Operator Definition and 1000 Threads

I'm trying to understand gradient as an operator in Bra-Ket notation, does the following make sense? <ψ|∇R |ψ> = 1/R where ∇R is the gradient operator. I mean do the ψ simply fall off in this case? Equally would it make any sense to use R as the wave function? <R|∇R |R> = 1/R

The Green's function is defined as follows, where ##\hat{L}_{\textbf{r}}## is a differential operator: \begin{equation} \begin{split} \hat{L}_{\textbf{r}} \hat{G}(\textbf{r},\textbf{r}_0)&=\delta(\textbf{r}-\textbf{r}_0) \end{split} \end{equation}However, I have seen the following...

So I am aware the charge conjugation operator changes the sign of all internal quantum numbers. But I was wondering how it acts on a state such as ## \left|\pi^{+} \pi^{-} \right>## when the individual ##\pi's## are not eigenstates of C. I believe the combination of the ##\pi's## has eigenvalue...

I am reading a proof of why \left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z Given a wavefunction \psi, \hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...

Hello! I am a bit confused about the physical meaning of time reversal operator (both in classical and quantum/QFT physics). Classically if we drop a ball, I can easily see why this is invariant under the translation operator, but I am not sure I understand how does it work with the time...

Hello! I read that the Klein-Gordon field can be viewed as an operator that in position space, when acted upon vacuum at position x creates a particle at position x: ##\phi(x) |0 \rangle \propto |x \rangle##. It make sense intuitively and the mathematical derivation is fine too, but I was...

Hello! I read that in Heisenberg picture the propagator from x to y is given by ##<0|\phi(x)\phi(y)|0>##, where ##\phi## is the Klein-Gordon field. I am not sure I understand why. I tried to prove it like this: ##|x>=\phi(x,0)|0>## and after applying the time evolution operator we have...

It is said that no Hermitian operator gives a time evolution where "I observed the spin to be both up and down" is a possible result. If you use non-Hermitian operator.. then it's possible.. and what operator is that where it is possible in principle where "I observed the spin to be both up and...

Homework Statement Show that if H is a hermitian operator, then U = eiH is unitary. Homework Equations UU† = I for a unitary matrix A†=A for hermitian operator I = identity matrix The Attempt at a Solution Here is what I have. U = eiH multiplying both by U† gives UU† = eiHU† then replacing U†...

I have written the following class: class Rectangle{ private: double length; double height; const char* name; public: Rectangle (){}; Rectangle (double l, double h, const char* n) : length(l), height(h), name(n) {}...

What the title says. Acting on a fermionic state with the number operator to a power is like acting with the fermionic operator itself. Does this allow us to define ## \hat{n}^k=\hat{n} ##? Or is there any picky mathematical reason not to do so?

There is an operator in a three-state system given by: 2 0 0 A_hat = 0 0 i 0 -i 0 a) Find the eigenvalues and Eigenvectors of the operator b) Find the Matrix elements of A_hat in the basis of the eigenvectors of B_hat c) Find the Matrix Elements of A_hat...

I'm having a little bit of trouble getting my head around the idea of the reduced density operator being used to tell us about the entanglement of a state. I understand that if you take the reduced density operator of any of the Bell states, you get a reduced density operator proportional to...

Homework Statement I really cannot seem to be able to follow the logic of how you would use the product rule when using 4 vector differential operator. ∂μ is the differential operator, Aμ is a scalar field and φ and φ* is it's complex conjugate scalar field. I have the answer, I'd just really...

Hey everyone. First I am new to programming (so my vocabulary and skills are not very proficient with C++), and I'm learning operator overloading. I have questions about line 1 and lines 5-8 of this code. Mod note: changed quote tags to code tags to preserve indentation. NumberArray&...

Homework Statement Let A and B be two unbounded operators, is what if A+B is densely defined then A and B are also densely defined? Homework Equations D(A+B)=D(A)∩D(B) The Attempt at a Solution Since A+B is densely defined then A and B are also densely defined because D(A+B)=D(A)∩D(B)

I'm working on this problem "Consider an experiment on a system that can be described using two basis functions. In this experiment, you begin in the ground state of Hamiltonian H0 at time t1. You have an apparatus that can change the Hamiltonian suddenly from H0 to H1. You turn this apparatus...

Homework Statement Write the density operator $$\rho=\frac{1}{3}|u><u|+\frac{2}{3}|v><v|+\frac{\sqrt{2}}{3}(|u><v|+|v><u|, \quad where <u|v>=0$$ In matrix form Homework Equations $$\rho=\sum_i p_i |\psi><\psi|$$ The Attempt at a Solution [/B] The two first factors ##\frac{1}{3}|u><u|##...

