Operators Definition and 1000 Threads

Hi, When one quantizes EM the resulting gauge boson, the photon, ends up being its own antiparticle. From what I read of gluons, they have anti particles. I can follow how anti particles come about quantizing a complex-valued field like that for electrons. For the spin 1/2 case non-interacting...

I know, $$ L=T-V \;\;\; \; \;\;\; [1]\;\;\; \; \;\;\; ( Lagrangian) $$ $$ H=T+V \;\;\; \; \;\;\;[2] \;\;\; \; \;\;\; (Hamiltonian)$$ and logically, this leads to the equation, $$ H - L= 2V \;\;\; \; \;\;\...

Hi, I have a general question. How do I show that an operator expressed in spherical coordinates is self adjoint ? e.g. suppose i have the operator i ∂/∂ϕ. If the operator was a function of x I know exactly what to do, just check <ψ|Qψ>=<Qψ|ψ> But what about dr, dphi and d theta

I don't really understand how their inverses work. For example, in solving 2nd order linear non-homogeneous differential equations. The particular solution is found by ## y_{pi} = \frac{p(x)}{f(D)} ## And they continue by expanding using maclaurin series. How do you treat an operator as a...

I study on quantum mechanics and I have question about operator. In one dimension. How do we know ## \hat{x} = x## and ## \hat{p}_{x} = -i \bar{h} \frac{d}{dx} ## When schrodinger was creating an equation, which later called "the schrodinger equation". How does he know momentum operator equal...

Hello! I read in many places the derivation of the representation for SU(2) using ladder operators and in all of the places they say that, due to the fact that we are looking for a finite dimensional representation, the ladder must end at a point, hence why we have an eigenvector of ##L_3##...

Homework Statement Homework Equations ##\hat{P}= -ih d/dx## The Attempt at a Solution To actually obtain ##\psi_{p_0}## I guess one can apply the momentum operator on the spatial wavefunction. If we consider a free particle (V=0) we can easily get obtain ##\psi = e^{\pm i kx}##, where ##k=...

Homework Statement After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin. 2. The attempt at a solution I tried to apply the...

Homework Statement Again, consider the two-dimensional vector space, with an orthonormal basis consisting of kets |1> and |2>, i.e. <1|2> = <2|1> = 0, and <1|1> = <2|2> = 1. Any ket in this space is a linear combination of |1> and |2>. a) [2pt] The operator A acts on the basis kets as A|1> =...

I was recently reading about annihilation and creation operators in particle physics using the model of an harmonic oscillator, and then quantizing it. This is fine. I can understand it. But how does this quantization of the energy of the harmonic oscillator manifest physically? Is it that only...

(a) and (b) are fairly traditional, but I have trouble understanding the phrasing of (c). What makes the infinite dimensionality in (c) different from (a) and (b)?

Homework Statement The possible (normalized) eigenstates of a photon in a given system are written as: $$|\psi_1>,|\psi_2>,...|\psi_m>$$ Let another state be $$|\phi> = \frac{|\psi_1>+|\psi_2>+...+|\psi_m>}{\sqrt{m}}$$ and denote: $$|n>=|\psi_1>|\psi_1>...|\psi_1>$$ which represent a state...

I don’t see how the definition of |an> transmorphs into the statement involving the kroneck delta functions.

Ok, so my question is "Does an irreducible representation acting on operators imply that the states also transform in an irreducible representation?" and what I mean by that is the following. If I have an operator transforming in an irreducible transformation of some group, I get a corresponding...

Homework Statement Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators ## \hat A, \hat A^\dagger ##, ## \hat x ## and ##\hat p ##. Homework Equations I know how to use all the above operators projecting them on...

In a PDF i was looking through i came about a question for the operator P = |a><b| find Px(adjoint) the adjoint was defined as <v|Px|u> = (<u|P|v>)* where u and v can be any bra and ket now for the question: (<u|a><b|v>)* = <v|Px|u> this is the confusing step , i thought conjugated simply...

> Operator $$\hat{A}$$ has two normalized eigenstates $$\psi_1,\psi_2$$ with > eigenvalues $$\alpha_1,\alpha_2$$. Operator $$\hat{B}$$ has also two > normalized eigenstates $$\phi_1,\phi_2$$ with eigenvalues > $$\beta_1,\beta_2$$. Eigenstates satisfy: > $$\psi_1=(\phi_1+2\phi_2)/\sqrt{5}$$ >...

