r)- V(r)\hat{p_y} )- y (\hat{p_x}V(r)+ \hat{p_x}V(r)).$$ The only way this can ...
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Translation invariance for scalar field [closed]

How can I see that for a scalar field $$\phi(x)=e^{i\hat{p}\cdot x}\phi(0)e^{-i\hat{p}\cdot x}$$ if we have translation invariance?
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Effect of Momentum Operator in Position Eigenstate [duplicate]

In Lectures on Quantum Mechanics by Steven Weinberg, section 3.5, he asserts that we can infer $$P_{j} \Phi_{\mathbf{x}}=i \hbar \frac{\partial}{\partial x_{j}} \Phi_{\mathbf{x}}\tag{3.5.11}$$ from ...
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Computing $f(x+\frac{d}{dx})$ operator using Taylor series [migrated]

According to Taylor's theorem, one can write $$f(x+h)=\sum_{k=0}^\infty\frac{f^k(x)}{k!}h^k,\tag{1}$$ where $f^k$ is the $k^{th}$ derivative of $f$ at $x$. Let us assume that the series converges for ...
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$SU(5)$ GUT Green's functions

Suppose I want to calculate two-point correlation functions (Green's functions) of gauge bosons in $SU(5)$ GUT theory, which field operators should I use?
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6 votes
2 answers
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Why does time reversal symmetry requires a real Hamiltonian?

I have some problems understanding the consequences of time reversal symmetry. If Hamiltonian $H$ is symmetric under time reversal, it satisfies: $$ \mathcal{T} H \mathcal{T}^{-1} = H \quad \mathrm{...
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Can you explain the relationship between spin operator $S_z$ and rotation of a vector field (or wave function)?

I was studying Rotational Invariance in Quantum Mechanics from R. Shankar's book. Where I found on problem 12.5.1 that if $\vec{\Psi}(x,y)$ is a vector as such $\vec{\Psi}(x,y)=\psi_x(x,y)\hat{x}+\...
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4 votes
1 answer
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How to calculate the OPE of the $X_L(z_1)X_L(z_2)$ in the free boson theory from the mode expansion?

From the polchinski page 238, given \begin{equation} [x_L,p_L] =[x_R,p_R]=i\tag{8.2.14} \end{equation} and the mode expansion $$\begin{equation} \begin{split} X_L(z) = x_L -i\frac{\alpha'}{2}p_L \ln z ...
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Number and phase operators in superconductors

It is stated in many texts that the number operator $N$ which counts the number of Cooper pairs and the phase operator $\phi$ which counts the superconducting order parameter's phase $\text{Arg}(\...
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3 votes
1 answer
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Recommendations for Algebraic quantum mechanics book

I am familiar with quantum mechanics and quantum information at the level of Sakurai and Preskill's lecture notes / Nielsen and Chuang. I want to study the $C^*$ algebraic formulation of quantum ...
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Bra ket notation in spherical coordinates

A position eigenket can be written using a tensor product of individual Cartesian eigenkets as $\mathbf x=|x\rangle \otimes|y\rangle \otimes |z\rangle$ Can I also using spherical coordinates write the ...
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