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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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What does the notation $\langle a|b|c\rangle$ mean? [on hold]

What does the notation $\langle a|b|c\rangle$ mean? I saw this in a Quantum Mechanics book and couldn’t understand it
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1answer
120 views

Need verification if this simple derivation of the Schrödinger equation is valid

By 1924 it was well observed that matter (as well as light) has wave-particle duality (later named quantum), and the wavelength-momentum-energy relation of quanta $$\lambda=\frac{h}{p}\;\;\...
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Doubt from Momentum operator in the position basis [duplicate]

J.J Sakurai shows in the section of ' Momentum operator in the position basis' as $$P \lvert\alpha\rangle = \int dx^{'}\lvert\ x{'}\rangle\Bigl(-i{h\over 2\pi} {\partial\over\partial x} \langle\ x{...
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27 views

Normal ordering of operators: Commutator or no commutator [duplicate]

My exercise for a quantum optics course tells me to "find a normal ordering" for an Operator $\hat{O}(\hat{a},\hat{a}^\dagger)$, which is given as a (rather complicated) string of $\hat{a}$'s and $\...
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1answer
24 views

Operators acting on a single subsystem within a combined system's state

I was reading over combined systems and operators acting on a single system within the combined system, and am confused by the math. For example, we have individual qubit states for subsystems $A$ ...
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28 views

Momentum Operator on a Riemannian Manifold

Consider a non-linear sigma model on a Riemannian manifold with metric $g_{ij}$ with the action $$S= \frac{1}{2} \int dt g_{ij}(X) \frac{dX^i}{dt} \frac{dX^j}{dt}.$$ The momentum operator is $$P_i= \...
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1answer
48 views

Questions about BRST symmetry [closed]

For a course about the standard model, I am writing a paper on BRST symmetry. For this I am mainly following the material developed in chapter 16.4 of Peskin and Schroeder. I am mostly done, however ...
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2answers
72 views

How to derive equation (N.15) in Ashcroft and Mermin's Solid State Physics?

They state in their book on page 792 the following: It can be proved, however, that if $A$ and $B$ are operators linear in the $u(R)$ and $P(R)$ of a harmonic crystal, then: $$\langle e^A e^B \...
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1answer
76 views

Do position operators commute at different time?

I had seen a general case that $\hat{q}(t)$ and $\hat{q}(t')$ doesn't commute at different time $t$ and $t'$, where $\hat{q}(t)$ and $\hat{q}(t')$ are Operators in Heisenberg's view. I tried to ...
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1answer
60 views

What are the eigenstates of $X^N$ operator?

The operator $X$ is the position operator with it's conjugate being the momentum operator: $$[X,P]=i$$ ($\hbar=1$). Eigenstates of the position operator is known as quadrature/position states: $$X|x\...
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29 views

Applying the adjoint of an operator

Consider the following inner product: $$ \langle x | Z\rho Z^\dagger | y\rangle$$ Here $Z$ is an operator and $Z^\dagger$ is it's conjugate. $\rho$ is a density matrix. Does this equal to the ...
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83 views

Why do we need creation and annihilation operators in QFT?

2. Why do we need creation and annihilation operators? Main point is that a particle can be created by creation operator and destroyed by ...
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1answer
89 views

Question about understanding quantum fields [closed]

1. How do we interpret colisions in QFT formalism? How did we know, when developing the theory, that we are getting the fields which describe creation of particles? How does one excite the field to ...
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2answers
69 views

Heisenberg Picture from $\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$

I have a question about the equation below: $$\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$$ Is this equation valid in the Schrödinger picture, Heisenberg picture, or in ...
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104 views

Is there a difference between a Hermitian operator and an observable?

My poorly written lecture notes say that any Hermitian operator does have a complete set of orthogonal eigenstates with real corresponding eigenvalues and is therefore an observable. In the article ...
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3answers
77 views

What does $X|n\rangle \propto |n+c\rangle$ mean?

$\renewcommand{\ket}[1]{\left \lvert #1 \right\rangle}$ I'm transcribing below (but see edit history for a scan) a calculation from pg 17 of this article on Lie groups and Lie algebras $$ [N,X]=cX....
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2answers
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Use of classical equations of motion inside correlation functions

I am reading this paper by Zamolodchikov about the expectation value of $T \bar{T}$ in $2d$ QFT and I don't understand how he uses the classical equations of motion. For instance, classically, in any ...
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1answer
49 views

Question about commutators acting on wavefunctions

Consider a commutator acting on a 1D wavefunction: $$[\frac{\hbar}{i} \frac{d}{dx},x]\psi(x)=(\frac{\hbar}{i} \frac{d}{dx}x-x\frac{\hbar}{i} \frac{d}{dx})\psi(x).$$ Now does this mean $\frac{\hbar}{...
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1answer
67 views