Take a projection operatorPm=|m><m| However if the ket of m is a column matrix of m x 1 and its bra the complex conjugate with 1 x m length therefore <m|m> = |m|2 since the m here is the same since projection operator is the same. if A is a matrix B = A A*B=B*A but Pm*Pm = Pm (Projection...

Hi EVERYBODY: General knowledge: The homogeneous linear Fredholm integral equation $\mu\ \varPsi(x)=\int_{a}^{b} \,k(x,s) \varPsi(s) ds$ (1) has a nontrivial solution if and only if $\mu$ is an eigenvalue of the integral operator $K$. By multiplying (1) by $k(x,s)$ and...

This may seem rather silly, but how would I go about enunciating Ehrenfest’s theorem? Also, does anyone know what this theorem implies for the relation between classical and quantum mechanics? Any suggestions or help is greatly appreciated!

Homework Statement Consider the action of ##T_2## acting on ##M_k(\Gamma_{0}(N)) ##, and show that ##\theta^4(n)+16F ## and ##F(t)## are both eigenfunctions. Functions are given by: Homework Equations For the Hecke Operators ##T_p## acting on ##M_k(\Gamma_{0}(N)) ##, the Hecke conguence...

Homework Statement Consider the eigenstates of a particle in an infinite well with walls at ##x=\pm a##. without explicitly evaluating any integrals, what is the expectation value of the following operator $$\hat{x}^2\hat{p_x}^3+3\hat{x}\hat{p_x}^3\hat{x}+\hat{p_x}^3\hat{x}^2$$ Homework...

Hello Everyone. I am searching for some clarity on this points. Thanks for your help: Based on Schrodinger wave mechanics formulation of quantum mechanics, the states of a system are represented by wavefunctions (normalizable or not) and operators (the observables) by instructions i.e...

Homework Statement I’m studying orthogonal curvilinear coordinates and practice calculating differential operators. However, I’ve run across an exercise where the coordinate system is only in 2D and I’m confused about how to proceed with the calculations. Homework Equations A point in the...

Hi, why do unbounded operators and bounded operators differ so much in terms of defining their spectra? 1. The unbounded operator requires a self-adjoint extension to define its spectrum. 2. A bounded one does not require a self-adjoint extension to define the spectral properties. 3. Still the...

Hello, Today I am studying complete set of commuting observables (CSCO) which is a set of commuting operators, pair by pair, whose eigenvalues completely specify the state of a system. For example, given 4 different commuting observables, there is a set of eigenstates which are eigenstates for...

Hi, from the books I have, it appears that some rules for operators, boundedness, positivity and possibly the definition of the spectrum regard real operators, and not complex operators. From the complex operator ##i\hbar d^3/dx^3 ## it appears that it can be defined as not bounded (unbounded)...

Hi all, so I'm not sure if what I'm asking is trivial or interesting, but is there any general or canonical way to interpret say, The follwing operator? (Specifically in the study of quantum mechanics): A = 1/(d/dx) (I do not mean d-1/dx-1, which is the antiderivative operator ) How would...

We know that for every symmetry transformation, we can define a linear, unitary operator (or antiunitary, anti linear operator) that takes a physical state into another state. My question is if there exists unitary operators that act in this way that do not correspond to any symmetry? Would a...

I'm reading Weinberg's QFT volume 1. At the end of section 2.4 he is deriving the Inonu-Wigner contraction where he reduces the Poincaré group to the Euclidean one by taking the low velocity limit. In analyzing how the operators depend on velocity there are some I understand and some I don't. I...

Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint ##P^{\dagger}=-\hbar D## has the same formal expression, but it acts on a different space of functions." Regarding the two main operators, X and D, each has...

I am interested in defining Krauss operators which allow you to define quantum measurements peaked at some basis state. To this end I am considering the Normal Distribution. Consider a finite set of basis states ##\{ |x \rangle\}_x## and a set of quantum measurement operators of the form $$A_C =...

Hi, I have in a previous thread discussed the case where: \begin{equation} TT' = T'T \end{equation} and someone, said that this was a case of non-linear operators. Evidently, they commute, so their commutator is zero and therefore they can be measured at the same time. What makes them however...

Hi, what is the true meaning and usefulness of the commutator in: \begin{equation} [T, T'] \ne 0 \end{equation} and how can it be used to solve a parent ODE? In a book on QM, the commutator of the two operators of the Schrödinger eqn, after factorization, is 1, and this commutation relation...

Hi, I noticed that the raising and lowering operators:\begin{equation} A =\frac{1}{\sqrt{2}}\big(y+\frac{d}{dy}\big) \end{equation}\begin{equation} A^{\dagger}=\frac{1}{\sqrt{2}}\big(y-\frac{d}{dy}\big) \end{equation}can be used to solve the eqn HY = EY However I am curious about something...

