Operator Definition and 1000 Threads

I'm just trying to follow the below And I understand all, I think, except what's happened to the term when A hits 1: [A,1] ? If I'm correct basically we're just hitting on the first operator so reducing the power by one each time of the operator in the right hand bracket thanks

Trying to prove Hermiticity of the operator AB is not guaranteed with Hermitian operators A and B and this is what I got: $$<\Psi|AB|\Phi> = <\Psi|AB\Phi> = ab<\Psi|\Phi>=<B^+A^+\Psi|\Phi>=<BA\Psi|\Phi>=b^*a^*<\Psi|\Phi>$$ but since A and B are Hermitian eigenvalues a and b are real, Therefore...

In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem. This should be value...

Hi all- I am trying to obtain eigenvalues for an equation that has a very simple second order linear differential operator L acting on function y - so it looks like : L[y(n)] = Lambda (n) * y(n) Where y(n) can be written as a sum of terms in powers of x up to x^n but I find L is non self...

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: Please see image [2] below. I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...

I think the solution to this problem is a straightforward calculation and I think I was able to make reasonable progress, but I'm not sure how to finish this... $$\begin{align*} \vec{P}&=-\int dx^3 \pi \nabla \phi\\ &= -\int\int\int dx^3\frac{dp^3}{(2\pi)^3 2e(p)} \frac{du^3}{(2\pi)^3}...

If i had a bosonic field ##\phi(x)## and I took the exponential in the following way to get the operator $$W=e^{\imath f \phi(x)}$$ where ##f## is a parameter what effect would this have when acting on the vacuum ##|0\rangle##? Is it analogous to the space translation operator? Will it transform...

I have written the equation, with an unknown constant $$a^\dagger \lvert n\rangle = C_{n+1}\lvert n+1 \rangle$$ I then take the adjoint to get $$\langle n \rvert a = \langle n+1 \rvert C_{n+1}^\text{*}$$ I then multiply them to get $$\langle n \rvert aa^\dagger \lvert n \rangle = \langle n+1...

While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...

Please see this page and give me an advice. https://physics.stackexchange.com/questions/499269/simultanious-eigenstate-of-hubbard-hamiltonian-and-spin-operator-in-two-site-mod Known fact 1. If two operators ##A## and ##B## commute, ##[A,B]=0##, they have simultaneous eigenstates. That means...

Hey guys, what would you say are the most important aspects of the job of being a nuclear reactor operator?

I'm just curious what physical processes these operators represent. Since particles are created/destroyed in pairs, do they have to applied twice to describe an actual process?

I can't think of a counterexample.

Continue reading...

The momentum operator for one spation dimension is -iħd/dx (which isn't a vector operator) but for 3 spatial dimensions is -iħ∇ which is a vector operator. So is it a vector or a scalar operator ?

Look, I am sorry for not being able to post any LaTeX. But I am stuck at a place where I feel I should not be stuck. I can not figure out how to correctly do this. I can't seem to recreate the Pauli matrices with that form using the 3 2-dimensional bases representing x, y, and z spin up/down...

here delete, thread, and const are obviously keywords, I saw this when the system libraries threw a breakpoint, not my code, I'm new at this and that line of code makes very little sense to me, afaik, (const thread&) should be a conversion operator, how can you set an operator to a statement...

I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25. How do I derive the given equations?

I have a 4x4 operator O. I apply it on a 4x1 vector A. Let's say A =[0.7; 0.4 ; 0.4; 0.3]. When O acts on A, I get B. Let's say B=[0.74 ; 0.56; 0.08 ; 0.36]. The problem is I don't know how to find O. Can you please help me. My basis are [1 ; 0 ; 0; 0], [0;1 ; 0 ;0] ... and so on. Thanks...

I am reading a PHD thesis online "A controlled quantum system of individual neutral atom" by Stefan Kuhr. In it on pg46, he has a Hamiltonian I am also reading a book by L. Allen "optical resonance and two level atoms" in it on page 34 he starts with a Hamiltonian where the perturbation is...

hey :) assume I have an operator A with |ai> eigenstates and matching ai eigenvalues, and assume my system is in state |Ψ> = Σci|ai> I know that applying the measurement that corresponds to A will collapse the system into one of the |ai>'s with probability |<Ψ|ai>|2. with that being...