Expectation value of derivative of operator

I was given the following question: Let $A(\lambda)$ be a Hermitian operator, which is dependent on some real parameter $\lambda$. Let us denote the eigenvalues and corresponding eigenstates of $A$ ...
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1answer
62 views

Question about field quantization

I'm reading a paper by Rubin, Klyshko, Shih and Sergienko titled "Theory of two-photon entanglement in type_II optical parametric down-conversion", 1994 (link to the paper). I'm stumped by equation ...
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3answers
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Does the set of the degenerated eigenfunctions of hamiltonian forms a subspace?

I have read in a book that the set $\{ \Psi_{n}^{(\nu)} \in \mathcal{H} | \ \ \hat{H}\Psi_{n}^{(\nu)} = E_{n}\Psi_{n}^{(\nu)} \}$ (that is, the set of all eigenfunctions of the hamiltonian with the ...
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1answer
51 views

Problem with expansion of normal ordering

I am reading normal ordering..and far now I'm able to understand. I am stuck in third line from second expression in the book Lectures On Quantum Field Theory By Ashok Das in page no. 237. It is ...
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1answer
21 views

Questions about finding “top rung” and “bottom rung” of angular momentum operator (Proof in Griffiths)

The problem is like this: Let $$L_x = yp_z - zp_y, L_y = zp_x - xp_z, L_z = xp_y- yp_x, \\ L^2 = {L_x}^2 + {L_y}^2 + {L_z}^2 \\ L_\pm \equiv L_x \pm iL_y $$ We wish to find a "top rung" $f_t$ and ...
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Rising/lowering operators and trigonometric functions

I've just started learning about angular momentum and spin theory, and when I came across the definitions of the rising and lowering operators, I noticed the inverse form looks suspiciously like the ...
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1answer
88 views

What is Wick's theorem and what this is use for? [closed]

I am reading Wick's theorem but although I look for it to clearly understand in some textbooks and youtube videos but still it is unclear to me. I cannot get my head over what is normal ordering ...
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3answers
100 views

What are the eigenvector's of the $\hat a^2$ operator?

Since $\hat a^2$ and $\hat a$ commute, then one of the eigenvectors of $\hat a^2$ will be, the coherent state $|\alpha\rangle$. Are there others states as well?
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2answers
126 views

Rigorously why there should be an operator product expansion in conformal field theory?

This is probably something quite trivial I'm not getting. I'm studying CFT (conformal field theory) through David Tong's lecture notes and on page 9 he says: We now define the operator product ...
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37 views

Which orderings are commonly used in modern physics?

I am curious as to which mathematical orderings are used in contemporary theoretical physics, and in what context/situations. So far, I have encountered the following: Time ordering: Commonly used ...
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2answers
116 views

Transformation connecting two representations - Quantum mechanics [duplicate]

I am working on Dirac's paper The Lagrangian in Quantum Mechanics. He looks for the analogy between a classical transformation between two sets of coordinates and momenta $p_r$, $q_r$ and $P_r$, $Q_r$ ...
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2answers
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Why does the number operator gives the number of excitations?

Could somebody explain why the number operator (for a simple harmonic oscillator) gives the number of excitations? I understand its definition and its relation to the Hamiltonian, but I just can't ...
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1answer
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Derivating operator acting on ket

I'm deducing a formula, and I used the "product rule" $\frac{\partial}{\partial t}(A|\phi>)=(\frac{\partial A}{\partial t})|\phi>+A\frac{\partial}{\partial t}|\phi>$. I'm actually getting the ...
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1answer
39 views

Quantum mechanics angular momentum spherical tensor components

In Sakurai Quantum Mechanics, problem 3.25b we imagine $J_z^2$ as the component of a tensor with components $T_{ij} = J_iJ_j$. $J_z^2 = \frac{1}{3}\pmb{J}^2 + (J_z^2 - \frac{1}{3}\pmb{J}^2) $ The ...
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2answers
129 views

$T \bar{T}$ OPE

In page 157 of Di Francesco (Conformal Field Theory) it is said that the holomorphic and antiholomorphic components of the energy-momentum tensor have the trivial OPE $T(z) \bar{T}(\bar{w}) \sim 0$....
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1answer
57 views

QFT Complex scalar field and commutators

The conserved charge is $$Q=i\int\ d^3x(\phi\pi-\phi^\dagger\pi^\dagger)$$ Expressing this in terms of creation annihilation operators gives $$Q=i\int d^3 x \frac{d^3p d^3k}{(2\pi)^3(2\pi)^3}\frac{i\...
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OPE Kac-Moody Currents

We have the following operators: \begin{align} J^a(z) = \frac{1}{2}\psi_s^{\dagger}(z)\sigma^a_{s s'}\psi_{s'}(z), \hspace{10 mm} \bar{J}^a(z) = \frac{1}{2}\psi_s^{\dagger}(\bar{z})\sigma^a_{s s'}\...
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35 views

Does the dictionary always map the bulk operator to the CFT operator?