Hi, I found in Kreyszig that if for any ##x_1\ and\ x_2\ \in \mathscr{D}(T)## then an injective operator gives: ##x_1 \ne x_2 \rightarrow Tx_1 \ne Tx_2 ## and ##x_1 = x_2 \rightarrow Tx_1 = Tx_2 ##If one has an operator T, is there an inequality or equality one can deduce from this, in...

Hi, I am reading QFT by Lancaster and Blundell. In chapter 4 of the book the field operators are introduced: "Now, by making appropriate linear combinations of operators, specifically using Fourier sums, we can construct operators, called field operators, that create and annihilate particles...

Hi thanks to George, I found the following criteria for boundedness: \begin{equation} \frac{||Bf(x)||}{||f(x)||} < ||Bf(x)|| \end{equation} If one takes f(x) = x, and consider B = (h/id/dx - g), where g is some constant, then B is bounded in the interval 0-##\pi##. However, given that I...

The following is a somewhat mathematical question, but I am interested in using the idea to define a set of quantum measurement operators defined as described in the answer to this post. Question: The Poisson Distribution ##Pr(M|\lambda)## is given by $$Pr(M|\lambda) =...

Hi, I have an operator which does not obey the following condition for boundedness: \begin{equation*} ||H\ x|| \leqslant c||x||\ \ \ \ \ \ \ \ c \in \mathscr{D} \end{equation*} where c is a real number in the Domain D of the operator H. However, this operator is also not really unbounded...

I can't follow the solution given in my textbook to the following problem. The solution goes right off the rails on the first step. Consider a system whose Hamiltonian is given by \hat H = \alpha \left( {\left. {\left| {{\phi _1}} \right.} \right\rangle \left\langle {\left. {{\phi _2}}...

In Hartree-Fock method, I saw the Fock operator has two integrals: Coulomb integral and exchange integral. One can define two operator. "The exchange operator is no local operator" why? Whats de diference: local and no local operator? And why do the operators have singularities? thanks

Given a wave function $$\Psi(r,\theta,\phi)=f(r)\sin^2(\theta)(2\cos^2(\phi)-1-2i*\sin(\phi)\cos(\phi))$$ we are trying to find what a measurement of angular momentum of a particle in such wave function would yield. Attempts were made using the integral formula for the Expectation Value over a...

Suppose ##A## is a linear operator ##V\to V## and ##\mathbf{x} \in V##. We define a non-linear operator ##\langle A \rangle## as $$\langle A \rangle\mathbf{x} := <\mathbf{x}, A\mathbf{x}>\mathbf{x}$$ Can we say ## \langle A \rangle A = A\langle A \rangle ##? What about ## \langle A \rangle B =...

Hello, In the case of 2D vector spaces, every vector member of the vector space can be expressed as a linear combination of two independent vectors which together form a basis. There are infinitely many possible and valid bases, each containing two independent vectors (not necessarily...

So the statement which the proof's about is: For every linear transformation ##A##(between finite dimension spaces), the product ##A^*A## is self-adjoint. So, the proof is: ##(A^*A)^*=A^*A^{**}=A^*A## What i don't understand is why ##(A^*A)^*=A^*A^{**}##. Isn't that true only if ##A## and...

Hello! I am reading this paper and on page 18 it states that "in (2 + 1)D electrodynamics, p−form Maxwell equations in the Fourier domain Σ are written as: ##dE=i \omega B ##, ##dB=0##, ##dH=-i\omega D + J##, ##dD = Q## where H is a 0-form (magnetizing field), D (electric displacement field)...

I have a few issues with understanding a section of Griffiths QM regarding Hermitian Operators and would greatly appreciate some help. It was first stated that, ##\langle Q \rangle = \int \Psi ^* \hat{Q} \Psi dx = \langle \Psi | \hat{Q} \Psi \rangle## and because expectation values are real...

Homework Statement A) Show that <x>=<p>=0 hint: use orthogonality B) Use the raising and lowering operators to evaluate an expression for < x2 > Homework Equations Also A- and A+ will definitely come in handy The Attempt at a Solution I tried setting up the equations for <x> and <p> but I...

It's known that the time-translation operator is ##\exp(-i Ht)## and the space-translation operator is ##\exp(i (p \cdot x))##. The former causes a time-translation for a state vector whereas the latter causes a space-translation. Can we operate with the two operators on the state vector? Like...