Is there a relationship between the momentum operator matrix elements and the following: <φ|dH/dkx|ψ> where kx is the Bloch wave number such that if I have the latter calculated for the x direction as a matrix, I can get the momentum operator matrix elements from it?

Hi PF! What is meant by the spectrum of a linear operator ##A##? I read somewhere that if ##0## belongs in the spectrum, then ##A## is not invertible. Can anyone finesse this for me? I read the wikipedia page, but this was tough for me to understand. Perhaps illustrating with a simple example...

Eigenvalues ##\lambda## for some operator ##A## satisfy ##A f(x) = \lambda f(x)##. Then $$ Af(x) = \lambda f(x) \implies\\ xf(x) = \lambda f(x)\implies\\ (\lambda-x)f(x) = 0.$$ How do I then show that no eigenvalues exist? Seems obvious one doesn't exists since ##\lambda-x \neq 0## for all...

The Feynman propagator: $$D_{F}(x,y) = <0|T\{\phi_{0}(x) \phi_{0}(y)\}|0> $$ is the Green's function of the operator (except maybe for a constant): $$ (\Box + m^2)$$ In other words: $$ (\Box + m^2) D_{F}(x,y) = - i \hbar \delta^{4}(x-y)$$ My question is: Which is the operator that...

Hello, I know we have the parity operator for inversion in quantum mechanics and for rotations we have the exponentials of the angular momentum/spin operators. But what if I want to write the operator that represent a reflection for example just switching y to -y, the matrix in real space...

That's my attempting: first I've wrote ##e## in terms of the power series, but then I don't how to get further than this $$ \sum_{n=0}^\infty (-1)^n \frac {Â^n} {n!} \hat B \sum_{n=0}^\infty \frac {Â^n} {n!} = \sum_{n=0}^\infty (-1)^n \frac {Â^2n} {\left( n! \right) ^2} $$. I've alread tried to...

Hello everybody! I have a doubt in using the chiral projection operators. In principle, it should be ##P_L \psi = \psi_L##. $$ P_L = \frac{1-\gamma^5}{2} = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} $$ If I consider ##\psi = \begin{pmatrix}...

In quantum mechanics, I can write the hamiltonian as ##\hat{H} = \hat{p}^{2}/2m + \hat{V}##. I am confusing with the definition of the operator ##\hat{V}##, who represents the potential energy. If the potential energy depend only on the position, is it correct write ##\hat{V} = V(\hat{x})##...

The definition of coherent state $$|\phi\rangle =exp(\sum_{i}\phi_i \hat{a}^\dagger_i)|0\rangle $$ How can I show that the state is eigenstate of annihilation operator a? i.e. $$\hat{a}_i|\phi\rangle=\phi_i|\phi\rangle$$

1) To show that ##K## is compact let ##\{ f_{n} \}_{n=1}^{\infty}## be a bounded sequence in ##L^{2}[0,1]## with ##\|f_{n}\| \le M##. For every ##\epsilon > 0##, there exists ##\delta > 0## such that ##|k(x,y)-k(x',y')| < \epsilon## whenever ##|x-x'|+|y-y'| < \delta##. Therefore, ##\{ Kf_{n}\}##...

Dear all, I've been reading and got confused of the concept below have two questions question 1) For <ψ|HA|ψ> = <Hψ|A|ψ>, why does the Hamiltonian operator acting on the bra state and <ψ|AH|ψ> in this configuration it will act on the ket state? question 2) what does it mean for H|ψ> = |Hψ>...

Homework Statement Given a Hilbert space $$V = \left\{ f\in L_2[0,1] | \int_0^1 f(x)\, dx = 0\right\},B(f,g) = \langle f,g\rangle,l(f) = \int_0^1 x f(x) \, dx$$ find the minimum of $$B(u,u)+2l(u)$$. Homework Equations In my text I found a variational theorem stating this minimization problem...

Homework Statement The tensor force operator between 2 nucleons is defined as ##S_{12}=3\sigma_1\cdot r\sigma_2\cdot r - \sigma_1\cdot \sigma_2##. Where r is the distance between the nucleons and ##\sigma_1##and ##\sigma_2## are the Pauli matrices acting on each of the 2 nucleons. Rewrite...