The Hilbert space for a free relativistic particle has inner product (in the momentum representation) $$ \langle \chi | \phi \rangle = \int \frac {d \vec k^3} {(2 \pi)^3 2 \sqrt{\vec k^2 + m^2}} \chi (\vec k) * \phi (\vec k)$$ States undergo time evolution $$i∂t|ψ \rangle = H_0 | ψ \rangle$$...

I am familiar with the usual solution of the hydrogen atom using the associated legendre functions and spherical harmonics, but my question is: is it possible to extend the hamiltonian of the hydrogen atom to naturally encompass half integer spin? My guess is that spin only pops in naturally in...

Would the action of the position operator on a wave function ##\psi(x)## look like this? $$\psi(x) \ =\ <x|\psi>$$ $${\bf \hat x}<x|\psi>$$ Question 2: the position operator can act only on the wave function?

Apparently, in QM, the photon does not have a position operator. Why is this so? As usual, thanks in advance.

In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian...

Homework Statement Hi, so I have been given the following operator in terms of 3 orthonormal states |Φi> A = |Φ2><Φ2| + |Φ3><Φ3| - i|Φ1><Φ2| - |Φ1><Φ3| + i|Φ2><Φ1| - |Φ3><Φ1| So I need to determine whether A is unitary and/or Hermitian and/or a projector and then calculate the eigenvalues and...

<mod note: moved to homework> Calculate the spectrum of the linear operator ##T## on ##B(\ell^1)##. $$T(x_1,x_2,x_3,\dots)=(\sum_{n=2}^\infty x_n, x_1, x_2, x_3, \dots)$$I think the way to do it is to find the point spectrums of ##T## and its adjoint ##T^*##. But I don't know how to calculate...

I notice that in Quantum Mechanics when extending an infinitesimal operation to a finite one, we should end with the exponential. For example: (rf. Sakurai, Modern Quantum Mechanics) $$ D(\boldsymbol{\hat n}, d\phi) = 1 - \frac{i}{\hbar} ( \boldsymbol {J \cdot \hat n})d\phi $$ This is the...

I've been thinking about it since yesterday and have noticed this pattern: We have, the first order derivative of a function ##f(x)## is: $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} ...(1)$$ The second order derivative of the same function is: $$f''(x)=\lim_{h\rightarrow...

Hi. I have been looking at the proof that the parity operator is hermitian in 3-D in the QM book by Zettili and I am confused by the following step ∫ d3r φ*(r) ψ(-r) = ∫ d3r φ*(-r) ψ(r) I realize that the variable has been changed from r to -r. In 3-D x,y,z this is achieved by taking the...

Homework Statement Given \hat{P}_r\psi=-i\hbar\frac{1}{r}\frac{\partial}{\partial{r}}(r\psi), show \hat{H}=\frac{1}{2m}(\hat{P}^2_r+\frac{\hat{L}^2}{r^2}) Homework EquationsThe Attempt at a Solution The solution starts out with...

Hello, I'm a bit confused about the eigenvalues of the second quantized fermionic field operators \psi(x)_a. Since these operators satisfy the condition \{\psi(x)_a, \psi(y)_b\} = 0 the eigenvalues should also anti-commute? Does this mean that the eigenvalues of \psi(x)_a are...

In classical mechanics, angular momentum, L = r x p, depends of r. For a given momentum p, the bigger r is, the bigger is the angular momentum. Event in spherical coordinates, r still appears in the classical angular momentum. However, the angular momentum operator in QM has no r dependence...

I have read many textbooks and googled google times for a clear explanation, but I could not find one. How does raising and lowering -annihilation/ creation-(is that energy or particle number?) translate to transition probabilities of path integral.

Hello everybody, My question today is: Given a molecule that has an internal degree of liberty ( let's take the ethane molecule with its internal rotation as an example), how to write the kinetic energy operator by means of the corresponding internal coordinate? Thank you guys. Konte

Hello Forum, I understand that in order to calculate the average of a certain operator (observable), whatever that observable may be that we are interested in, we need to prepare many many many identical copies of the same state and apply the operator of interest to those identical state. By...

Hi I have some questions about operator and experimental action 1, For each experimental action(no matter how trivial or complex), can they ALWAYS be described by some corresponded operator? how to proof? For example, adding some energy to excit a particle can be described in operator language...

