Operator Definition and 1000 Threads

I don't know what is the answer, so i am not sure when to stop the computation or not. The far i reached was ## <k|j> \sum_{j} c_{j}|k>##. That is, the action of the projector operator is, obviously, project the state in |k>. Now, the coefficients was changed. So now what i have to do?

I mentioned in another thread that I've been playing with a C++ class for 3D vectors, to review overloaded operators, constructors, etc. All those functions display output so I can follow which ones get called, and when. The class generates a serial number for each vector so I can tell them...

In operator overload I often see the parameter is always a reference parameter. I want to find out why. I read the Copy Constructor in the book. I want to verify whether I understand this correctly as it's really important. I use "xx" just to represent some operator like =, >> etc. class ABC...

I want to start a thread on Operator overloading for related question. I watched this youtube video and I like the way Jamie King write the operator overloading function: This is the program I copied from him: #include <iostream> using namespace std; struct Vector {int x; int y;}; //Below is...

Suppose, there is an electro-optical modulator that can couple the neighboring modes in an optical ring resonator. The Hamiltonian for the system looks something like this^^ (see the attached image). Here we sum over all modes m and 𝜙0 is a parameter. What will be a good set of basis for the...

In the simple harmonic oscillator, I was told to use the raising and lowering operator to generate the excited states from the ground state. However, I am just thinking that how do we confirm that the raising operator doesn't miss some states in between. For example, I can define a raising...

I understand the Frechet derivative of a bounded linear operator is a bounded linear operator if the Frechet derivative exists, but is the result always the same exact linear operator you started with? Or, is it just "a" bounded linear operator that may or may not be known in the most general case?

Divergence & curl are written as the dot/cross product of a gradient. If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator. is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation

I think I get the approach. We first need to evaluate the term ##\dot A_{\mu} \nabla A^{\mu}## and then evaluate the 3D space integral; we may need to take the limit ##V \rightarrow \infty## (i.e ##\sum_{\vec k} (2 \pi)^3/V \rightarrow \int d^3 \vec k##) at some point. The mode expansions of...

My trouble might be from how I interpret the problem. Alice and Bob are entangled. After Alice makes the measurement both of their states should collapse to one of these states with a certain probability. (Unless my understand of how entanglement is wrong.) The way I am understand the question...

The basis he is talking about: {1,x,x²,x³,...} I don't know how to answer this question, the only difference i can see between this hermitians and the others we normally see, it is that X is acting on an infinite space, and, since one of the rules involving Hermitian fell into decline in the...

Given that the normal vector cross product is rotational invariant, that is $$\mathbf R(a\times b) = (\mathbf R a)\times(\mathbf R b),$$ where ##a, b \in \mathbb{R}^3## are two arbitrary (column) vectors and ##\mathbf R## is a 3x3 rotation matrix, and given the cross product matrix operator...

Is the following true if the momentum operator changes the direction in which it acts? \langle \phi | p_\mu | \psi \rangle = -\langle \phi |\overleftarrow{p}_\mu| \psi \rangle My reasoning: \langle \phi | p_\mu | \psi \rangle = -i\hbar \langle \phi | \partial_\mu | \psi \rangle \langle...

I am just solving the equation $$\frac{h}{2\pi i}\frac{\partial F}{\partial x} = pF$$, finding $$F = e^{\frac{ipx2\pi }{h}}C_{1}$$, and$$ \int_{-\infty }^{\infty }C_{1}^2 = 1$$, which gives me $$C_{1} = \frac{1}{(2\pi)^{1/2} }$$, so i am getting the answer without the h- in the denominator...

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...

I have a question for anyone on here that has pursued the nuclear energy field. I'm interested in becoming an equipment technician or operator at a nuclear power plant, but I am already 26 years old. I understand that many of the operators and technicians were former Navy nukes. I looked into...

I learned that the energy operator is ##\hat{E} = i\hbar \frac{\partial}{\partial t} ## and the Hamiltonian is ##\hat{H} = \frac{-\hbar^2}{2m}\nabla^2+V(r,t)## If the Hamiltonian represents the total energy of the system. I expect the two should be the same. Did I misunderstand the concept of...

My attempt is below. Could somebody please check if everything is correct? Thanks in advance!

Hello. Since I learned the least action principle several years ago, I cannot figure out the difference between the variational operator ##\delta## in ##\delta S=0## and the differential operator ##d## in, say ##dS##. Everytime I encountered the variational operator, I just treated it as a...

I am confused about the problem. I thought operators do not act on bra vectors, and the problem is equivalent to ##a^{\dagger} \left | \alpha \right > = \left ( \alpha ^{*} + \frac {\partial} {\partial \alpha} \right ) \left | \alpha \right > ##. Then, strangely, ##\left < \alpha \right |##...

I have some problems when calculating the operators in Heisenberg picture. First, ##\frac {dx} {dt} = \frac {1} {i \hbar} \left [ x, H \right ] = \frac {p} {m}##. Similarly, ##\frac {dp} {dt} = \frac {1} {i \hbar} \left [ p, H \right ] = - m \omega ^ 2 x##. These are coupled equations. I...

Hi All, I've been going through Shankar's 'Principles of Quantum Mechanics' and I don't quite understand the point the author is trying to make in this exercise. I get that this wavefunction is not a solution to the Schrodinger equation as it is not continuous at the boundaries and neither is...

Hello guys, I struggle with one step in a calculation to show a quantum operator equality .It would be nice to get some help from you.The problematic step is red marked.I make a photo of my whiteboard activities.The main problem is the step where two infinite sums pops although I work...