Using the (extrapolate) dictionary, one can map a bulk field to a boundary CFT operator. The mapped operator is always a CFT operator? How is it guaranteed?
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Why is an operator the quantum mechanical analogue of an observable?

I used to think because that, if objects are treated as waves, then using operators is the necessary thing to do in order to "retrieve" the observable from a given wavefunction. For example, in ...
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65 views

How to efficiently compute the commutator $[\hat{r},\nabla^2]$?

Given a system with Hamiltonian $ \hat{H} = \frac {\hat{p} ^2}{2m} + \hat{V}(r)$ in a certain state $|\psi \rangle$, I want to find if $\langle r \rangle$ varies with time. From $$ i \hbar\frac {d ...
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1answer
48 views

How to derive the analytical expression for the retarded Green's function with quadratic Hamiltonian?

For two operators, $A(t)$ and $B(t)$ the retarded Green’s function is defined as \begin{equation} G^R(t,t') \equiv \langle \langle A(t)|B(t) \rangle \rangle^R = -i\theta(t-t')\langle \{A(t),B(t')\} \...
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1answer
62 views

Proof for $\langle i[A,B]\rangle$ [closed]

I have to prove the following equation: $$ \langle i[A,B]\rangle = 2\mathfrak{Im}\left[\int dV(\overline{B\psi)}(A\psi)\right]\,,$$ where A,B are hermitian operators. Here is my calculation, but I don'...
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37 views

How to understand notation in “Introduction to Quantum Mechanics (3rd Edition)” by David Griffiths, Chapter 3.6.2?

In the 3rd edition, on page 118, the projection operator is introduced as $$\hat{P}=|\alpha\rangle\langle\alpha|.$$ Then Griffiths says that when $\hat{P}$ acts on another vector, it looks like ...
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2answers
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Finding the Green function of an operator in QFT

I'm working on some quantum field theory and have to operate on a field with the following operator: $$ (x^\mu \partial_\mu + 1)^{-1} $$ I've been trying to find an explicit form of this operator, ...
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1answer
62 views

What is the physical interpretation of the derivative of a particle field?

I am learning quantum field theory, specifically the quantization of the electromagnetic field. We have this Laplacian $$ \mathcal{L} = -\frac{1}{2} \partial_\mu A_\nu \partial^\mu A^\nu -j_\mu A^\mu $...
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21 views

Source for mathematical methods [duplicate]

I am just curious about any question and example sources for linear vector spaces, bra-ket notation, operators, commutators and hilbert spaces.
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1answer
64 views

Well-definedness of Holstein-Primakoff transformation

In many-body physics, Holstein-Primakoff transformation is defined as follows: \begin{align} S_i^+ &= \sqrt{2S}(1-a_i^\dagger a_i/2S)^{1/2}a_i, \\ S_i^- &=\sqrt{2S}(1-a_i^\dagger a_i/2S)^{1/2}...
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3answers
75 views

Action of unitary operator on operator: two different definitions?

Defnition 1: Ref: Section 2.4.9, Quantum Mechanics: Concepts and Applications By Nouredine Zettili Ket $|\psi \rangle $ transform as $\underline{|\psi' \rangle = \hat{U} |\psi \rangle }$. Given ...
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1answer
50 views

Domain of the infinite square well hamiltonian

I am reading the book by Gitman et al. 'self-adjoint extensions in quantum mechanics'. In the book, they give a precise definition of the domain of the hamiltonian of an infinite square well. For ...
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1answer
55 views

Is positron creation operator times electron creation operator equal to the ground state?

This is part of a larger problem, but the important part is that at one point I have: $$ bb^\dagger+bd+d^\dagger b^\dagger + d^\dagger d + b^\dagger b +db + b^\dagger d^\dagger+d d^\dagger $$ where $...
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1answer
113 views

Expressing the Schrödinger equation in 2nd quantised language

For times sake, I will only write about the non-interacting part of the Hamiltonian, $$H_0=\sum_{j=1}\left(-\frac{\hbar^2}{2m}\frac{\partial}{\partial x_j^2}+U(x_j)\right)$$ where $U(x_j)$ is some ...
7
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173 views

What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics?

Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group. Now several subgroups of this group play an important role ...