If I have the operator, ##e^{a\partial_p}## acting on ##f(p)##, I know that $$e^{a\partial_p}f(p)=f(p+a)\,.$$ If I have ##e^{a\partial_p^2}f(p)##, this is just the Weierstrass transform of ##f(p)##. However, what happens if I have a general operator, ##e^{g(p)\partial_p}## or...

Homework Statement So, I'm doing this problem from Townsend's QM book 6.2[/B] Show that <p|\hat{x}|\psi> = i\hbar \frac{\partial}{\partial p}<p|\psi> Homework Equations |\psi(p)> = \int_\infty^{-\infty} dp |p><p|\psi> The Attempt at a Solution So, <p|\hat{x}|\psi> = <p|\hat{x}...

Homework Statement The energy operator for a time-dependent system is iħ d/dt. A possible eigenfunction for the system is Ψ(x,y,z,t)=ψ(x,y,z)e-2πiEt/h Show that the probability density is independent of time Homework Equations ĤΨn(x) = EnΨn The Attempt at a Solution [/B] I understand the...

Hi Guys, at the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be self-adjoint (or in many cases hermitian what maybe means symmetric or maybe self-adjoint). Which condition do we need for our observables...

This is not part of my coursework but a question from a past paper (that we don't have solutions to). 1. Homework Statement Construct the matrix ##\sigma_{-} = \sigma_{x} - i\sigma_{y}## and show that the states resulting from ##\sigma_{-}## acting on the eigenstates of ##\sigma_{z} ## are...

I am looking at an explanation of the gradient operator acting on a scalar function ## \phi ##. This is what is written: In the steps 1.112 and 1.113 it is written that ## \frac {\partial x'_k} {\partial x'_i} ## is equivalent to the Kronecker delta. It makes sense to me that if i=k, then...

In chapter 10 of his book "Quantum Field Theory of Point Particles and Strings", Hatfield treats what he calls the Schrodinger representation of QFT. He starts with a free scalar field and introduces field operators ## \hat \varphi(\vec x) ## and its eigenstates ## \hat \varphi(\vec...

The Kraus operator is defined as, $$A_{k}(t)={\sum_{\{k_i\}}^{k}}'\langle\{k_i\}|U(t)|\{0\}\rangle$$ is given in eqn(5) in the [Arxiv link](https://arxiv.org/pdf/quant-ph/0407263.pdf) the matrix representation of $A_k(t)$ is given in eqn (7) as...

Is there any concept in mathematics called twist operator or twist field? It should somehow be related to branch points and Riemann sheets in complex analysis. Thanks

Hello everybody, A special problem constrain me to make a variable change in my Hamiltonian operator, so with the kinetic energy operator, I have a doubt. The variable change is: ## \theta \longrightarrow (\theta + k) ## (where ##k## is a constant). And the kinetic energy operator...

Which of the operators T:C[0,1]\rightarrow C[0,1] are compact? $$(i)\qquad Tx(t)=\sum^\infty_{k=1}x\left(\frac{1}{k}\right)\frac{t^k}{k!}$$ and $$(ii)\qquad Tx(t)=\sum^\infty_{k=0}\frac{x(t^k)}{k!}$$ ideas for compactness of the operator: - the image of the closed unit ball is relatively...

Homework Statement Obtain the matrix representation of the ladder operators ##J_{\pm}##. Homework Equations Remark that ##J_{\pm} | jm \rangle = N_{\pm}| jm \pm 1 \rangle## The Attempt at a Solution [/B] The textbook states ##|N_{\pm}|^2=\langle jm | J_{\pm}^\dagger J_{\pm} | jm \rangle##...

Hi I have seen an example of commutator of the Parity operator of the x-coordinate , Px and angular momentum in the z-direction Lz calculated as [ Px , Lz ] ψ(x , y) = -2Lz ψ (-x , y) I have tried to calculate the commutator without operating on a wavefunction and just by expanding...

Hi, I was wondering how you can formally determine whether a given operator is degenerate. I undertand you can produce the 'usual equation' det(Q-##\lambda ##)=0 and solve for ##\lambda ##, where Q is our operator. But if Q is a differential (for example ##\frac{p^2}{2m}= - \bar{h}...

Homework Statement This is an ordinary differential equation using the differential operator. Given the system: d^2x/dt - x + d^2y/dt^2 + y = 0 and dx/dt + 2x + dy/dt + 2y = 0 find x and y equation Answer: x = 5ce^(-2t) y = -3ce^(-2t) Homework EquationsThe Attempt at a Solution We change...