Hi everyone, I'm new to PF and this is my second post, I'm taking a QFT course this semester and my teacher asked us to obtain: $$[\Phi(x,t), \dot{\Phi}(y,t) = iZ\delta^3(x-y)]$$ We're using the Otto Nachtman: Elementary Particle Physics but I've seen other books use this notation: $$[\Phi(x,t)...

Hi everyone, I'm taking a QFT course this semester and we're studying from the Otto Nachtman: Texts and Monographs in Physics textbook, today our teacher asked us to get to the equation: [Φ(x,t),∂/∂tΦ(y,t)]=iZ∂3(x-y) But I am unsure of how to get to this, does anyone have any advice or any...

My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions. I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero...

I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...

I am attaching an image from David J. Griffith's "Introduction to Quantum Mechanics; Second Edition" page 205. In the scenario described (the Hamiltonian treats the two particles identically) it follows that $$PH = H, HP = H$$ and so $$HP=PH.$$ My question is: what are the necessary and...

I'm trying to come up with a proof of the operator identity typically used in the Mori projector operator formalism for Generalized Langevin Equations, e^{tL} = e^{t(1-P)L}+\int_{0}^{t}dse^{(t-s)L}PLe^{s(1-P)L}, where L is the Liouville operator and P is a projection operator that projects...

Hi All, I try to prove the following commutator operator Identity used in Harmonic Oscillator of Quantum Mechanics. In the process, I do not know how to proceed forward. I need help to complete my proof. Many Thanks.

In an Introduction to Quantum Mechanics by Griffiths (pg. 180), he claims that "P and H are compatible observables, and hence we can find a complete set of functions that are simultaneous eigenstates of both. That is to say, we can find solutions to the Schrodinger equation that are either...

The eigenvalue equation is $$\frac{d^2}{d \phi^2} f(\phi) = q f(\phi)$$ This is a second order linear homogeneous differential equation. The second order polynomial associated to it is $$\lambda ^2 - q = 0 \rightarrow \lambda = \pm \sqrt{q}$$ As both roots are real and distinct, the...

$$<f|\hat H g> = \int_{-\infty}^{\infty} f^*\Big(-\frac{\hbar}{2m} \frac{d^2}{dx^2} + V(x) \Big) g dx$$ Integrating (twice) by parts and assuming the potential term is real (AKA ##V(x) = V^*(x)##) we get $$<f|\hat H g> = -\frac{\hbar}{2m} \Big( f^* \frac{dg}{dx}|_{-\infty}^{\infty} -...

I feel like I'm going around in circles trying to do something with the expression ## tr( \rho *log(\rho)) ##. I thought about a Taylor expansion, but I don't think there's a useful one here because of the logarithm. We learned the Jacobi's formula in class, but I don't think I want a derivative...

Hello, I am a senior undergrad doing research in quantum optics, and I am trying to work out at the moment the output state of sending a coherent state through one input port and a squeezed vacuum state through the other, just to see what happens tbh. The problem I have constantly been running...

in fact the answer is given in the book (written by philippe Martin). we have $$ (\tau_1| A^{-1} | \tau_2) = 2D \ min(\tau_1 ,\tau_2) = 2D(\tau_1 \theta (\tau_2 -\tau_1)+\tau_2 \theta (\tau_1 -\tau_2))$$ So $$-1/2D \frac{d^2}{d\tau_1^2} (\tau_1| A^{-1} | \tau_2) = \delta( \tau_1 - \tau_2) $$...

How does the isospin operator I_3 act on a state |ud>, where u ist an up- and d a Down quark?

I have found various formulations for the Laplacian and I want to check that they are all really the same. Two are from Wikipedia and the third is from Sean Carroll. They are: A Wikipedia formula in ##n## dimensions: \begin{align} \nabla^2=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial...

I have in my notes the charge conjugation operator converts the spinnor into its complex conjugate , ## C\begin{pmatrix} \varepsilon \\ \eta \end{pmatrix}=\begin{pmatrix} \varepsilon^{*}{} \\ \eta ^{*} \end{pmatrix}##when applied to gamma matrix from dirac equation does it do the same...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need yet further help in fully understanding the proof of Proposition 8.7 ...Proposition...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some further help in fully understanding the proof of Proposition 8.7 ...Proposition...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the proof of Proposition 8.7 ...Proposition 8.7 and...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the concepts in Proposition 8.6 ...Proposition 8.6...

I am reading Michael Field's book: "Essential Real Analysis" ... ... I am currently reading Chapter 9: Differential Calculus in $$\mathbb{R}^m$$ and am specifically focused on Section 9.2.1 Normed Vector Spaces of Linear Maps ... I need some help in fully understanding Theorem 9.2.9 (3) ...

The attempt ##\int_{-\infty}^{\infty} |ψ^*(x)\, \hat x\,\psi(x)|\, dxˆ## Using ˆxψ(x) ≡ xψ(x) =##\int_{-\infty}^{\infty} |ψ^*(x)\,x\,\psi(x)|\, dxˆ## =##\int_{-\infty}^{\infty} |ψ^*(x)\,\psi(x)\,x|\, dxˆ## =##\int_{-\infty}^{\infty} |x\,ψ^2(x)|\, dxˆ## I'm pretty sure this is not the...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding some remarks by Browder after Lemma 8.4 pertaining to...

First some background, then the actual question... Background: (a) Very simple example: if we take ##Asin(x+ϕ)+0.1##, the average is obviously 0.1, which we can express as the integral over one period of the sine function. (assume that we know the period, but don't know the phase or other...

I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the differences between Andrew Browder and Michael...

Homework Statement: In the case of the quantum harmonic oscillator in 3D , does the z-component of the angular momentum of a particle commute with the Hamiltonian? Does the fundamental state has a well defined value of L_z (variance = 0) ? If you said no , why? If you said yes , what is the...