Was not sure weather to post, this here or in differential geometry, but is related to a GR course, so... I am having some trouble reproducing a result, I think it is mainly down to being very new to tensor notation and operations. But, given the metric ##ds^2 = -dudv + \frac{(v-u)^2}{4}...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 12: Multilinear Algebra ... ... I need some help in order to fully understand the proof of Proposition 12.2 on pages 277 - 278 ... ...Proposition 12.2 and its proof read as...

I've been reading this book, in which the author expresses the vacuum projection operator ##\vert 0\rangle\langle 0\vert## in terms of the number operator ##\hat{N}=\hat{a}^{\dagger}\hat{a}##, where ##\hat{a}^{\dagger}## and ##\hat{a}## are the usual creation and annihilation operators...

Is the following true? ψ*∇^2 ψ = ∇ψ*⋅∇ψ It seems like it should be since you can change the direction of operators.

Hi. I have just looked at a question concerning a free particle on a circle with ψ(0) = ψ(L). The question asks to find a self-adjoint operator that commutes with H but not p. Because H commutes with p , i assumed there was no such operator. The answer given , was the parity operator. It acts...

Homework Statement Let ##T## be a distribution in ##\mathcal{D}(\mathbb{R}^2)## such that: $$T(\phi) = \int_{0}^{1}dr \int_{0}^{\pi} \phi(r, \Phi)d\Phi$$ $$\phi \in \mathcal{D}(\mathbb{R}^2)$$ calculate ##r \frac{\partial{}}{\partial{r}} \frac{\partial{}}{\partial{\Phi}}T##. Homework...

Homework Statement I'd like to calculate the form of Liouville operator in a Robertson Walker metric. Homework Equations The general form is $$ \mathbb{L} = \dfrac{\text{d} x^\mu}{\text{d} \lambda} \dfrac{\partial}{\partial x^\mu} - \Gamma^{\mu}_{\nu \rho} p^{\nu} p^{\rho}...

In QM, states evolve in time by action of the Time Evolution Unitary Operator, U(t,t0). Without the action of this operator, states do not move forward in time. Yet even stationary states, like an eigenstate of energy, still contain a time variable – they oscillate in time at a fixed...

So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ | \hat A ψ \rangle = \langle ψ | \hat A ^* \hat A | ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counter-intuitive...

Hi everyone, I am beginner in python programming. So many doubts are being generted in the learing process Can anyone please explain me how the bitwise NOT (~) operator actually works on values. I have attached a screen short of my textbook (unofficial) with this post and I am confused how...

My question is given an orthonormal basis having the basis elements Ψ's ,matrix representation of an operator A will be [ΨiIAIΨj] where i denotes the corresponding row and j the corresponding coloumn. Similarly if given two dimensional harmonic oscillator potential operator .5kx2+.5ky2 where x...

Homework Statement This isn't exactly a problem but rather a problem in understanding the derivation of the phenomenon, or more precisely, one step in the derivation. In the following we will consider the EPR pair of two spin ##1/2## particles, where the state can be written as $$ \vert...

Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...

1. The problem statement, all variables and given/k Find the adjoint operator L* to the first order differential operator L = curl defined to domain omega. The full problem is attached. Homework EquationsThe Attempt at a Solution I've checked online. I am getting two different answers. Is the...

Homework Statement Consider the state $$\psi_\alpha = Ne^{\alpha \hat a^\dagger}\phi_0, $$ where ##\alpha## can be complex, and ##N = e^{-\frac{1}{2}|\alpha|^2}## normalizes ##\psi_\alpha##. Find ##\hat N \psi_\alpha##. Homework Equations $$\hat N = \hat a^\dagger \hat a$$ $$\hat a\phi_n =...

Homework Statement Prove Sturm-Liouville differential operator is self adjoint when subjected to Dirichlet, Neumann, or mixed boundary conditions. Homework Equations l = -(d/dx)[p(x)(d/dx)] + q(x) The Attempt at a Solution I have no idea. If someone can give me a place to start that would...